personal draft detail concise

# Diagonal cubic forms and the large sieve

Victor Y. Wang

###### Abstract.

Let $F(\bm{x})$ be a diagonal integer-coefficient cubic form in
$m\in\\{4,5,6\\}$ variables. Excluding rational lines if $m=4$, we bound the
number of integral solutions $\bm{x}\in[-X,X]^{m}$ to $F(\bm{x})=0$ by
$O_{F,\epsilon}(X^{3m/4-3/2+\epsilon})$, conditionally on an “optimal large
sieve inequality” (in a specific range of parameters) for “approximate
Hasse–Weil $L$-functions” of smooth hyperplane sections
$F(\bm{x})=\bm{c}\cdot\bm{x}=0$ as $\bm{c}\in\mathbb{Z}^{m}$ varies in natural
boxes.

When $m$ is even, these results were previously established conditionally
under Hooley’s Hypothesis HW. Our $\ell^{2}$ large sieve approach requires
that certain bad factors be roughly $1$ on average in $\ell^{2}$, while the
$\ell^{\infty}$ Hypothesis HW approach only required the bound in $\ell^{1}$.
Furthermore, the large sieve only accepts uniform vectors; yet our “initially
given” vectors are only approximately uniform over $\bm{c}$, due to variation
in bad factors and in the archimedean component. Nonetheless, after some
bookkeeping, partial summation, and Cauchy, the large sieve will still apply.

In an appendix, we suggest a framework for non-diagonal cubics, up to Hessian
issues.

###### Contents

  1. 1 Introduction
  2. 2 Delta method ingredients
  3. 3 Contribution from smooth hyperplane sections: Cauchy or counting
  4. 4 Contribution from the singular series
  5. 5 Contribution from singular hyperplane sections
  6. A General cubic forms: a tentative axiomatic framework
  7. B Other open problems of a similar nature
  8. C Natural modularity questions
  9. D “Correctly” defining discriminants of hyperplane sections
  10. E An example of a desirable but non-standard conjecture
  11. F Using Jutila’s circle method instead of the delta method

## Conventions and notes

### On series references

This is the first paper in a series of three, which we refer to as _Papers
I–III_. (We refer the reader to the web address tinyurl.com/hooley33 for
details, and for any errata in I–III that may be discovered.111In case this
web address fails, one can email<EMAIL_ADDRESS>for help, or go to the
webpage wangyangvictor.github.io/thesis_links.html for direct links to all
relevant files.) The present Paper I is logically independent of II–III. But
in the present paper, we may occasionally _mention_ II–III when they go beyond
(or perhaps supersede) Paper I in some _specific_ technical aspect.

### On notation

We write $f\ll_{S}g$ to mean $\lvert f\rvert\lesssim_{S}g$, i.e. $\lvert
f\rvert\leq Cg$ for some $C=C(S)>0$.

For bookkeeping reasons, $\lVert\bm{x}\rVert$ will (by default) refer to
$\lVert\bm{x}\rVert_{\infty}$ when $\bm{x}$ is a vector. And in the context of
indices, $[n]$ will denote the set $\\{1,2,\dots,n\\}$ when
$n\in\mathbb{Z}_{\geq 0}$.

We let $e(t)\colonequals e^{2\pi it}$ (if $t$ “makes sense” in
$\mathbb{R}/\mathbb{Z}$), and $e_{r}(t)\colonequals e(t/r)$ (for
$r\in\mathbb{R}^{\times}$).

Everywhere in the present paper, we will let
$\mathop{{\sum\nolimits^{\mathrlap{\prime}}}}_{\bm{c}\in\ast}\cdots$ denote
$\sum_{\bm{c}\in\ast\;\textnormal{with}\;F^{\vee}(\bm{c})\neq 0}\cdots$,
unless specified otherwise. (We will define $F^{\vee}$ in Proposition-
Definition 1.8.)

##  1\. Introduction

###  1.1. The setting

Let $F(\bm{x})$ be a diagonal cubic form in $m\geq 4$ variables with (nonzero)
integer coefficients. For convenience, let $m_{\ast}\colonequals m-3$ refer to
a certain dimension to appear below. (This is not to be confused with the
“Hardy–Littlewood exponent” $m-\deg{F}$.)

######  Definition 1.1.

Given $\bm{c}\in\mathbb{Z}^{m}$, let $\mathcal{V}$ and $\mathcal{V}_{\bm{c}}$
denote the proper schemes over $\mathbb{Z}$ defined respectively by the
equations $F(\bm{x})=0$ and $F(\bm{x})=\bm{c}\cdot\bm{x}=0$. Then let
$V\colonequals\mathcal{V}_{\mathbb{Q}}$ and
$V_{\bm{c}}\colonequals(\mathcal{V}_{\bm{c}})_{\mathbb{Q}}$ denote the generic
fibers of $\mathcal{V}$ and $\mathcal{V}_{\bm{c}}$, respectively.

For convenience, we also generalize the notation
$\mathcal{V}_{\bm{c}},V_{\bm{c}}$ as follows: given a ring $R$, an $m$-tuple
$\bm{c}$ that “makes sense” in $R^{m}$, and a closed subscheme
$W\subseteq\mathbb{P}^{m-1}_{R}$, define $W_{\bm{c}}$ to be the scheme-
theoretic intersection $W\cap\\{\bm{c}\cdot\bm{x}=0\\}$.

######  Remark 1.2.

Here $V=V(F)/\mathbb{Q}$ is a smooth projective hypersurface in
$\mathbb{P}^{m-1}_{\mathbb{Q}}$. And if $\bm{c}\in\mathbb{Q}^{m}$, then
$V_{\bm{c}}=V(F,\bm{c}\cdot\bm{x})/\mathbb{Q}$ is a hypersurface in
$(\mathbb{P}^{m-1}_{\mathbb{Q}})_{\bm{c}}$, where
$(\mathbb{P}^{m-1}_{\mathbb{Q}})_{\bm{c}}\cong\mathbb{P}^{m-2}_{\mathbb{Q}}$
if $\bm{c}\neq\bm{0}$.

Fix a smooth compactly supported weight $w(\bm{x})$ on $\mathbb{R}^{m}$ such
that $\lVert\nabla{F}\rVert$ is bounded away from $0$ on
$\operatorname{Supp}{w}\colonequals\overline{\\{w\neq 0\\}}$; equivalently
(since $V_{\mathbb{R}}$ is smooth), we require
$\bm{0}\notin\operatorname{Supp}{w}$.

######  Definition 1.3.

For $X\geq 1$, define the weighted point count
$N_{F,w}(X)\colonequals\sum_{\bm{x}\in\mathbb{Z}^{m}}w(\bm{x}/X)\cdot\bm{1}_{F(\bm{x})=0}$
for $F(\bm{x})=0$. If $m=4$, let $N^{\prime}_{F,w}(X)$ denote the
corresponding restricted point count over $\bm{x}\in\mathbb{Z}^{m}$ not lying
on any rational line of the cubic surface $V/\mathbb{Q}$.

Finally, for expository convenience, let $N_{F}(X)$ and $N^{\prime}_{F}(X)$
denote the analogous _unweighted_ counts over $[-X,X]^{m}$ (i.e. with
$\bm{1}_{\bm{x}\in[-X,X]^{m}}$ in place of $w(\bm{x}/X)$).

_Temporarily_ (for a paragraph) assume $F=x_{1}^{3}+\dots+x_{6}^{3}$. Then
[hooley_greaves_harman_huxley_1997] and [heath1998circle] (independently, at
least modulo [hooley1986HasseWeil]) proved
$N_{F}(X)\ll_{F,\epsilon}X^{3+\epsilon}$, _assuming_ a certain unproved
“Hypothesis HW” (practically amounting to “automorphy and GRH”) for the
standard Hasse–Weil $L$-functions $L(s,V_{\bm{c}})$ associated to _smooth_
$V_{\bm{c}}$’s. But the best unconditional result is still more or less Hua’s
bound $N_{F}(X)\ll_{F,\epsilon}X^{3.5+\epsilon}$ (a special case of
[hua1938waring]*Main Lemma); obtaining an exponent of $3.5-\delta$ remains an
open problem today [vaughan2020some].

Since automorphy (in full generality)—and especially GRH—might not be proven
for some time, it is therefore natural to ask whether “Hypothesis HW” can be
weakened. (Regarding the question of automorphy, see Appendix C for more
details in the specific setting of $L(s,V_{\bm{c}})$’s above.)

In the present paper, we will first show that “approximate elementary GRH on
average in $\ell^{2}$” suffices—as a _complete replacement_ for “Hypothesis
HW”—in the main results of [hooley1986HasseWeil,
hooley_greaves_harman_huxley_1997, heath1998circle]. We will then show that in
“sufficiently standard” situations, such an $\ell^{2}$-hypothesis would be
implied by an “optimal” $\ell^{2}$ large sieve inequality.

We will not attempt to prove such $\ell^{2}$-hypotheses here; for now we
simply note that even _partial_ results (hopefully easier than GRH itself)—if
sufficiently strong (e.g. “quasi-GRH up to the line $\sigma=5/6-\delta$ on
average in $\ell^{2}$”)—could help to “beat Hua” in the exponent.

######  Remark 1.4.

From the perspective of the _series_ I–III, the above “standard conjectural”
approach to the “Hua problem” is the most significant. But what about from the
perspective of “Hua” itself? See Appendix E for an alternative and “less
standard” conditional approach. See also Appendix F for another alternative
approach based on Jutila’s circle method, rather than [duke1993bounds,
heath1996new]’s delta method (or Hooley’s “upper bound precursor” thereof).
These alternative approaches may or may not be worth considering.

###  1.2. Basic definitions

We now recall some natural definitions from [hooley1986HasseWeil,
hooley_greaves_harman_huxley_1997, heath1998circle].

######  Definition 1.5.

If $n\in\mathbb{Z}_{\geq 1}$, then for each $m$-tuple $\bm{c}$ that “makes
sense” in $(\mathbb{Z}/n)^{m}$, let

$S_{\bm{c}}(n)\colonequals\sum_{a\in(\mathbb{Z}/n)^{\times}}\sum_{\bm{x}\in(\mathbb{Z}/n)^{m}}e_{n}(aF(\bm{x})+\bm{c}\cdot\bm{x})$

and $\widetilde{S}_{\bm{c}}(n)\colonequals n^{-(m+1)/2}S_{\bm{c}}(n)$. Then
given $\bm{c}\in\mathbb{Z}^{m}$, let $\Phi(\bm{c},s)\colonequals\sum_{n\geq
1}\widetilde{S}_{\bm{c}}(n)n^{-s}$.

[hooley1986HasseWeil] observed several properties of the sums $S_{\bm{c}}(n)$,
originating from the _homogeneity_ of $F$ (crucial in our analysis, and
likewise in [hooley1986HasseWeil, hooley_greaves_harman_huxley_1997,
heath1998circle]).

  1. (1)

The function $n\mapsto S_{\bm{c}}(n)$ is _multiplicative_ , i.e.
$S_{\bm{c}}(n_{1}n_{2})=S_{\bm{c}}(n_{1})S_{\bm{c}}(n_{2})$ if
$(n_{1},n_{2})=1$.

  2. (2)

The function $(F,\bm{c})\mapsto S_{\bm{c}}(n)$ is _scale-invariant_ , e.g.
$S_{\bm{c}}(n)=S_{\lambda\bm{c}}(n)$ if $\lambda\in(\mathbb{Z}/n)^{\times}$.

In particular, (2) suggests that $S_{\bm{c}}$ might only depend on
_homogeneous geometric_ invariants of $(F,\bm{c})$. This is indeed the case!
We now recall some of the main invariants involved.

######  Definition 1.6.

For a prime power $q$, and an $m$-tuple $\bm{c}$ that “makes sense” in
$\mathbb{F}_{q}^{m}$, let $\rho(q),\rho_{\bm{c}}(q)$ be the respective
$\mathbb{F}_{q}$-point counts of
$\mathcal{V}_{\mathbb{F}_{q}},(\mathcal{V}_{\mathbb{F}_{q}})_{\bm{c}}$.
Normalize the “errors” $E(q)\colonequals\rho(q)-(q^{m-1}-1)/(q-1)$ and
$E_{\bm{c}}(q)\colonequals\rho_{\bm{c}}(q)-(q^{m-2}-1)/(q-1)$ to get
$\widetilde{E}(q)\colonequals q^{-(1+m_{\ast})/2}E(q)$ and
$\widetilde{E}_{\bm{c}}(q)\colonequals q^{-m_{\ast}/2}E_{\bm{c}}(q)$.

######  Remark 1.7.

Perhaps somewhat confusingly, $E_{\bm{0}}(q)\neq E(q)$.

######  Proposition-Definition 1.8.

Up to scaling, there is a _unique_
$F^{\vee}\in\mathbb{Z}[\bm{c}]\setminus\\{0\\}$ of degree $3\cdot 2^{m-2}$
such that if $\bm{c}\in\mathbb{C}^{m}\setminus\\{\bm{0}\\}$, then
$F^{\vee}(\bm{c})=0$ if and only if $(V_{\mathbb{C}})_{\bm{c}}$ is singular.

Now fix $F^{\vee}$. Then $F^{\vee}$ is homogeneous, and irreducible over
$\mathbb{C}$. Informally, we call $F^{\vee}$ a (geometric) _discriminant
form_. Furthermore, we may choose $F^{\vee}$ so that for all
$\bm{c}\in\mathbb{Z}^{m}$ and primes $p\nmid F^{\vee}(\bm{c})$, the special
fiber $(\mathcal{V}_{\bm{c}})_{\mathbb{F}_{p}}$ is smooth of dimension
$m_{\ast}$.

###### Proof sketch.

This is classical (and discussed in greater geometric depth in Paper II). But
since $F$ is diagonal, a simple explicit treatment is possible.

In fact, say $F=F_{1}x_{1}^{3}+\dots+F_{m}x_{m}^{3}$, and define
$G\in\mathbb{Z}[\bm{x}]$ by the formula [heath1998circle]*(4.2). Then $G$ is
homogeneous of degree $3\cdot 2^{m-2}$. Also, a short calculation (using the
Jacobian criterion) shows that if
$K\in\\{\mathbb{C}\\}\cup\\{\mathbb{F}_{p}:p\nmid 3(F_{1}\cdots F_{m})\\}$ and
$\bm{c}\in K^{m}$, then $G(\bm{c})\neq 0$ (in $K$) if and only if
$(\mathcal{V}_{K})_{\bm{c}}$ is smooth of dimension $m_{\ast}$.

In particular,
$\\{\bm{c}\in\mathbb{C}^{m}:G(\bm{c})=0\\}=\\{\bm{0}\\}\cup\\{\bm{c}\in\mathbb{C}^{m}\setminus\\{\bm{0}\\}:(V_{\mathbb{C}})_{\bm{c}}\;\textnormal{is
singular}\\}$. But $G$ is irreducible over $\mathbb{C}$ (by a “multi-quadratic
Galois theory” argument), since $m\geq 3$. So for degree reasons, $F^{\vee}$
_can and must_ be a scalar multiple of $G$ (by Hilbert’s nullstellensatz).
Furthermore, the (safe) choice $F^{\vee}\colonequals 3(F_{1}\cdots F_{m})\cdot
G$ has all the desired properties (including that required in the last
sentence of Proposition-Definition 1.8). ∎

######  Example 1.9.

If $F=x_{1}^{3}+\dots+x_{m}^{3}$, then we may take
$F^{\vee}=3\prod(c_{1}^{3/2}\pm c_{2}^{3/2}\pm\dots\pm c_{m}^{3/2})$.

######  Remark 1.10.

In Proposition-Definition 1.8, one can in fact choose $F^{\vee}$ so that for
all $\bm{c}\in\mathbb{Z}^{m}$ and primes $p$, the conditions $p\nmid
F^{\vee}(\bm{c})$ and “$(\mathcal{V}_{\bm{c}})_{\mathbb{F}_{p}}$ is smooth of
dimension $m_{\ast}$” are _equivalent_. This property does not necessarily
_uniquely specify_ “$F^{\vee}$ up to $\pm 1$”, but does come fairly close to
doing so. See Appendix D for the relevant theory, valid even for non-diagonal
$F$. (But see Definition D.6 for one possible explicit choice of $F^{\vee}$ in
our diagonal setting.)

As we will explain in §2, we know—essentially by [hooley1986HasseWeil]—that

  1. (1)

$\widetilde{S}_{\bm{c}}(p)=\widetilde{E}_{\bm{c}}(p)-p^{-1/2}\widetilde{E}(p)$
if $p\nmid\bm{c}$ (e.g. if $p\nmid F^{\vee}(\bm{c})$), and

  2. (2)

$\widetilde{S}_{\bm{c}}(p^{l})=0$ for $l\geq 2$ if $p\nmid F^{\vee}(\bm{c})$.

But if $\bm{c}\in\mathbb{Z}^{m}$ and $p\nmid F^{\vee}(\bm{c})$, then
$(\mathcal{V}_{\bm{c}})_{\mathbb{F}_{p}}$ may be viewed, non-canonically, as a
smooth projective cubic hypersurface over $\mathbb{F}_{p}$ of dimension
$m_{\ast}$. By the theory of $\ell$-adic cohomology (including the
Grothendieck–Lefschetz fixed-point theorem and the resolution of the Weil
conjectures), we can thus make the following definition:

######  Definition 1.11.

Fix $\bm{c}\in\mathbb{Z}^{m}$ with $F^{\vee}(\bm{c})\neq 0$, i.e. with
$V_{\bm{c}}$ smooth of dimension $m_{\ast}$. Then for each prime $p\nmid
F^{\vee}(\bm{c})$, define the _analytically normalized local factor_

$L_{p}(s,V_{\bm{c}})\colonequals\exp\left((-1)^{m_{\ast}}\sum_{r\geq
1}\widetilde{E}_{\bm{c}}(p^{r})\frac{(p^{-s})^{r}}{r}\right)=\prod_{j=1}^{\dim_{m_{\ast}}}(1-\tilde{\alpha}_{\bm{c},j}(p)p^{-s})^{-1}\equalscolon\sum_{l\geq
0}\tilde{\lambda}_{\bm{c}}(p^{l})p^{-ls}$

of degree $\dim_{m_{\ast}}\colonequals\dim
H^{m_{\ast}}_{\operatorname{prim}}(V_{\bm{c}}\times\mathbb{C})=\frac{(-1+\deg{F})^{m_{\ast}+2}+(-1)^{m_{\ast}}(-1+\deg{F})}{\deg{F}}=\frac{2^{m-1}+2(-1)^{m-3}}{3}$,
where $H^{m_{\ast}}_{\operatorname{prim}}$ denotes a Betti cohomology group
(_primitive_ if $m_{\ast}\equiv 0\bmod{2}$). Here
$\tilde{\alpha}_{\bm{c},j}(p)$ denote certain “normalized” Frobenius
eigenvalues (known to satisfy $\lvert\tilde{\alpha}_{\bm{c},j}(p)\rvert=1$).

In particular, if $p\nmid F^{\vee}(\bm{c})$, then
$(-1)^{m_{\ast}}\widetilde{E}_{\bm{c}}(p)=\sum_{j}\tilde{\alpha}_{\bm{c},j}(p)=\tilde{\lambda}_{\bm{c}}(p)$,
so $\widetilde{E}_{\bm{c}}(p)\ll_{m}1$. (Similarly, we have
$\widetilde{E}(p)\ll_{F}1$ uniformly over _all_ primes $p$.) So for each
_fixed_ $\bm{c}\in\mathbb{Z}^{m}$ with $F^{\vee}(\bm{c})\neq 0$, we roughly
expect an approximation of the form

$\Phi(\bm{c},s)\approx\prod_{p\nmid
F^{\vee}(\bm{c})}L_{p}(s,V_{\bm{c}})^{(-1)^{m_{\ast}}}\qquad\textnormal{(``to
leading order'')}.$

###  1.3. Fixing approximate Hasse–Weil coefficients

Our work is based on the intuitive notion of a “first-order approximation” of
a Dirichlet series. One could imagine many different precise
definitions—perhaps useful for different purposes. We will use the following
definition:

######  Definition 1.12.

Fix a family of Dirichlet series $\Psi_{1}(\bm{c},s)$ indexed by
$\\{\bm{c}\in\mathbb{Z}^{m}:F^{\vee}(\bm{c})\neq 0\\}$. For each $\bm{c}$, let
$b_{\bm{c}}(n),a_{\bm{c}}(n),a^{\prime}_{\bm{c}}(n)$ be the $n^{-s}$
coefficients of $\Psi_{1},\Psi_{1}^{-1},\Phi(\bm{c},s)/\Psi_{1}$,
respectively. Suppose that

  1. (1)

$b_{\bm{c}}$ is multiplicative, i.e. $\Psi_{1}$ has a (formal) Euler product;

  2. (2)

$b_{\bm{c}}(n),a^{\prime}_{\bm{c}}(n)\ll_{\epsilon}n^{\epsilon}\sum_{d\mid
n}\lvert\widetilde{S}_{\bm{c}}(d)\rvert$ holds uniformly over $\bm{c},n$; and

  3. (3)

$a^{\prime}_{\bm{c}}(p)\cdot\bm{1}_{p\nmid F^{\vee}(\bm{c})}\ll p^{-1/2}$
holds uniformly over $\bm{c},p$ with $p$ prime.

Then we call $\Psi_{1}$ a _(one-sided, first-order, Euler-product)
approximation of $\Phi$_.

######  Example 1.13.

Suppose one of the following holds.

  1. (1)

For each $\bm{c}$, we let $\Psi_{1}(\bm{c},s)\colonequals\Phi(\bm{c},s)$.

  2. (2)

For each $\bm{c}$, we let $\Psi_{1}(\bm{c},s)\colonequals\prod_{p\nmid
F^{\vee}(\bm{c})}\Phi_{p}(\bm{c},s)$.

  3. (3)

For each $\bm{c}$, we let $\Psi_{1}(\bm{c},s)\colonequals\prod_{p\nmid
F^{\vee}(\bm{c})}L_{p}(s,V_{\bm{c}})^{(-1)^{m_{\ast}}}$.

  4. (4)

$m\in\\{4,6\\}$, and for each $\bm{c}$, we let $\Psi_{1}(\bm{c},s)\colonequals
L(s,V_{\bm{c}})^{-1}$, with the standard $L$-function $L(s,V_{\bm{c}})$
defined as in [hooley1986HasseWeil] (for $F=x_{1}^{3}+\dots+x_{6}^{3}$) or
[heath1998circle] (in general).

Then $\Psi_{1}$ is an approximation of $\Phi$, in the sense of Definition
1.12.

We will “prove” Example 1.13 later, in §2.

######  Remark 1.14.

Example 1.13(4) should be valid for all $m\geq 3$, but our understanding of
Hasse–Weil $L$-functions at bad primes might not be fully satisfactory for all
values of $m$.

###  1.4. Conditional results and discussion of hypotheses

Fix an approximation $\Psi_{1}$ of $\Phi$, in the sense of Definition 1.12.
Our main conditional result, Theorem 1.28, will assume that _either_ of two
certain hypotheses on $\Psi_{1}$ holds. To motivate the precise forms of these
hypotheses (involving certain real parameters $Y,Z>0$ with $Y\leq Z^{3}$), we
need the following definition:

######  Definition 1.15.

As in [heath1998circle] (based on [duke1993bounds, heath1996new]), set
$Y\colonequals X^{(\deg F)/2}=X^{3/2}$ in the delta method to be, roughly, the
largest modulus used—and given $\epsilon_{0}>0$, set
$Z=Z_{\epsilon_{0}}\colonequals Y/X^{1-\epsilon_{0}}=X^{1/2+\epsilon_{0}}$.

######  Remark 1.16.

Then only the region $\lVert\bm{c}\rVert\ll_{\epsilon_{0}}Z$, say, really
contributes to the “dual” sum (over $\bm{c}$) in the delta method setup of
[heath1998circle].

######  Remark 1.17 (Intuition).

If $\bm{x}\ll X$, then $F(\bm{x})\ll X^{3}$. So it is natural to try using a
total of $\asymp X^{3}$ harmonics to “detect” the condition $F(\bm{x})=0$ over
$\bm{x}\ll X$. Since the delta method morally uses $\asymp Y^{2}$ harmonics
(corresponding to proper reduced fractions with denominator $\ll Y$), this
suggests setting $Y^{2}\asymp X^{3}$.

(Smaller choices of $Y$ could in principle also be worth considering; see the
quartic analysis of [marmon2019hasse]. But we do not explore such $Y$’s in the
present paper.)

One then encounters certain “pseudo-exponential sums to modulus $n\ll Y$” over
$\bm{x}\ll X$. These sums are typically “incomplete” (since $X=o(Y)$). By
“completing” these sums using Poisson summation, we end up with “dual sums”
morally of length $\lessapprox Y/X\asymp X^{1/2}$.

####  1.4.1. A second-moment hypothesis

We first state a hypothesis that can morally be thought of _either as_ an
elementary GRH-on-average statement, _or as_ an elementary manifestation of a
zero-density hypothesis, if $m$ is _even_ —and as an elementary GLH-on-average
statement, _plus_ a technical bound on
$\\{\bm{c}:L(s,V_{\bm{c}})\;\textnormal{has a pole at}\;s=1\\}$, if $m$ is
_odd_.

######  Definition 1.18.

We define the _second-moment hypothesis_ (for $\Psi_{1}$) to be the statement
that for all $(Z,\beta)\in\mathbb{R}_{>0}^{2}$, for all (positive reals)
$Y\leq Z^{3}$, for all (positive reals) $N\leq\beta Y$, and for all (real)
intervals $I\subseteq[N/2,2N]$, we have

$\sideset{}{{}^{\prime}}{\sum}_{\bm{c}\in[-Z,Z]^{m}}\left\lvert\sum_{n\in
I}b_{\bm{c}}(n)\right\rvert^{2}\ll_{\beta,\epsilon}Z^{\epsilon}\max(Z^{m},Y)\cdot
N\qquad\textnormal{(uniformly over $Z,Y,N,I$)},$

where we restrict the $\bm{c}$-sum to
$\\{\bm{c}\in\mathbb{Z}^{m}:F^{\vee}(\bm{c})\neq 0\\}$. (Here $Z^{m}\gg Y$,
since $m\geq 3$. But we write $\max(Z^{m},Y)$ in connection with the large
sieve to be discussed soon.)

######  Remark 1.19 (Counting, or “density”, interpretation).

Since $m\geq 4$, the family $\mathcal{F}\approx[-Z,Z]^{m}$ is in fact
_strictly_ larger than the range of moduli, $Y$. Given $N,I$ in Definition
1.18, we tolerate roughly $\lvert\mathcal{F}\rvert/N^{2\sigma-1}$
contributions of size $N^{\sigma-1/2}$. In particular, we are OK with roughly
$\lvert\mathcal{F}\rvert/N$ “extremely exceptional” $\bm{c}$’s with
$\lvert\sum_{n\in I}b_{\bm{c}}(n)\rvert\gg N$ matching the “heuristic trivial
bound”.222In the full generality of Definition 1.12, we cannot always
“actually trivially bound” $\lvert\sum_{n\in I}b_{\bm{c}}(n)\rvert$. Contrast
with “almost covering” problems (about e.g. small primes modulo $c$), where—in
analogy with the “Sarnak–Xue density philosophy”—an “expander property” (e.g.
a sufficiently strong prime number theorem) is often needed near $\sigma=1$.

######  Remark 1.20.

If $m\in\\{4,6\\}$, and we define $\Psi_{1}$ as in Example 1.13(4), then the
second-moment hypothesis for $\Psi_{1}$ would easily follow from Hypothesis
HW, or—up to smoothing issues—from automorphy and Remark 1.29’s density
hypothesis for “somewhat low-lying” zeros.

######  Remark 1.21.

When _applying_ the second-moment hypothesis towards Theorem 1.28, we will
_fix_ $\beta$ in terms of $F,w$ (roughly based on the implied constant in
Proposition 2.1).

####  1.4.2. A large-sieve hypothesis

Temporarily (for a paragraph) assume $2\mid m$. Then Definition 1.18 is
perhaps unattractive in that $b_{\bm{c}}(n)$ “morally contains” a Möbius
factor $\mu(n)$. But the “Bombieri–Vinogradov philosophy” suggests that at
least in favorable situations, the second-moment hypothesis for
$\Psi_{1}=\sum_{n\geq 1}b_{\bm{c}}(n)n^{-s}$ can be reduced to a large-sieve
hypothesis involving the “friendlier” Dirichlet coefficient family
$\bm{c}\mapsto(a_{\bm{c}}(n))_{n\geq 1}$.

In general, we might thus hope to “efficiently relate” the families of
Dirichlet coefficient vectors $(a_{\bm{c}}(n))_{n},(b_{\bm{c}}(n))_{n}$. This
task is most _easily_ achieved in the situation of the following definition:

######  Definition 1.22.

Call $\Psi_{1}$ _standard_ if
$b_{\bm{c}}(n),a_{\bm{c}}(n)\ll_{\epsilon}n^{\epsilon}$ holds uniformly over
$\bm{c},n$.

######  Example 1.23.

Say $\Psi_{1}$ is defined as in (3) or (4) of Example 1.13. Then $\Psi_{1}$ is
standard. (Here the hypothesis $m\in\\{4,6\\}$ in (4) is conjecturally
unnecessary.)

We now come to our “main” hypothesis: a large sieve in certain ranges.

######  Definition 1.24.

If $\Psi_{1}$ is standard, then let
$\gamma\colonequals\bm{1}_{n=\operatorname{rad}(n)}\cdot a$, or let
$\gamma\colonequals b$; if $\Psi_{1}$ is non-standard, then let
$\gamma\colonequals b$. We define the _large-sieve hypothesis_ (for $\gamma$)
to be the statement that for all $(Z,\beta)\in\mathbb{R}_{>0}^{2}$, and for
all (positive reals) $Y\leq Z^{3}$, we have

$\sideset{}{{}^{\prime}}{\sum}_{\bm{c}\in[-Z,Z]^{m}}\left\lvert\sum_{n\leq
2\beta
Y}v_{n}\cdot\gamma_{\bm{c}}(n)\right\rvert^{2}\ll_{\beta,\epsilon}Z^{\epsilon}\max(Z^{m},Y)\cdot\sum_{n\leq
2\beta Y}\lvert v_{n}\rvert^{2}\quad\textnormal{for all
$\bm{v}\in\mathbb{R}^{\lfloor 2\beta Y\rfloor}$},$

uniformly over $Z,Y$. (Here $Z^{m}\gg Y$, since $m\geq 3$. But we write
$\max(Z^{m},Y)$ to avoid potential confusion with large sieves in other ranges
of parameters.)

######  Remark 1.25.

The large-sieve hypothesis for $\gamma\colonequals a$ (though excluded from
Definition 1.24 for expository convenience) would directly imply that for
$\gamma\colonequals\bm{1}_{n=\operatorname{rad}(n)}\cdot a$. The factor
$\bm{1}_{n=\operatorname{rad}(n)}$ is simply “restriction to square-free
moduli $n$”.

######  Remark 1.26.

For concreteness, say $\Psi_{1}$ is defined as in (3) or (4) of Example 1.13.
Say $2\mid m$ and $\gamma\colonequals\bm{1}_{n=\operatorname{rad}(n)}\cdot a$,
or $2\nmid m$ and $\gamma\colonequals b$. Then
$\bm{c}\mapsto(\gamma_{\bm{c}}(n))_{n\geq 1}$ is genuinely a family of
Hasse–Weil coefficients, up to bad primes and
$\bm{1}_{n=\operatorname{rad}(n)}$. It remains open to prove, for at least one
choice of $\Psi_{1},\gamma$, the associated large sieve (or satisfying partial
results towards it, if true). One approach might be to dualize and try to
adapt [louvel2011distribution], at least over $\mathbb{Z}[\zeta_{3}]$.
Alternatively, one could try elementary geometric and analytic arguments.

######  Remark 1.27.

For a discussion of large sieves in general (and what one can or cannot expect
to be true), we refer the reader to [iwaniec2004analytic]*§7 and
[djankovic2012larger]*§1. From the point of view of _geometric_ families, a
nice example of a large sieve is [heath1995mean]’s quadratic-character large
sieve (“optimal” if one restricts to square-free moduli, as is OK for most
applications), applied in [perelli1997averages] to quadratic twist families of
elliptic curves. (Interestingly, the conductors in the latter setting grow
significantly faster than those in the former, but [heath1995mean] applies
equally well to the two settings, as far as the large sieve is concerned.)

Optimistically, one might hope that a “natural” geometric family—especially
one with “big monodromy” (in some sense)—would have a good chance of
satisfying an “optimal” large sieve. For a discussion of the (closely related)
“expected random matrix symmetry type” of our present $V_{\bm{c}}$ families
(at least when $2\mid m$), we refer the reader to Paper III.

####  1.4.3. The main result

Fix an approximation $\Psi_{1}$ of $\Phi$, in the sense of Definition 1.12.

######  Theorem 1.28.

Fix $\Psi_{1}$ as above. If $\Psi_{1}$ is standard, then let
$\gamma\colonequals\bm{1}_{n=\operatorname{rad}(n)}\cdot a$, or let
$\gamma\colonequals b$; if $\Psi_{1}$ is non-standard, then let
$\gamma\colonequals b$. Given $\Psi_{1},\gamma$, assume _either_ (i) the
_large-sieve hypothesis_ for $\gamma$, or (ii) the _second-moment hypothesis_
for $\Psi_{1}$. Then

  1. (1)

$N^{\prime}_{F,w}(X)\ll_{\epsilon}X^{3m/4-3/2+\epsilon}=X^{1.5+\epsilon}$ if
$m=4$;

  2. (2)

$N_{F,w}(X)\ll_{\epsilon}X^{3m/4-3/2+\epsilon}=X^{2.25+\epsilon}$ if $m=5$;

  3. (3)

$N_{F,w}(X)\ll_{\epsilon}X^{3m/4-3/2+\epsilon}=X^{3+\epsilon}$ if $m=6$; and

  4. (4)

$N_{F,w}(X)=\sigma_{\infty,w}\mathfrak{S}X^{m-3}+O_{\epsilon}(X^{3m/4-3/2+\epsilon})$
if $m\geq 7$ (with $\sigma_{\infty,w},\mathfrak{S}$ as defined in §4).

Here the weight can be removed for $m\leq 6$. Furthermore, if $m=6$, then the
Fermat case $F=x_{1}^{3}+\dots+x_{6}^{3}$ is logically the “hardest” (among
diagonal cubics).

######  Remark 1.29 (Historical motivation).

To paraphrase [hooley1986HasseWeil]*bottom of p. 51: “The removal [of Riemann]
is an obvious desideratum. [Some weakening] is certainly possible either by
substituting some [zero density requirement] or [a quasi-GRH sufficient for
partial results]. Yet…we cannot predict the precise form of the first
serviceable alternative [to GRH].”

In fact, by [density_beats_hua_when], it suffices to assume automorphy, along
with a “density hypothesis for zeros of height $\lessapprox 1$”—an upper-bound
statement on the exceptional counts
$N(\sigma,\mathcal{F},T)\colonequals\\#\\{\pi\in\mathcal{F}:L(s,\pi)\;\textnormal{has
a zero in}\;[\sigma,1]\times[-T,T]\\}$—of the form

$N(\sigma,\mathcal{F},T)\ll_{\epsilon}T^{O(1)}\lvert\mathcal{F}\rvert^{2(1-\sigma)+\epsilon},\quad\textnormal{uniformly
over $T,\lvert\mathcal{F}\rvert\geq 1$ and $\sigma\in[1/2,1]$}$

(“positive-linearly optimal” in the $\lvert\mathcal{F}\rvert$-aspect, but
“arbitrarily poor” in the $T$-aspect). Such a hypothesis is morally provable
under an “optimal” large sieve inequality over $\mathcal{F}$, i.e.
“approximate $\ell^{2}$ orthogonality” of the matrix $\mathbb{R}^{\lvert
I\rvert}\to\mathbb{R}^{\lvert\mathcal{F}\rvert}$ defined by
$n\mapsto(\lambda_{\pi}(n))_{\pi\in\mathcal{F}}$—but currently a “Lindelöf on
average” hypothesis over $\mathcal{F}$ also plays a role in rigorous proof
[axiomatize_bombieri-huxley]. (For instance, [density_beats_hua_when]*§6’s
density hypothesis, with $l(\sigma)\colonequals 2(1-\sigma)+\epsilon$ is
“close” to being known for the Bombieri–Vinogradov family
$\\{\chi\bmod{q}:q\leq Q\\}$.)

However, Theorem 1.28 shows that a large sieve _by itself_ —if true—would
suffice for the original goal of “recovering” the results of
[hooley1986HasseWeil, hooley_greaves_harman_huxley_1997, heath1998circle].

######  Remark 1.30.

Appendix A discusses what would need to be done to generalize Theorem 1.28 to
arbitrary smooth cubic forms (under our methods).

######  Remark 1.31.

The philosophy behind Theorem 1.28 should also apply to other problems; see
Appendix B for some potential examples in the spirit of the problems discussed
so far. In geometric settings in particular, one might also wonder about the
structure of the conjecturally relevant automorphic objects (perhaps with the
aim of eventually applying automorphic techniques); Appendix C discusses a
basic preliminary question in this direction for the present varieties
$V_{\bm{c}}$ when $m=6$ and $F=x_{1}^{3}+\dots+x_{6}^{3}$.

###### Proof of Theorem 1.28.

First we prove that (i) implies (ii). If $\gamma\colonequals b$, this is
trivial: just take $v_{n}\colonequals\bm{1}_{n\in I}$ for certain intervals
$I\subseteq(0,2\beta Y]$. Now suppose $\Psi_{1}$ is standard and
$\gamma\colonequals\bm{1}_{n=\operatorname{rad}(n)}\cdot a$. Then by
Proposition 1.34 below, (i) implies (ii).

Now _assume_ (ii) for the rest of the proof. In the delta method (to be set up
in §2), we will then address the “smooth contribution” from
$F^{\vee}(\bm{c})\neq 0$ in §3 (conditionally on (ii), for some _constant_
$\beta\ll_{F,w}1$), the “singular series” from $\bm{c}=\bm{0}$ in §4
(unconditionally), and the remaining “singular contribution” from
$F^{\vee}(\bm{c})=0$ in §5 (unconditionally).

The results for $m\leq 6$ are merely (polynomial) upper bounds on various
homogeneous sets of integral points, so dyadic decomposition lets us pass
freely between the weighted and unweighted versions of the problem.

Furthermore, a simple Hölder argument shows that out of all diagonal cubics in
a given even number of variables, the Fermat cubic has the greatest number,
$\int_{0}^{1}d\theta\,(\sum_{\lvert x\rvert\leq X}e(\theta x^{3}))^{m}$, of
integral solutions $\bm{x}\in[-X,X]^{m}$. For $m=6$ we seek an upper bound on
the number of such points, so we may reduce the general diagonal case to the
case of the Fermat cubic $F(\bm{x})=x_{1}^{3}+\dots+x_{m}^{3}$—though we will
give a direct general treatment. ∎

######  Remark 1.32.

For the $m=6$ Fermat cubic, another Hölder argument allows us to restrict
attention to points with $X\ll\lvert x_{1}\rvert,\dots,\lvert x_{m}\rvert\ll
X$, i.e. with $\bm{x}/X$ localized away from the vanishing locus of the
Hessian of $F$. As essentially explained in [hooley2014octonary]*pp. 246–252,
§§4–5, our integral estimates in Lemma 2.10 would then take a simpler
(stronger) form. The uniformity gained would let us treat some lacunary
$\bm{c}$’s more easily in certain places.

######  Remark 1.33.

After reducing to the Fermat case for $m=6$, we could alternatively use
Hooley’s old “positive” approach [hooley1986HasseWeil,
hooley_greaves_harman_huxley_1997], without the smooth delta function
technology of [duke1993bounds, heath1996new]. In the “analog” of the notation
above, we could then reduce (by Hölder again) to the study of
$\bm{c}\in\mathbb{Z}^{m}$ with $\lvert c_{1}\rvert\asymp\dots\asymp\lvert
c_{m}\rvert\neq 0$; this would give comparable simplifications.

####  1.4.4. A first-order (square-free-)restriction-type conversion between
standard $b,\mu a$

######  Proposition 1.34.

_Assume_ $\Psi_{1}$ is standard. _Fix_ $\beta>0$. Let $Z,Y,d,N$ denote
_positive_ variables. Given $Z$, let
$\mathop{{\sum\nolimits^{\mathrlap{\prime}}}}_{\bm{c}}$ denote a sum
restricted to $\bm{c}\in[-Z,Z]^{m}$ with $F^{\vee}(\bm{c})\neq 0$. Then the
following statements, (1)–(2), are equivalent.

  1. (1)

$\mathop{{\sum\nolimits^{\mathrlap{\prime}}}}_{\bm{c}}\left\lvert\sum_{n\in
I}b_{\bm{c}}(n)\right\rvert^{2}\ll_{\beta,\epsilon}Z^{m+\epsilon}N$ holds
uniformly over (positive) reals $Z$, reals $Y\leq Z^{3}$, reals $N\leq\beta
Y$, and (real) intervals $I\subseteq[N/2,2N]$.

  2. (2)

$\mathop{{\sum\nolimits^{\mathrlap{\prime}}}}_{\bm{c}}\left\lvert\sum_{n\in
I}\bm{1}_{d\perp
n}\mu(n)a_{\bm{c}}(n)\right\rvert^{2}\ll_{\beta,\epsilon}Z^{m+\epsilon}N$
holds uniformly over (positive) reals $Z$, reals $Y\leq Z^{3}$, integers
$d\leq 2\beta Y$, reals $N\leq\beta Y$, and (real) intervals
$I\subseteq[N/2,2N]$.

Furthermore, (1)–(2) are implied by the large-sieve hypothesis for
$\gamma\colonequals\bm{1}_{n=\operatorname{rad}(n)}\cdot a$.

######  Remark 1.35.

Fix $\bm{c}$. Then $\sum_{n\geq 1}\bm{1}_{d\perp n}\mu(n)a_{\bm{c}}(n)n^{-s}$
need not be a first-order approximation of $\sum_{n\geq 1}b_{\bm{c}}(n)n^{-s}$
in the sense suggested by Definition 1.12; this is because
$(1-a_{\bm{c}}(p)p^{-s})^{-1}$ need not have bounded coefficients. But it can
still be thought of as a first-order approximation in a “coefficient
restriction” sense; this is why (2) implies (1) in Proposition 1.34.

###### Proof.

Since $\mu(n)a_{\bm{c}}(n)$ is supported on square-free integers $n$, the
large-sieve hypothesis clearly implies (2). So it remains to prove the
equivalence of (1)–(2).

First, $a_{\bm{c}}(p)=-b_{\bm{c}}(p)$ by definition. So given $n$, if we let
$d\colonequals\operatorname{sq}(n)$ denote the square-full part of $n$, and
let $n_{1}\colonequals n/d$ (so that $n_{1}\perp d$ and
$n_{1}=\operatorname{rad}(n_{1})$), then we get

$b_{\bm{c}}(n)=\sum_{n_{1}d=n}\bm{1}_{d=\operatorname{sq}(d)}\bm{1}_{d\perp
n_{1}}\mu(n_{1})a_{\bm{c}}(n_{1})b_{\bm{c}}(d),$

with $d,n_{1}$ “unrestricted” variables. (Here all previous “restrictions”
have been “absorbed” into the factor
$\bm{1}_{d=\operatorname{sq}(d)}\bm{1}_{d\perp n_{1}}\mu(n_{1})$.)

Now let $B_{\bm{c}}(I)\colonequals\sum_{n\in I}b_{\bm{c}}(n)$ and
$A_{\bm{c}}(d,J)\colonequals\sum_{n_{1}\in J}\bm{1}_{d\perp
n_{1}}\mu(n_{1})a_{\bm{c}}(n_{1})$. Fix $N\leq\beta Y$ and
$I\subseteq[N/2,2N]$. Then by Cauchy over _square-full_ $d\leq 2N$, we
get—since $\Psi_{1}$ is standard and
$\mathop{{\sum\nolimits^{\mathrlap{\prime}}}}_{d\leq
2N}d^{-1/2}\ll_{\epsilon}N^{\epsilon}$—that

$\displaystyle\sideset{}{{}^{\prime}}{\sum}_{\bm{c}}\lvert
B_{\bm{c}}(I)\rvert^{2}$
$\displaystyle\leq\sideset{}{{}^{\prime}}{\sum}_{\bm{c}}\left(\sideset{}{{}^{\prime}}{\sum}_{d\leq
2N}\lvert
b_{\bm{c}}(d)\rvert^{2}d^{-1/2}\right)\left(\sideset{}{{}^{\prime}}{\sum}_{d\leq
2N}d^{1/2}\lvert A_{\bm{c}}(d,I/d)\rvert^{2}\right)$
$\displaystyle\ll_{\epsilon}N^{\epsilon}\sideset{}{{}^{\prime}}{\sum}_{\bm{c}}\sideset{}{{}^{\prime}}{\sum}_{d\leq
2N}d^{1/2}\lvert
A_{\bm{c}}(d,I/d)\rvert^{2}=N^{\epsilon}\sideset{}{{}^{\prime}}{\sum}_{d\leq
2N}d^{1/2}\sideset{}{{}^{\prime}}{\sum}_{\bm{c}}\lvert
A_{\bm{c}}(d,I/d)\rvert^{2}.$

Because $N/d\leq\beta Y$ and $d\leq 2\beta Y$ hold for all integers
$d\in[1,2N]$, we find that (2) implies (1): if (2) holds, then

$\sideset{}{{}^{\prime}}{\sum}_{\bm{c}}\lvert
B_{\bm{c}}(I)\rvert^{2}\ll_{\epsilon}N^{2\epsilon}\sideset{}{{}^{\prime}}{\sum}_{d\leq
2N}d^{1/2}[Z^{m+\epsilon}(N/d)]\ll_{\epsilon}N^{3\epsilon}Z^{m+\epsilon}N.$

On the other hand, $\Psi_{1}^{-1}\cdot\prod_{p\nmid d}(1+b_{\bm{c}}(p)p^{-s})$
has coefficients $a_{\bm{c}}\ast(\bm{1}_{d\perp-}\cdot\mu^{2}\cdot
b_{\bm{c}})$, which vanish on $\\{n\perp d:n=\operatorname{rad}(n)\\}$ (by a
computation at primes $p\nmid d$), and which are $O_{\epsilon}(n^{\epsilon})$
in general (since $\Psi_{1}$ is standard). So the identity

$\prod_{p\nmid d}(1-a_{\bm{c}}(p)p^{-s})=\prod_{p\nmid
d}(1+b_{\bm{c}}(p)p^{-s})=\Psi_{1}\cdot\Psi_{1}^{-1}\prod_{p\nmid
d}(1+b_{\bm{c}}(p)p^{-s})$

implies

$\bm{1}_{d\perp n}\mu(n)a_{\bm{c}}(n)=\bm{1}_{d\perp
n}\mu(n)^{2}b_{\bm{c}}(n)=\sum_{n_{1}e=n}b_{\bm{c}}(n_{1})\cdot[a_{\bm{c}}\ast(\bm{1}_{d\perp-}\cdot\mu^{2}\cdot
b_{\bm{c}})](e),$

where we sum with “no restrictions” on $n_{1},e$. But the sum is still
effectively restricted to $e$ square-full away from $d$—and there are only
$O_{\epsilon}(d^{\epsilon}E^{1/2+\epsilon})$ such integers $e\asymp E$—so by a
similar Cauchy argument as before, the bound (1) now implies (2). ∎

######  Remark 1.36.

The equivalence of (1)–(2) in Proposition 1.34 remains true (with the same
proof) if one replaces the factor $Z^{m+\epsilon}$ with
$(ZY)^{\epsilon}C(Z,Y)$, and the condition $Y\leq Z^{3}$ with $P(Y,Z)$, where
$C(Z,Y)$ is _increasing in $Y$_ and $P(Y,Z)$ is _stable under decreasing $Y$_.
(For instance, $C(Z,Y)$ could be $\max(Z^{m},Y)$, and $P$ could be the empty
condition.) But to maintain focus, we have written Proposition 1.34 as is.

##  2\. Delta method ingredients

###  2.1. The basic setup

Recall $Y,Z$ from Definition 1.15. By [heath1996new]*Theorem 2, (1.2) (based
on [duke1993bounds])—or rather, [heath1996new]*(1.2), up to easy manipulations
from §3 involving the formula (3.3) and the switching of $n,\bm{c}$—we have

$(1+O_{A}(Y^{-A}))\cdot N_{F,w}(X)=Y^{-2}\sum_{n\geq
1}\sum_{\bm{c}\in\mathbb{Z}^{m}}n^{-m}S_{\bm{c}}(n)I_{\bm{c}}(n),$

where $S_{\bm{c}}(n)$ is defined as in Definition 1.5, and where

$I_{\bm{c}}(n)\colonequals\int_{\bm{x}\in\mathbb{R}^{m}}d\bm{x}\,w(\bm{x}/X)h(n/Y,F(\bm{x})/Y^{2})e_{n}(-\bm{c}\cdot\bm{x}),$

for $\bm{c}\in\mathbb{Z}^{m}$. Dimensional analysis suggests the normalization

$\widetilde{I}_{\bm{c}}(n)\colonequals
X^{-m}I_{\bm{c}}(n)=\int_{\tilde{\bm{x}}\in\mathbb{R}^{m}}d\tilde{\bm{x}}\,w(\tilde{\bm{x}})h(n/Y,F(\tilde{\bm{x}}))e_{n/X}(-\bm{c}\cdot\tilde{\bm{x}})$

(using $Y^{2}=X^{\deg F}$ to get $F(\bm{x})/Y^{2}=F(\tilde{\bm{x}})$ for
$\tilde{\bm{x}}\colonequals\bm{x}/X$). Thus

$(1+O_{A}(X^{-A}))\cdot N_{F,w}(X)=X^{m-3}\sum_{n\geq
1}\sum_{\bm{c}\in\mathbb{Z}^{m}}n^{-(m-1)/2}\widetilde{S}_{\bm{c}}(n)\widetilde{I}_{\bm{c}}(n).$

We have referred to the switching of $n,\bm{c}$ as “easy” because the compact
support of $w$ guarantees that $I_{\bm{c}}(n)$ is supported on finitely many
moduli $n$, and is rapidly decaying in $\lVert\bm{c}\rVert$ for each $n$.
These _qualitative_ facts—important for [heath1996new]*§3, proof of Theorem 2
(which at one point switches $n,\bm{c}$ to get to [heath1996new]*(1.2)), and
for us—follow from the following two standard results, which will soon begin
to play an important _quantitative_ role as well.

######  Proposition 2.1 (Modulus cutoff).

The functions $n\mapsto I_{\bm{c}}(n)$ are supported on $n\ll_{F,w}Y$,
uniformly over $\bm{c}\in\mathbb{Z}^{m}$.

###### Proof.

See e.g. [heath1996new]*p. 180, par. 1 of §7. The vanishing of $I_{\bm{c}}(n)$
for $n\gg_{F,w}Y$ sufficiently large is a consequence of the choice $Y\asymp
X^{(\deg F)/2}$ and the definition of $h(-,-)$. ∎

######  Lemma 2.2 (Decay for large $\bm{c}$).

If $\lVert\bm{c}\rVert\geq Z$ and $n\geq 1$, then
$I_{\bm{c}}(n)\ll_{\epsilon_{0},A}\lVert\bm{c}\rVert^{-A}$.

###### Proof.

See e.g. [heath1998circle]*Lemma 3.2, (3.9). ∎

Proposition 2.1 and Lemma 2.2 easily (and comfortably) imply the _absolute_
bound

$Y^{-2}\sum_{n\geq 1}\sum_{\lVert\bm{c}\rVert\geq Z}n^{-m}\lvert
S_{\bm{c}}(n)\rvert\cdot\lvert
I_{\bm{c}}(n)\rvert\ll_{F,w,\epsilon_{0},A}X^{-A}$

(even if we only use the trivial bound $\lvert S_{\bm{c}}(n)\rvert\leq
n^{1+m}$). To analyze $N_{F,w}(X)$ as $X\to\infty$, it thus precisely remains
to understand

$X^{m-3}\sum_{n\geq 1}\sum_{\lVert\bm{c}\rVert\leq
Z}n^{-(m-1)/2}\widetilde{S}_{\bm{c}}(n)\widetilde{I}_{\bm{c}}(n).$

(Here $\widetilde{I}_{\bm{c}}(n)=\widetilde{I}_{\bm{c}}(n)\cdot\bm{1}_{n\ll
Y}$ for a suitable factor $\bm{1}_{n\ll Y}$. But it is more convenient to keep
the factor implicit, to allow for more flexible technique later on—especially
in §3.)

###  2.2. Exponential sums as linear combinations of approximate coefficients

Recall that $S_{\bm{c}}$ is multiplicative. So $\widetilde{S}_{\bm{c}}$ is
too. Now we recall some formulas at prime powers.

######  Proposition 2.3 (Cf. [hooley1986HasseWeil]*p. 69, (47)).

For all tuples $\bm{c}\in\mathbb{Z}^{m}$ and primes $p\nmid\bm{c}$, we have
$S_{\bm{c}}(p)=p^{2}E_{\bm{c}}(p)-pE(p)$.

###### Proof sketch.

(Although [hooley1986HasseWeil] assumes $p\nmid F^{\vee}(\bm{c})$ and
$F=x_{1}^{3}+\dots+x_{6}^{3}$, the underlying arguments work more generally;
cf. [hooley2014octonary]*p. 254, Lemma 7.)

Because $F$ is homogeneous, we have $S_{\bm{c}}(p)=S_{\lambda\bm{c}}(p)$ for
all $\lambda\in\mathbb{F}_{p}^{\times}$. So

$\displaystyle(p-1)S_{\bm{c}}(p)$
$\displaystyle=\sideset{}{{}^{\prime}}{\sum}_{\lambda,a\bmod{p}}\sum_{\bm{x}\bmod{p}}e_{p}(aF(\bm{x})+\lambda\bm{c}\cdot\bm{x})$
$\displaystyle=\sum_{\bm{x}\bmod{p}}(p\cdot\bm{1}_{p\mid
F(\bm{x})}-1)(p\cdot\bm{1}_{p\mid\bm{c}\cdot\bm{x}}-1)=p^{2}(p-1)E_{\bm{c}}(p)-p(p-1)E(p)$

if $p\nmid\bm{c}$. (The factor $p^{2}(p-1)$ comes from $\sum_{\lambda,a}$ and
the “cone operation” over $(\mathcal{V}_{\mathbb{F}_{p}})_{\bm{c}}$.) ∎

######  Remark 2.4.

If $p\mid\bm{c}$, then $S_{\bm{c}}(p)=S_{\bm{0}}(p)=p^{2}E(p)-pE(p)$ instead.

In particular,
$\widetilde{S}_{\bm{c}}(p)=\widetilde{E}_{\bm{c}}(p)-p^{-1/2}\widetilde{E}(p)$
at primes $p\nmid F^{\vee}(\bm{c})$. Here $\widetilde{E}(p)\ll 1$, by Weil’s
diagonal hypersurface bound (after “absorbing” bad primes for $F$ into the
implied constant); this will be essentially negligible for our purposes.

######  Proposition 2.5 (Cf. [hooley1986HasseWeil]*pp. 65–66, Lemma 7).

Say $p\nmid F^{\vee}(\bm{c})$. Then $S_{\bm{c}}(p^{l})=0$ for all integers
$l\geq 2$.

###### Proof sketch.

(Although [hooley1986HasseWeil] assumes $F=x_{1}^{3}+\dots+x_{6}^{3}$, the
underlying proof immediately generalizes; see [hooley2014octonary]*p. 255,
Lemma 10.)

This time, scalar symmetry in $\bm{c}$ (using homogeneity of $F$) gives

$\phi(p^{l})S_{\bm{c}}(p^{l})=\sum_{\bm{x}\in(\mathbb{Z}/p^{l})^{m}}[-p^{l-1}\cdot\bm{1}_{p^{l-1}\mid\bm{c}\cdot\bm{x}}+p^{l}\cdot\bm{1}_{p^{l}\mid\bm{c}\cdot\bm{x}}][-p^{l-1}\cdot\bm{1}_{p^{l-1}\mid
F(\bm{x})}+p^{l}\cdot\bm{1}_{p^{l}\mid F(\bm{x})}].$

So $S_{\bm{c}}(p^{l})=0$ is equivalent to statements about point counts, which
are proven by Hensel lifting. The lifting calculus follows dimension
predictions, because $p\nmid F^{\vee}(\bm{c})$ implies (by Proposition-
Definition 1.8) that the $\mathbb{F}_{p}$-variety
$(\mathcal{V}_{\mathbb{F}_{p}})_{\bm{c}}$ is smooth of codimension $2$. ∎

We can now establish the validity of Example 1.13.

###### Proof of validity of Example 1.13(1).

Trivial, since
$(b_{\bm{c}},a^{\prime}_{\bm{c}})=(\widetilde{S}_{\bm{c}},\bm{1}_{n=1})$. ∎

###### Proof for Example 1.13(2).

Trivial, since
$(b_{\bm{c}},a^{\prime}_{\bm{c}})=(\widetilde{S}_{\bm{c}}\cdot\bm{1}_{n\perp
F^{\vee}(\bm{c})},\widetilde{S}_{\bm{c}}\cdot\bm{1}_{n\mid
F^{\vee}(\bm{c})^{\infty}})$. ∎

###### Proof for Example 1.13(3).

Here $b_{\bm{c}}(n),a_{\bm{c}}(n)\ll_{\epsilon}n^{\epsilon}$ by Deligne. But
$a^{\prime}_{\bm{c}}=\widetilde{S}_{\bm{c}}\ast a_{\bm{c}}$. So the condition
1.12(2) holds. Furthermore, if $p\nmid F^{\vee}(\bm{c})$, then
$a^{\prime}_{\bm{c}}(p)=\widetilde{S}_{\bm{c}}(p)+a_{\bm{c}}(p)=-p^{-1/2}\widetilde{E}(p)$,
since
$a_{\bm{c}}(p)=-b_{\bm{c}}(p)=-(-1)^{m_{\ast}}\tilde{\lambda}_{\bm{c}}(p)=-\widetilde{E}_{\bm{c}}(p)$.
So the condition 1.12(3) also holds. ∎

###### Proof for Example 1.13(4).

Again $b_{\bm{c}}(n),a_{\bm{c}}(n)\ll_{\epsilon}n^{\epsilon}$, but not “just”
by Deligne; at bad primes, we need additional input, which is luckily known
for $m\in\\{4,6\\}$. Otherwise, the proof for Example 1.13(3) carries over
directly. ∎

Now fix $\Psi_{1}$ in Theorem 1.28. For each $\bm{c}$ with
$F^{\vee}(\bm{c})\neq 0$, the following result expresses
$\widetilde{S}_{\bm{c}}(n)$ as a reasonably well-behaved linear combination of
coefficients of $\Psi_{1}(\bm{c},s)$.

######  Proposition 2.6.

Fix $\bm{c}$ with $F^{\vee}(\bm{c})\neq 0$. Then
$\widetilde{S}_{\bm{c}}=a^{\prime}_{\bm{c}}\ast b_{\bm{c}}$. Here
$a^{\prime}_{\bm{c}}$ is multiplicative, and

$\displaystyle a^{\prime}_{\bm{c}}(p)\cdot\bm{1}_{p\nmid F^{\vee}(\bm{c})}\ll
p^{-1/2}\quad\textnormal{and}\quad
a^{\prime}_{\bm{c}}(p^{k})\cdot\bm{1}_{p\nmid F^{\vee}(\bm{c})}$
$\displaystyle\ll_{\epsilon}p^{k\epsilon}\quad\textnormal{and}\quad
a^{\prime}_{\bm{c}}(p^{k})\ll_{\epsilon}p^{k\epsilon}\sum_{d\mid
p^{k}}\lvert\widetilde{S}_{\bm{c}}(d)\rvert$
$\displaystyle\quad\textnormal{and}\quad b_{\bm{c}}(p^{k})\cdot\bm{1}_{p\nmid
F^{\vee}(\bm{c})}$
$\displaystyle\ll_{\epsilon}p^{k\epsilon}\quad\textnormal{and}\quad
b_{\bm{c}}(p^{k})\ll_{\epsilon}p^{k\epsilon}\sum_{d\mid
p^{k}}\lvert\widetilde{S}_{\bm{c}}(d)\rvert$

hold uniformly over $\bm{c},p,k$.

###### Proof.

From $\Phi=(\Phi/\Psi_{1})\cdot\Psi_{1}$, we obtain
$\widetilde{S}_{\bm{c}}=a^{\prime}_{\bm{c}}\ast b_{\bm{c}}$. It remains to
address $a^{\prime}_{\bm{c}},b_{\bm{c}}$.

By definition, $a_{\bm{c}}\ast b_{\bm{c}}=\bm{1}_{n=1}$ and
$a^{\prime}_{\bm{c}}=\widetilde{S}_{\bm{c}}\ast a_{\bm{c}}$. Since
$b_{\bm{c}},\widetilde{S}_{\bm{c}}$ are multiplicative, it follows that
$a_{\bm{c}},a^{\prime}_{\bm{c}}$ are too. It remains to bound
$a^{\prime}_{\bm{c}}(p^{k}),b_{\bm{c}}(p^{k})$. When $p\mid F^{\vee}(\bm{c})$,
there is nothing to prove—the condition 1.12(2) already gives what we want. So
assume $p\nmid F^{\vee}(\bm{c})$. Then the condition 1.12(3) gives
$a^{\prime}_{\bm{c}}(p)\ll p^{-1/2}$, while the condition 1.12(2) gives

$b_{\bm{c}}(p^{k}),a^{\prime}_{\bm{c}}(p^{k})\ll_{\epsilon}p^{k\epsilon}\sum_{d\mid
p^{k}}\lvert\widetilde{S}_{\bm{c}}(d)\rvert\ll_{\epsilon}p^{2k\epsilon},$

since
$\widetilde{S}_{\bm{c}}(p)=\widetilde{E}_{\bm{c}}(p)-p^{-1/2}\widetilde{E}(p)\ll
1$ and $\widetilde{S}_{\bm{c}}(p^{l})\cdot\bm{1}_{l\geq 2}=0\ll 1$. ∎

###  2.3. Main unconditional general pointwise bounds

For technical reasons, we will analyze the $\bm{c}$’s in groups depending on
how many coordinates are zero. It would be interesting to find a similar
notion for non-diagonal forms $F$.

######  Definition 2.7.

Given $\mathcal{I}\subseteq[m]$ of size $r\leq m$, we call the set
$\mathcal{R}\subseteq[-Z,Z]^{m}$ of $\bm{c}\in\mathbb{Z}^{m}$ with
$c_{j}\in[-Z,Z]\setminus\\{0\\}$ for $j\in\mathcal{I}$ and $c_{j}=0$ for
$j\notin\mathcal{I}$ a (uniform) $r$-dimensional _deleted box_.

Let $\mathcal{R}\subseteq[-Z,Z]^{m}$ be a deleted box with
$\lvert\mathcal{I}\rvert=r\in[0,m]$. Then we can bound the sums
$\widetilde{S}_{\bm{c}}(n)\colonequals n^{-(m+1)/2}S_{\bm{c}}(n)$ using the
somewhat crude but _general_ pointwise bound (available since $F$ is diagonal)
given by the following result:

######  Proposition 2.8 ([hooley1986HasseWeil, heath1998circle]).

For all $\bm{c}\in\mathbb{Z}^{m}$ and integers $n\geq 1$, we have

$n^{-1/2}\lvert\widetilde{S}_{\bm{c}}(n)\rvert\ll_{F}O(1)^{\omega(n)}\prod_{j\in[m]}\gcd(\operatorname{cub}(n)^{1/6},\gcd(\operatorname{cub}(n),\operatorname{sq}(c_{j}))^{1/4}),$

where $\operatorname{cub}(\star)$ and $\operatorname{sq}(\star)$ denote the
cube-full and square-full parts of $\star$, respectively. Here we interpret
$\gcd(-,-)$ formally in terms of exponents that are allowed to be rational.

###### Proof.

In general $\widetilde{S}_{\bm{c}}(n)\colonequals n^{-(m+1)/2}S_{\bm{c}}(n)$
by definition, and for diagonal $F$ we have

$\lvert S_{\bm{c}}(p^{l})\rvert\ll(p^{\infty},O_{F}(1))^{O(1)}\cdot
p^{l(1+m/2)}\prod_{j\in[m]}\gcd(\operatorname{cub}(p^{l})^{1/6},\gcd(\operatorname{cub}(p^{l}),\operatorname{sq}(c_{j}))^{1/4})$

by [heath1998circle]*p. 682, (5.1)–(5.2) for $l\geq 2$ and [heath1983cubic]*p.
238, Lemma 11 for $l=1$. ∎

######  Remark 2.9.

We have stated Proposition 2.8 uniformly over $\bm{c}\in\mathbb{Z}^{m}$. But
given $\mathcal{R}$, Proposition 2.8 _implies_ that for all
$\bm{c}\in\mathcal{R}$ and integers $n\geq 1$, we have

$n^{-1/2}\lvert\widetilde{S}_{\bm{c}}(n)\rvert\ll_{F}O(1)^{\omega(n)}\operatorname{cub}(n)^{(m-r)/6}\prod_{j\in\mathcal{I}}\gcd(\operatorname{cub}(n),\operatorname{sq}(c_{j}))^{1/4}.$

It is this simpler statement that we use in the present paper.

Now we turn to the _integrals_ $\widetilde{I}_{\bm{c}}(n)\colonequals
X^{-m}I_{\bm{c}}(n)$, assuming $r\geq 1$. (We cover $\bm{c}=\bm{0}$ in §4.)
The statement [heath1998circle]*p. 678, Lemma 3.2 covers the essential cases
$k=0,1$ in the next result (which, like Proposition 2.8, is available since
$F$ is diagonal). We state a generalization to all $k\geq 0$, just in case it
comes in handy for some (future) smoothing purposes.

######  Lemma 2.10 (Main $n$-aspect bounds).

Uniformly over $\bm{c}\in\mathcal{R}$ and $n\in[1/2,\infty)$, we have

$n^{k}\lvert\partial_{n}^{k}\widetilde{I}_{\bm{c}}(n)\rvert\ll_{k,\epsilon}X^{\epsilon}\left(\frac{X\lVert\bm{c}\rVert}{n}\right)^{1-(m+r)/4}\prod_{i\in\mathcal{I}}\left(\frac{\lVert\bm{c}\rVert}{\lvert
c_{i}\rvert}\right)^{1/2}\quad\textnormal{for}\;k=0,1,2,\dots.$

Furthermore, if $B\in C^{\infty}_{c}(\mathbb{R}_{>0})$ is supported on
$[1/2,1]$, then
$n^{k+1}\lvert\partial_{n}^{k}[y^{-1}B(n/y)\widetilde{I}_{\bm{c}}(n)]\rvert$
satisfies the same bound for all $n\in(0,\infty)$, uniformly as $y\geq 1$
varies.

###### Proof.

By [density_beats_hua_when]*Remark 4.1 and Lemma 4.9, we know that

$n^{k}\cdot\partial_{n}^{k}I_{\bm{c}}(n)\ll_{k,\epsilon}\left(\frac{X\lVert\bm{c}\rVert}{n}\right)X^{m+\epsilon}\prod_{i=1}^{m}\min[(n/X\lvert
c_{i}\rvert)^{1/2},(n/X\lVert\bm{c}\rVert)^{1/4}]\quad\textnormal{for all
$\bm{c}\in\mathbb{R}^{m}\setminus\\{\bm{0}\\}$},$

and that $n^{k+1}\cdot\partial_{n}^{k}[y^{-1}B(n/y)I_{\bm{c}}(n)]$ satisfies
the same bound. Now “replace” $\min[-,-]$ with $(n/X\lvert c_{i}\rvert)^{1/2}$
for each $i\in\mathcal{I}$, and with $(n/X\lVert\bm{c}\rVert)^{1/4}$ for each
$i\in[m]\setminus\mathcal{I}$; this bounds the right-hand side by
$X^{m+\epsilon}(X\lVert\bm{c}\rVert/n)^{1-(m-r)/4}\prod_{i\in\mathcal{I}}(n/X\lvert
c_{i}\rvert)^{1/2}$, which (after dividing by $X^{m}$) simplifies to what we
want (since $\widetilde{I}_{\bm{c}}(n)\colonequals X^{-m}I_{\bm{c}}(n)$). ∎

######  Remark 2.11.

For $m=6$, in the Fermat reduction described in Theorem 1.28, we may replace
the bound with $X^{\epsilon}\min(1,(X\lVert\bm{c}\rVert/n)^{1-m/2})\ll
X^{\epsilon}(X\lVert\bm{c}\rVert/n)^{1-m/2}$, by [hooley2014octonary]*p. 252.

######  Remark 2.12.

We should emphasize that the above bounds on $I_{\bm{c}}(n)$ are likely only
valid (as written) for the usual (and present) setting of the parameter
$Y\asymp X^{(\deg F)/2}=X^{3/2}$.

######  Remark 2.13.

It would be interesting to know the optimal asymptotics, e.g. whether or not
the bound remains true with a smaller power of $X\lVert\bm{c}\rVert/n$. At
least for generic $\bm{c}$, one might expect to reduce the exponent $1-m/2$
using a deeper stationary phase analysis (implemented to some extent in Paper
III).

###  2.4. Main unconditional second-moment bounds

We need the following bound for one half of Cauchy in §3.

######  Lemma 2.14.

For $N_{0},N\ll X^{O(1)}$, the square root of the second moment of

$\star=N^{-(m-1)/2}\sup_{n\asymp
N}\left(\lvert\widetilde{I}_{\bm{c}}(n)\rvert,N\cdot\lvert\partial_{n}\widetilde{I}_{\bm{c}}(n)\rvert\right)\cdot\sum_{n_{0}\asymp
N_{0}}\lvert a^{\prime}_{\bm{c}}(n_{0})\rvert$

over $\bm{c}\in\mathcal{R}$ with $F^{\vee}(\bm{c})\neq 0$ is at most
$O_{\epsilon}$ of

$X^{\epsilon}N^{-(m-1)/2}N_{0}^{1/2+(m-r)/6}\left[(XZ/N)^{1-(m+r)/4}Z^{r/2}+(X/N)^{1-(m+r)/4}\right].$

###### Proof.

It suffices to restrict attention to $\bm{c}$ in a dyadic box
$\mathcal{C}\subseteq\mathcal{R}$, with $\lvert c_{i}\rvert\asymp C_{i}$ for
$i\in\mathcal{I}$, say. Let $C\colonequals\max(C_{i})$. Then Lemma 2.10
implies

$\sup_{n\asymp
N}\left(\lvert\widetilde{I}_{\bm{c}}(n)\rvert,N\cdot\lvert\partial_{n}\widetilde{I}_{\bm{c}}(n)\rvert\right)\ll_{\epsilon}X^{\epsilon}(XC/N)^{1-(m+r)/4}\prod_{i\in\mathcal{I}}(C/C_{i})^{1/2}$

uniformly over $\bm{c}\in\mathcal{C}$, and we reduce to bounding the second
moment of $\sum_{n_{0}\asymp N_{0}}\lvert a^{\prime}_{\bm{c}}(n_{0})\rvert$.
By Proposition 2.16 below, we get (upon cancellation of the $C_{i}$ factors) a
final estimate of

$\sqrt{\sideset{}{{}^{\prime}}{\sum}_{\bm{c}\in\mathcal{C}}(\star)^{2}}\ll_{\epsilon}X^{2\epsilon}N^{-(m-1)/2}N_{0}^{1/2+(m-r)/6}(XC/N)^{1-(m+r)/4}\prod_{i\in\mathcal{I}}C^{1/2}.$

Plugging in the endpoints of the range $1\ll C\ll Z$ gives the desired result.
∎

######  Remark 2.15.

In the $m=6$ Fermat reduction of Theorem 1.28,
$(XC/N)^{1-(m+r)/4}\prod_{i\in\mathcal{I}}C^{1/2}$ can be improved to
$(XC/N)^{1-m/2}\prod_{i\in\mathcal{I}}C_{i}^{1/2}\ll[(XZ/N)^{1-m/2}Z^{r/2}+(X/N)^{1-m/2}]$.

######  Proposition 2.16.

For $N_{0}\ll X^{O(1)}$, the second moment of $\sum_{n_{0}\asymp N_{0}}\lvert
a^{\prime}_{\bm{c}}(n_{0})\rvert$ over $\bm{c}\in\mathcal{C}$ with
$F^{\vee}(\bm{c})\neq 0$ is
$O_{\epsilon}(X^{\epsilon}N_{0}^{1+(m-r)/3}\prod_{i\in\mathcal{I}}C_{i})$.

###### Proof.

For $N_{0}\ll 1$ sufficiently small, everything vanishes, so suppose $N_{0}\gg
1$.

If $n_{\bm{c}}\colonequals\prod_{p\mid F^{\vee}(\bm{c})}p^{v_{p}(n_{0})}$ and
$n_{2}\colonequals\operatorname{sq}(n_{0}/n_{\bm{c}})$—so that
$\operatorname{rad}(n_{\bm{c}})\mid F^{\vee}(\bm{c})\neq 0$ and $n_{2}$ is
square-full—then by multiplicativity of $a^{\prime}_{\bm{c}}$, together with
Proposition 2.6’s bounds at primes $p\nmid F^{\vee}(\bm{c})$ (applied to the
“good part” $n_{0}/n_{\bm{c}}$ of $n_{0}$), we get

$\displaystyle\sum_{n_{0}\asymp N_{0}}\lvert a^{\prime}_{\bm{c}}(n_{0})\rvert$
$\displaystyle\ll_{\epsilon}\sideset{}{{}^{\prime}}{\sum}_{n_{\bm{c}}n_{2}\ll
N_{0}}\frac{N_{0}}{n_{\bm{c}}n_{2}}\cdot(N_{0}/n_{\bm{c}}n_{2})^{-1/2+\epsilon}n_{2}^{\epsilon}\lvert
a^{\prime}_{\bm{c}}(n_{\bm{c}})\rvert$ $\displaystyle\ll
N_{0}^{1/2+\epsilon}\sideset{}{{}^{\prime}}{\sum}_{n_{\bm{c}}n_{2}\ll
N_{0}}(n_{\bm{c}}n_{2})^{-1/2}\lvert a^{\prime}_{\bm{c}}(n_{\bm{c}})\rvert$
$\displaystyle\ll_{\epsilon}N_{0}^{1/2+2\epsilon}\sideset{}{{}^{\prime}}{\sum}_{n_{\bm{c}}\ll
N_{0}}n_{\bm{c}}^{-1/2}\lvert
a^{\prime}_{\bm{c}}(n_{\bm{c}})\rvert\ll_{\epsilon}N_{0}^{1/2+2\epsilon}(N_{0}C)^{\epsilon}\sideset{}{{}^{\prime}}{\max}_{n_{\bm{c}}\ll
N_{0}}n_{\bm{c}}^{-1/2}\lvert a^{\prime}_{\bm{c}}(n_{\bm{c}})\rvert,$

since $\mathop{{\sum\nolimits^{\mathrlap{\prime}}}}_{n_{2}\leq
z}n_{2}^{-1/2}\ll_{\epsilon}z^{\epsilon}$ and there are few $n_{\bm{c}}\ll
N_{0}$ with $\operatorname{rad}(n_{\bm{c}})\mid F^{\vee}(\bm{c})\neq 0$.

Now recall the general bound
$a^{\prime}_{\bm{c}}(n_{\bm{c}})\ll_{\epsilon}n_{\bm{c}}^{\epsilon}\sum_{d\mid
n_{\bm{c}}}\lvert\widetilde{S}_{\bm{c}}(d)\rvert$ (from the condition
1.12(2)). Divisors of $n_{\bm{c}}$ are sparse, so up to $N_{0}^{\epsilon}$, we
may replace $a^{\prime}_{\bm{c}}(n_{\bm{c}})$ with
$\widetilde{S}_{\bm{c}}(n_{\bm{c}})$ in the $\max^{\prime}$ above. By
Proposition 2.8, the second moment of the resulting $\max^{\prime}$ is

$\sideset{}{{}^{\prime}}{\sum}_{\bm{c}\in\mathcal{C}}\sideset{}{{}^{\prime}}{\max}_{n_{\bm{c}}\ll
N_{0}}n_{\bm{c}}^{-1}\lvert\widetilde{S}_{\bm{c}}(n_{\bm{c}})\rvert^{2}\ll_{\epsilon}N_{0}^{(m-r)/3+2\epsilon}\sideset{}{{}^{\prime}}{\sum}_{\bm{c}\in\mathcal{C}}\prod_{j\in\mathcal{I}}\operatorname{sq}(c_{j})^{1/2}\leq
N_{0}^{(m-r)/3+2\epsilon}\sum_{\bm{c}\in\mathcal{C}}\prod_{j\in\mathcal{I}}\operatorname{sq}(c_{j})^{1/2}$

since
$\gcd(\operatorname{cub}(n_{\bm{c}}),\operatorname{sq}(c_{j}))^{1/4}\leq\operatorname{sq}(c_{j})^{1/4}$.
But $\sum_{\lvert c\rvert\leq
z}\operatorname{sq}(c)^{1/2}\ll\mathop{{\sum\nolimits^{\mathrlap{\prime}}}}_{d\leq
z}(z/d)d^{1/2}\ll_{\epsilon}z^{1+\epsilon}$ since there are $O(D^{1/2})$
square-full $d\asymp D$. Now the final desired result easily follows. ∎

######  Remark 2.17.

The above method “barely” works. Is there a more robust way to bound

$\sideset{}{{}^{\prime}}{\sum}_{\bm{c}\in\mathcal{C}}\sideset{}{{}^{\prime}}{\max}_{n_{\bm{c}}\ll
N_{0}}n_{\bm{c}}^{-1}\lvert\widetilde{S}_{\bm{c}}(n_{\bm{c}})\rvert^{2}\ll\sideset{}{{}^{\prime}}{\sum}_{\bm{c}\in\mathcal{C}}\sideset{}{{}^{\prime}}{\sum}_{n\ll
N_{0}}n^{-1}\lvert\widetilde{S}_{\bm{c}}(n)\rvert^{2}=\sideset{}{{}^{\prime}}{\sum}_{n,\bm{c}}n^{-1}\lvert\widetilde{S}_{\bm{c}}(n)\rvert^{2},$

perhaps using more precise or conceptual information about the sums
$\widetilde{S}_{\bm{c}}(n)$? (See Proposition A.1 for equivalent formulations
of the problem.)

###  2.5. A “trivial” improvement of our assumed second-moment hypothesis

For Cauchy in §3, we will also use the bound

$\sideset{}{{}^{\prime}}{\sum}_{\bm{c}\in\mathcal{R}}\left\lvert\sum_{n_{1}\in
I}b_{\bm{c}}(n_{1})\right\rvert^{2}\ll_{\beta,\epsilon}X^{\epsilon}\min\left(Z^{m}N_{1},Z^{r}N_{1}^{\max(2,1+(m-r)/3)}\right)$

for $N_{1}\leq\beta Y$ and $I\subseteq[N_{1}/2,2N_{1}]$, where
$\beta\colonequals 100\cdot(\textnormal{Proposition~{}\ref{PROP:rigorous-
modulus-cutoff-Y}'s implied constant})\asymp_{F,w}1$, say. Here the bound
$X^{\epsilon}Z^{m}N_{1}$ follows from our assumption in Theorem 1.28;
meanwhile, $X^{\epsilon}Z^{r}N_{1}^{\max(2,1+(m-r)/3)}$ comes from the
following lemma:

######  Lemma 2.18.

For $N_{1}\ll X^{O(1)}$, the second moment of $\sum_{n_{1}\asymp N_{1}}\lvert
b_{\bm{c}}(n_{1})\rvert$ over $\bm{c}\in\mathcal{R}$ with
$F^{\vee}(\bm{c})\neq 0$ is $O_{\epsilon}(X^{\epsilon})\cdot
N_{1}^{\max(2,1+(m-r)/3)}\cdot Z^{r}$.

###### Proof.

Fix a dyadic box $\mathcal{C}\subseteq\mathcal{R}$, say with $\lvert
c_{i}\rvert\asymp C_{i}$ for $i\in\mathcal{I}$. Then $1\ll\max(C_{i})\ll Z$.
By Proposition 2.19 below, the desired result follows upon dyadic summation. ∎

######  Proposition 2.19.

For $N_{1}\ll X^{O(1)}$, the second moment of $\sum_{n_{1}\asymp N_{1}}\lvert
b_{\bm{c}}(n_{1})\rvert$ over $\bm{c}\in\mathcal{C}$ with
$F^{\vee}(\bm{c})\neq 0$ is $O_{\epsilon}(X^{\epsilon})\cdot
N_{1}^{\max(2,1+(m-r)/3)}\cdot\prod_{i\in\mathcal{I}}C_{i}$.

###### Proof.

We mimic the proof of Proposition 2.16. For $N_{1}\ll 1$ sufficiently small,
everything vanishes, so suppose $N_{1}\gg 1$. If
$n_{\bm{c}}\colonequals\prod_{p\mid F^{\vee}(\bm{c})}p^{v_{p}(n_{1})}$—so that
$\operatorname{rad}(n_{\bm{c}})\mid F^{\vee}(\bm{c})\neq 0$—then by
multiplicativity of $b_{\bm{c}}$, together with Proposition 2.6’s bounds at
primes $p\nmid F^{\vee}(\bm{c})$ (applied to the “good part”
$n_{1}/n_{\bm{c}}$ of $n_{1}$), we get

$\sum_{n_{1}\asymp N_{1}}\lvert
b_{\bm{c}}(n_{1})\rvert\ll_{\epsilon}\sideset{}{{}^{\prime}}{\sum}_{n_{\bm{c}}\ll
N_{1}}\frac{N_{1}}{n_{\bm{c}}}\cdot(N_{1}/n_{\bm{c}})^{\epsilon}\lvert
b_{\bm{c}}(n_{\bm{c}})\rvert\ll_{\epsilon}N_{1}^{1+\epsilon}(N_{1}C)^{\epsilon}\sideset{}{{}^{\prime}}{\max}_{n_{\bm{c}}\ll
N_{1}}n_{\bm{c}}^{-1}\lvert b_{\bm{c}}(n_{\bm{c}})\rvert,$

since there are few $n_{\bm{c}}\ll N_{1}$ with
$\operatorname{rad}(n_{\bm{c}})\mid F^{\vee}(\bm{c})\neq 0$.

Now recall the general bound
$b_{\bm{c}}(n_{\bm{c}})\ll_{\epsilon}n_{\bm{c}}^{\epsilon}\sum_{d\mid
n_{\bm{c}}}\lvert\widetilde{S}_{\bm{c}}(d)\rvert$ (from the condition
1.12(2)). Divisors of $n_{\bm{c}}$ are sparse, so up to $N_{1}^{\epsilon}$, we
may replace $b_{\bm{c}}(n_{\bm{c}})$ with $\widetilde{S}_{\bm{c}}(n_{\bm{c}})$
in the $\max^{\prime}$ above. By Proposition 2.8, the second moment of the
resulting $\max^{\prime}$ is

$\displaystyle\sideset{}{{}^{\prime}}{\sum}_{\bm{c}\in\mathcal{C}}\sideset{}{{}^{\prime}}{\max}_{n_{\bm{c}}\ll
N_{1}}n_{\bm{c}}^{-2}\lvert\widetilde{S}_{\bm{c}}(n_{\bm{c}})\rvert^{2}$
$\displaystyle\ll_{\epsilon}N_{1}^{2\epsilon}\max(1,N_{1}^{-1+(m-r)/3})\sideset{}{{}^{\prime}}{\sum}_{\bm{c}\in\mathcal{C}}\prod_{j\in\mathcal{I}}\operatorname{sq}(c_{j})^{1/2}$
$\displaystyle\ll_{\epsilon}(N_{1}C)^{2\epsilon}\max(1,N_{1}^{-1+(m-r)/3})\prod_{j\in\mathcal{I}}C_{j}$

(cf. the proof of Proposition 2.16). Now the final desired result easily
follows. ∎

##  3\. Contribution from smooth hyperplane sections: Cauchy or counting

In this section, we will show—conditionally on the results of §2—that

$X^{m-3}\sideset{}{{}^{\prime}}{\sum}_{\bm{c}\in[-Z,Z]^{m}}\sum_{n\geq
1}n^{-(m-1)/2}\widetilde{S}_{\bm{c}}(n)\widetilde{I}_{\bm{c}}(n)\ll_{\epsilon}X^{3m/4-3/2+\epsilon}.$

######  Remark 3.1.

In Paper III, we obtain more precise bounds under stronger hypotheses.

Writing $n=n_{0}n_{1}$ and expanding
$\widetilde{S}_{\bm{c}}(n)=(a^{\prime}_{\bm{c}}\ast b_{\bm{c}})(n)$ for $n\geq
1$, we seek to bound

$\sideset{}{{}^{\prime}}{\sum}_{\lVert\bm{c}\rVert\leq Z}\sum_{n_{0}n_{1}\geq
1}n^{-(m-1)/2}a^{\prime}_{\bm{c}}(n_{0})b_{\bm{c}}(n_{1})\widetilde{I}_{\bm{c}}(n)=\sideset{}{{}^{\prime}}{\sum}_{\lVert\bm{c}\rVert\leq
Z}\sum_{n_{0}\geq 1}a^{\prime}_{\bm{c}}(n_{0})\sum_{n_{1}\geq
1}(n_{0}n_{1})^{-(m-1)/2}\widetilde{I}_{\bm{c}}(n_{0}n_{1})b_{\bm{c}}(n_{1}).$

By Proposition 2.1, we have $\widetilde{I}_{\bm{c}}(n)=0$ for all $\bm{c}$
when $n\gg Y$. Hence the _double sum_ over $n_{0},n_{1}$ breaks into _dyadic
pieces_ $\Diamond_{\bm{c},N_{0},N_{1}}$ running over
$(n_{0},n_{1})\in[N_{0},2N_{0})\times[N_{1},2N_{1})$, with $N\colonequals
N_{0}N_{1}\ll Y$. Now, given $n_{0}$, let

$\diamond_{\bm{c},n_{0},N_{1}}\colonequals\sum_{n_{1}\in[N_{1},2N_{1})}(n_{0}n_{1})^{-(m-1)/2}\widetilde{I}_{\bm{c}}(n_{0}n_{1})b_{\bm{c}}(n_{1}).$

For convenience, let $B_{\bm{c}}(J)\colonequals\sum_{n_{1}\in
J}b_{\bm{c}}(n_{1})$ (as in the proof of Proposition 1.34) for each real
interval $J$. Then we have the following lemma:

######  Lemma 3.2.

As the _dyadic_ parameters $N_{0},N_{1}\geq 1$ vary, there exist probability
measures $\nu=\nu_{N_{0},N_{1}}$ supported on $[N_{1},2N_{1}]$, and only
depending on $N_{0},N_{1}$, such that

$\diamond_{\bm{c},n_{0},N_{1}}\ll N^{-(m-1)/2}\left(\sup_{n\asymp
N}\left(\lvert\widetilde{I}_{\bm{c}}(n)\rvert,N\cdot\lvert\partial_{n}\widetilde{I}_{\bm{c}}(n)\rvert\right)\right)\int_{x\in[N_{1},2N_{1}]}d\nu(x)\,\lvert
B_{\bm{c}}([N_{1},x))\rvert$

holds, with an absolute implied constant, for all $\bm{c}\in\mathbb{Z}^{m}$
and $n_{0}\in[N_{0},2N_{0})$.

###### Proof.

It suffices to construct a measure $\nu$ of absolutely bounded mass $O(1)$.
Fix $\bm{c}\in\mathbb{Z}^{m}$ and $n_{0}\in[N_{0},2N_{0})$, and let
$I(n)=n^{-(m-1)/2}\widetilde{I}_{\bm{c}}(n)$. By partial summation,

$\diamond_{\bm{c},n_{0},N_{1}}=\sum_{n_{1}\in[N_{1},2N_{1})}I(n_{0}n_{1})\cdot
b_{\bm{c}}(n_{1})=\sum_{n\in[n_{0}N_{1},2n_{0}N_{1})}I(n)\cdot\bm{1}_{n_{0}\mid
n}b_{\bm{c}}(n/n_{0})$

is bounded in magnitude by

$\lVert I(n)\rVert_{L^{\infty}([N,4N])}\cdot\lvert
B_{\bm{c}}([N_{1},2N_{1}))\rvert+\lVert
I^{\prime}(n)\rVert_{L^{\infty}([N,4N])}\cdot\sum_{z\in[n_{0}N_{1},2n_{0}N_{1})}\lvert
B_{\bm{c}}([N_{1},z/n_{0}))\rvert.$

The $z$-sum simplifies to
$n_{0}N_{1}\cdot\mathbb{E}_{x\in[N_{1},2N_{1})}\lvert
B_{\bm{c}}([N_{1},x))\rvert\asymp N\cdot\mathbb{E}_{x\in[N_{1},2N_{1})}\lvert
B_{\bm{c}}([N_{1},x))\rvert$, where the expectation $\mathbb{E}_{x}$ is
defined using a Haar measure either on $\mathbb{Z}$ or $\mathbb{R}$. (Note
that $n_{0},N_{0},N_{1}$ are all positive _integers_.) Thus we may choose
$\nu=\delta_{2N_{1}}+N_{1}^{-1}\omega$ with $\omega$ the usual unit-normalized
Haar measure on $\mathbb{R}$. ∎

Thus $\lvert\Diamond_{\bm{c},N_{0},N_{1}}\rvert\leq\sum_{n_{0}\asymp
N_{0}}\lvert
a^{\prime}_{\bm{c}}(n_{0})\cdot\diamond_{\bm{c},n_{0},N_{1}}\rvert$ is (by
Lemma 3.2) at most

$N^{-(m-1)/2}\left(\sup_{n\asymp
N}\left(\lvert\widetilde{I}_{\bm{c}}(n)\rvert,N\cdot\lvert\partial_{n}\widetilde{I}_{\bm{c}}(n)\rvert\right)\right)\left(\sum_{n_{0}\asymp
N_{0}}\lvert
a^{\prime}_{\bm{c}}(n_{0})\rvert\right)\cdot\int_{x\in[N_{1},2N_{1}]}d\nu(x)\,\lvert
B_{\bm{c}}([N_{1},x))\rvert.$

######  Remark 3.3.

It is plausible that one could do double summation by parts in $n_{0},n_{1}$,
but for a given $\bm{c}$ (with $F^{\vee}(\bm{c})\neq 0$), the dominant terms
in the sum $\sum_{n_{0}\geq 1}a^{\prime}_{\bm{c}}(n_{0})$ are sparse enough
that it may be difficult to get additional cancellation.

Now
$\mathop{{\sum\nolimits^{\mathrlap{\prime}}}}_{\bm{c}\in\mathcal{R}}\lvert\Diamond_{\bm{c},N_{0},N_{1}}\rvert$,
over a deleted box $\mathcal{R}\subseteq[-Z,Z]^{m}$, is, by Cauchy–Schwarz, at
most

$O_{\epsilon}(X^{\epsilon})N^{-(m-1)/2}N_{0}^{1/2+(m-r)/6}\max[(XZ/N)^{1-(m+r)/4}Z^{r/2},(X/N)^{1-(m+r)/4}]$

times

$\sqrt{\sideset{}{{}^{\prime}}{\sum}_{\bm{c}\in\mathcal{R}}\left(\int
d\nu\,\lvert B_{\bm{c}}\rvert\right)^{2}}\ll\sqrt{\int
d\nu\sideset{}{{}^{\prime}}{\sum}_{\bm{c}\in\mathcal{R}}\lvert
B_{\bm{c}}\rvert^{2}}\ll_{\epsilon}X^{\epsilon}\min\left(Z^{m}N_{1},Z^{r}N_{1}^{\max(2,1+(m-r)/3)}\right)^{1/2},$

where the two “halves” follow respectively from Lemma 2.14 and §2.5. (To
derive the first inequality in the second line, use
$\int_{[N_{1},2N_{1})}d\nu\ll 1$ and Cauchy to get $(\int d\nu\,\lvert
B_{\bm{c}}\rvert)^{2}\ll\int d\nu\,\lvert B_{\bm{c}}\rvert^{2}$; then switch
$\sum,\int$.)

Recall $Z=Z_{\epsilon_{0}}\colonequals X^{1/2+\epsilon_{0}}$. To avoid
cumbersome notation below, $\epsilon>0$ will denote a parameter of the form
$O(\epsilon_{0})$, with absolute implied constant varying from line to line.

######  Proposition 3.4.

For $r=m$, we get the conditional bound
$\mathop{{\sum\nolimits^{\mathrlap{\prime}}}}_{\bm{c}\in\mathcal{R}}\lvert\Diamond_{\bm{c},N_{0},N_{1}}\rvert\ll_{\epsilon}X^{3/2-m/4+\epsilon}$.

###### Proof.

Here $\min(-,-)\asymp Z^{m}N_{1}$. So up to $O_{\epsilon}(X^{\epsilon})$, the
product above is at most

$Z^{m/2}N_{1}^{1/2}\cdot
N^{-(m-1)/2}N_{0}^{1/2}\max[(XZ/N)^{1-m/2}Z^{m/2},(X/N)^{1-m/2}].$

Upon simplifying $N_{1}N_{0}=N$, we get an $N$-exponent of $1/2-(m-1)/2=1-m/2$
outside the $\max$, canceling the $N$ inside. Since $N\ll
Y=X^{1-\epsilon_{0}}Z$, we may replace $N$ with $Y$ to get

$Z^{m/2}Y^{1-m/2}\max[Z^{m/2},(1/Z)^{1-m/2}]\ll
Z^{m/2}Y^{1-m/2}Z^{m/2}=Z^{m}Y^{1-m/2}$

up to $X^{\epsilon}$, which (up to $X^{\epsilon}$) simplifies to
$X^{m/2}X^{3/2-3m/4}=X^{3/2-m/4}$, as desired. ∎

######  Remark 3.5.

Interestingly, the contribution remains constant over the range $1\ll N\ll Y$.

######  Proposition 3.6.

For $r=m-1$ and $N_{1}\gg Z$, we get a conditional bound of
$O_{\epsilon}(X^{3/2-m/4+\epsilon})$.

###### Proof.

Again $\min(-,-)\asymp Z^{m}N_{1}$. So this time, up to
$O_{\epsilon}(X^{\epsilon})$, the product is at most

$Z^{m/2}N^{1-m/2}N_{0}^{(m-r)/6}\max[(XZ/N)^{1-(m+r)/4}Z^{r/2},(X/N)^{1-(m+r)/4}].$

Since $(m-r)/6\geq 0$, we may replace $N_{0}=N/N_{1}$ with its upper bound
$N/Z$; note $N\gg N_{1}\gg Z$. The total exponent of $N$ (regardless of the
“case” of the $\max$) is then

$1-m/2+(m-r)/6-1+(m+r)/4=(r-m)/12<0,$

so we may replace $N$ with its lower bound $Z$ (thus setting $N_{0}\asymp 1$
and $N\asymp N_{1}\asymp Z$) to get

$Z^{m/2}Z^{1-m/2}\max[X^{1-(m+r)/4}Z^{r/2},(X/Z)^{1-(m+r)/4}],$

which (up to $X^{\epsilon}$) simplifies to

$X^{1/2}\max[X^{1-m/4},X^{1/2-(m+r)/8}]=\max[X^{3/2-m/4},X^{1-(m+r)/8}]\ll
X^{3/2-m/4}$

since $1-(m+r)/8\leq 1-m/4+1/8\leq 3/2-m/4$, as desired. ∎

######  Proposition 3.7.

If $1\leq r\leq m-2$, then we get $O_{\epsilon}(X^{3/2-m/4+\epsilon})$
unconditionally.

###### Proof.

This time, say we use the “unconditional choice”
$Z^{r}N_{1}^{\max(2,1+(m-r)/3)}$ in $\min(-,-)$. Then up to
$O_{\epsilon}(X^{\epsilon})$, the product of interest is at most

$Z^{r/2}N_{1}^{\max(1/2,(m-r)/6)}N^{1-m/2}N_{0}^{(m-r)/6}\max[(XZ/N)^{1-(m+r)/4}Z^{r/2},(X/N)^{1-(m+r)/4}].$

Since $N_{1}N_{0}=N$, we may then replace
$N_{1}^{\max(1/2,(m-r)/6)}N_{0}^{(m-r)/6}$ with its upper bound
$N^{\max(1/2,(m-r)/6)}$. The resulting total exponent for $N$ is
$(r-m)/4+\max(1/2,(m-r)/6)$, which is $\leq 0$ since $r-m\leq-2$. So we may
replace $N$ with its lower bound $1$ to get

$Z^{r/2}\max[(XZ)^{1-(m+r)/4}Z^{r/2},X^{1-(m+r)/4}],$

which (up to $X^{\epsilon}$) simplifies to

$X^{r/4}\max[X^{3/2-3(m+r)/8}X^{r/4},X^{1-(m+r)/4}]=\max[X^{3/2+(r-3m)/8},X^{1-m/4}]\ll
X^{3/2-m/4},$

as desired. ∎

######  Proposition 3.8.

If $r=m-1$ and $N_{1}\ll Z$, then we get $O_{\epsilon}(X^{3/2-m/4+\epsilon})$
unconditionally.

###### Proof.

Again, say we use the “unconditional choice” $Z^{r}N_{1}^{\max(2,1+(m-r)/3)}$
in $\min(-,-)$. Then up to $O_{\epsilon}(X^{\epsilon})$, the product of
interest is at most

$Z^{r/2}N_{1}^{\max(1/2,(m-r)/6)}N^{1-m/2}N_{0}^{(m-r)/6}\max[(XZ/N)^{1-(m+r)/4}Z^{r/2},(X/N)^{1-(m+r)/4}].$

Fix $N_{1}$, and write $N_{0}=N/N_{1}$; then the exponent for $N$ is
$(r-m)/12<0$ (as seen in our previous $r=m-1$ analysis), so we may replace $N$
with its lower bound $N_{1}$ to get

$Z^{r/2}N_{1}^{\max(1/2,(m-r)/6)}N_{1}^{1-m/2}\max[(XZ/N_{1})^{1-(m+r)/4}Z^{r/2},(X/N_{1})^{1-(m+r)/4}].$

Now the exponent for $N_{1}$ is $\max(1/2,(m-r)/6)+(r-m)/4=1/4>0$, so we may
replace $N_{1}$ with its upper bound $Z$ to get

$Z^{r/2}Z^{1/2}Z^{1-m/2}\max[X^{1-(m+r)/4}Z^{r/2},(X/Z)^{1-(m+r)/4}],$

which (up to $X^{\epsilon}$) simplifies to

$X^{3/4+(r-m)/4}\max[X^{1-m/4},X^{1/2-(m+r)/8}]=X^{1/2}\max[X^{1-m/4},X^{1/2-(m+r)/8}],$

which was shown to be $O(X^{3/2-m/4})$ in our previous $r=m-1$ analysis. ∎

We have bounded each dyadic piece
$\mathop{{\sum\nolimits^{\mathrlap{\prime}}}}_{\bm{c}\in\mathcal{R}}\lvert\Diamond_{\bm{c},N_{0},N_{1}}\rvert$
by $X^{3/2-m/4+\epsilon}$, so

$X^{m-3}\sideset{}{{}^{\prime}}{\sum}_{\bm{c}\in\mathcal{R}}\sum_{n\geq
1}n^{-(m-1)/2}\widetilde{S}_{\bm{c}}(n)\widetilde{I}_{\bm{c}}(n)\ll_{\epsilon}X^{m-3}X^{3/2-m/4+\epsilon}=X^{3m/4-3/2+\epsilon};$

summing over the $O(1)$ deleted boxes $\mathcal{R}$ yields the conditional
portion of Theorem 1.28.

######  Remark 3.9.

In the $m=6$ Fermat reduction described in Theorem 1.28, one can give an
easier proof of the $X^{3m/4-3/2+\epsilon}$ bound. This time, we bound
$\mathop{{\sum\nolimits^{\mathrlap{\prime}}}}_{\bm{c}\in\mathcal{R}}\lvert\Diamond_{\bm{c},N_{0},N_{1}}\rvert$,
up to $X^{\epsilon}$, by

$\displaystyle\min(Z^{m/2},Z^{r/2}N_{1}^{\max(1/2,(m-r)/6)})N^{1-m/2}N_{0}^{(m-r)/6}\max[(XZ/N)^{1-m/2}Z^{r/2},(X/N)^{1-m/2}]$
$\displaystyle=\min(Z^{m/2},Z^{r/2}N_{1}^{\max(1/2,(m-r)/6)})N_{0}^{(m-r)/6}\max[(XZ)^{1-m/2}Z^{r/2},X^{1-m/2}].$

Using $\min(-,-)\leq Z^{m/2}$ and $N_{0}\ll N$ for the first $\max[-,-]$ entry
gives

$N^{(m-r)/6}X^{1-m/2}Z^{1+r/2}\ll X^{1-m/2}Z^{1+m/2}\approx X^{3/2-m/4}$

upon $N\ll Y\ll Z^{3}$. The second $\max[-,-]$ entry can only strictly
dominate if $1-m/2+r/2<0$, i.e. if $r\leq m-3$, in which case the “choice”
$\min(-,-)\leq Z^{r/2}N_{1}^{\max(1/2,(m-r)/6)}$ suffices: here $(m-r)/6\geq
1/2$, so from $N_{1}N_{0}=N$ we get $Z^{r/2}N^{(m-r)/6}X^{1-m/2}$, which is
strictly easier than the previous case.

##  4\. Contribution from the singular series

######  Remark 4.1.

In Paper II, we implement a more efficient but more technical analysis.

Following [hooley2014octonary]*pp. 279–280, §16, we have
$n^{-m}S_{\bm{0}}(n)\ll_{\epsilon}n^{-(m-3)/3+\epsilon}=n^{1-m/3+\epsilon}$ by
[hooley1988nonary]*p. 95, (170), which guarantees absolute convergence of the
singular series

$\mathfrak{S}=\sum_{n\geq 1}n^{-m}S_{\bm{0}}(n)$

for $m\geq 7$. On the other hand, recall from [heath1996new]*p. 176, Lemma 11
that

$\sigma_{\infty,w}=\lim_{\epsilon\to 0}\,(2\epsilon)^{-1}\int_{\lvert
F(\bm{x})\rvert\leq\epsilon}d\bm{x}\,w(\bm{x})\ll_{F,w}1.$

And by [heath1996new]*p. 181, Lemma 13,
$\widetilde{I}_{\bm{0}}(n)=\sigma_{\infty,w}+O_{A}((n/Y)^{A})$ for all
sufficiently small $n\ll Y$—hence for all $n\geq 1$ (since
$\widetilde{I}_{\bm{0}}(n)\ll 1$ always, by [heath1996new]*p. 183, Lemma 16).

Here $\widetilde{I}_{\bm{0}}(n)=0$ for $n\gg Y$ (by Proposition 2.1), so the
previous paragraph implies that

$\sum_{n\geq
1}n^{-m}S_{\bm{0}}(n)\cdot\widetilde{I}_{\bm{0}}(n)=\mathfrak{S}\cdot\sigma_{\infty,w}+O_{A,\epsilon}\left(Y^{\epsilon}\sup_{N\ll
Y}N\cdot N^{-(m-3)/3+\epsilon}\cdot(N/Y)^{A}\right).$

If we choose $A=(m-3)/3$, then the $N$-exponent is positive, so the $\sup$
occurs at $N\asymp Y$, giving a final error term of
$O_{\epsilon}(Y^{1-(m-3)/3+\epsilon})=O_{\epsilon}(X^{3/2-(m-3)/2+\epsilon})$.

Since $n^{-m}S_{\bm{0}}(n)=n^{-(m-1)/2}\widetilde{S}_{\bm{0}}(n)$, it follows
that

$X^{m-3}\sum_{n\geq
1}n^{-(m-1)/2}\widetilde{S}_{\bm{0}}(n)\widetilde{I}_{\bm{0}}(n)=X^{m-3}\cdot[\sigma_{\infty,w}\mathfrak{S}+O_{\epsilon}(X^{3/2-(m-3)/2+\epsilon})].$

One checks that $3/2-(m-3)/2\leq 3/2-m/4$ for $m\geq 6$, which is what we need
for $m\geq 7$ in Theorem 1.28. (Recall $(m-3)+(3/2-m/4)=3m/4-3/2$.)

For $m\in\\{4,5,6\\}$, we just need an upper bound for the $\bm{c}=\bm{0}$
contribution. Using $p^{-lm}S_{\bm{0}}(p^{l})\ll p^{l(1-m/2)}$ for $l\leq 2$
and $p^{-lm}S_{\bm{0}}(p^{l})\ll p^{l(1-m/3)}$ for $l\geq 3$, together with
$\widetilde{I}_{\bm{0}}(n)\ll 1$ for $n\ll Y$ and
$\widetilde{I}_{\bm{0}}(n)=0$ for $n\gg Y$, we get

$\sum_{n\geq
1}n^{-m}S_{\bm{0}}(n)\cdot\widetilde{I}_{\bm{0}}(n)\ll\prod_{p\leq
O(Y)}\left(1+O\left(\sum_{1\leq l\leq 2}p^{l(1-m/2)}+\sum_{3\leq l\leq
O(1)+\log_{2}{Y}}p^{l(1-m/3)}\right)\right).$

Given $p$, the sum over $l$ is $O(p^{1-m/2})+O(p^{3(1-m/3)})=O(p^{1-m/2})$,
since $3-m\leq 1-m/2$ for $m\geq 4$. The product over $p$ is thus $O(1)$ if
$m\geq 5$, and at most $\exp(O(\sum_{p\leq O(Y)}p^{-1}))\ll(\log(2+Y))^{O(1)}$
if $m=4$, which is $O_{\epsilon}(X^{\epsilon})$ in either case.
(Alternatively, we could have bounded the $n$-sum by introducing a weight of
$(Y/n)^{\epsilon}$.) Thus

$X^{m-3}\sum_{n\geq
1}n^{-(m-1)/2}\widetilde{S}_{\bm{0}}(n)\widetilde{I}_{\bm{0}}(n)\ll_{\epsilon}X^{m-3+\epsilon},$

which suffices for Theorem 1.28 since $m-3\leq 3m/4-3/2$ for $m\leq 6$.

##  5\. Contribution from singular hyperplane sections

For each $m\geq 4$, it remains to address $\bm{c}\neq\bm{0}$ with
$F^{\vee}(\bm{c})=0$. Specifically, we must estimate (if $m=4$) or bound (if
$m\geq 5$)

$X^{m-3}\sum_{n\geq 1}\sideset{}{{}^{\prime}}{\sum}_{\lVert\bm{c}\rVert\leq
Z}\bm{1}_{\bm{c}\neq\bm{0}}\cdot
n^{-(m-1)/2}\widetilde{S}_{\bm{c}}(n)\widetilde{I}_{\bm{c}}(n)\qquad\textnormal{(where
we restrict to $\bm{c}$ with $F^{\vee}(\bm{c})=0$)}$

to the precision required for Theorem 1.28. For $m\in\\{4,6\\}$, this is done
in [heath1998circle]*p. 690, Lemmas 7.2 and 8.1.

######  Remark 5.1.

Paper II implements a more precise analysis for $m=6$.

For $m\geq 5$ in general, we combine ideas from [hooley1986HasseWeil] and
[heath1998circle]. Fix a deleted box $\mathcal{R}$ with
$\lvert\mathcal{I}\rvert=r\geq 1$. Then fix a dyadic box
$\mathcal{C}\subseteq\mathcal{R}$ with $\lvert c_{i}\rvert\asymp C_{i}$ for
all $i\in\mathcal{I}$. Let $C\colonequals\max(C_{i})$. Now consider
$\bm{c}\in\mathcal{C}$ with $F^{\vee}(\bm{c})=0$ (assuming such a $\bm{c}$
exists). In the notation of [heath1998circle]*p. 686, §7, write
$F=F_{1}x_{1}^{3}+\dots+F_{m}x_{m}^{3}$, and denote the nonempty fibers of the
set map $i\mapsto F_{i}c_{i}(\mathbb{Q}^{\times})^{2}$ by
$\mathcal{I}(k)\colonequals\\{i\in\mathcal{I}:F_{i}c_{i}\equiv
g_{k}\pmod{(\mathbb{Q}^{\times})^{2}}\\}$, where the (finitely many) $g_{k}$
denote (signed, nonzero) square-free integers.

Given $k$, we may write $c_{i}=g_{k}F_{i}^{-1}e_{i}^{2}$ (with $F_{i}\mid
g_{k}e_{i}^{2}$ implicitly understood) for each $i\in\mathcal{I}(k)$, with
$e_{i}\in\mathbb{Z}$ only determined up to sign. By the explicit product form
of $F^{\vee}(\bm{c})$ for diagonal $F$, and linear independence of square
roots (of distinct square-free integers), we may choose the signs of
$e_{1},\dots,e_{m}$ (non-uniquely) so that
$\sum_{i\in\mathcal{I}(k)}F_{i}(e_{i}/F_{i})^{3}=0$ for all $k$. Since
$c_{i}\neq 0$ implies $e_{i}\neq 0$, we immediately have
$\\#\mathcal{I}(k)\neq 1$, for each $k$. Thus $r\geq 2$.

Now recall that by Proposition 2.1, we may restrict attention to $n\ll Y$. To
bound $\sum_{n\ll
Y}\mathop{{\sum\nolimits^{\mathrlap{\prime}}}}_{\bm{c}\in\mathcal{C}}n^{-(m-1)/2}\widetilde{S}_{\bm{c}}(n)\widetilde{I}_{\bm{c}}(n)$,
we recall Proposition 2.8, and Lemma 2.10 for $\bm{c}\neq\bm{0}$, which
respectively imply

$\displaystyle\widetilde{S}_{\bm{c}}(n)$
$\displaystyle\ll_{\epsilon}n^{1/2+\epsilon}\operatorname{cub}(n)^{(m-r)/6}\prod_{i\in\mathcal{I}}\gcd(\operatorname{cub}(n),\operatorname{sq}(c_{i}))^{1/4}$
$\displaystyle\widetilde{I}_{\bm{c}}(n)$
$\displaystyle\ll_{\epsilon}X^{\epsilon}\left(\frac{X\lVert\bm{c}\rVert}{n}\right)^{1-(m-r)/4}\prod_{i\in\mathcal{I}}\left(\frac{n}{X\lvert
c_{i}\rvert}\right)^{1/2}.$

We will also repeatedly use the well-known bound

$\sum_{e\leq E}\gcd(n,e)\ll\lvert\\{d\leq E:d\mid n\\}\rvert\cdot
E\ll_{\epsilon}n^{\epsilon}E.$

Now fix $n$. And given $k$, fix $\mathcal{I}(k)\subseteq\mathcal{I}$. We
modify [hooley1986HasseWeil]*pp. 82–85, §10 slightly, using the bound
$\operatorname{sq}(gF_{i}^{-1}e^{2})\mid\operatorname{sq}(ge^{2})=\gcd(g,e)e^{2}\mid
e^{3}$ for square-free $g$, to get

$\displaystyle\sideset{}{{}^{\prime}}{\sum}_{g_{k}}\prod_{i\in\mathcal{I}(k)}\sum_{\lvert
e_{i}\rvert\ll(F_{i}C_{i}/\lvert
g_{k}\rvert)^{1/2}}\gcd(n,\operatorname{sq}(g_{k}F_{i}^{-1}e_{i}^{2}))^{1/4}$
$\displaystyle\ll\sum_{g_{k}}\prod_{i\in\mathcal{I}(k)}\sum_{\lvert
e_{i}\rvert\ll(F_{i}C_{i}/\lvert g_{k}\rvert)^{1/2}}\gcd(n,e_{i})^{3/4}$
$\displaystyle\ll_{\epsilon}\sum_{g_{k}}n^{\epsilon}\prod_{i\in\mathcal{I}(k)}(F_{i}C_{i}/\lvert
g_{k}\rvert)^{1/2},$

the sum over $g_{k}$ being restricted to square-free integers with $\lvert
g_{k}\rvert\ll\min_{i\in\mathcal{I}(k)}(F_{i}C_{i})$. In any case, the
exponent on $g_{k}$ in the end is $\\#\mathcal{I}(k)/2\geq 1$, so we get a
final bound of
$\ll_{\epsilon}n^{\epsilon}\prod_{i\in\mathcal{I}(k)}C_{i}^{1/2+\epsilon}$. In
particular, up to a combinatorial factor depending on $m$, we have

$\sideset{}{{}^{\prime}}{\sum}_{\bm{c}\in\mathcal{C}}\sum_{n\asymp
N}\operatorname{cub}(n)^{(m-r)/6}\prod_{i\in\mathcal{I}}\gcd(\operatorname{cub}(n),\operatorname{sq}(c_{i}))^{1/4}\ll_{\epsilon}N^{\epsilon}\max_{D\ll
N}(D^{1/3}(N/D)D^{(m-r)/6})\prod_{i\in\mathcal{I}}C_{i}^{1/2+\epsilon},$

since there are $O(D^{1/3})$ cube-full numbers $d\asymp D$, and $O(N/d)$
numbers $n\asymp N$ with cube-full part $d$, with each $n$ contributing
$\operatorname{cub}(n)^{(m-r)/6}n^{\epsilon}\prod_{i\in\mathcal{I}(k)}C_{i}^{1/2+\epsilon}$
after summing over $\bm{c}$. Upon plugging in the endpoints of the range $1\ll
D\ll N$, the $\max$ simplifies to $\max(N,N^{1/3+(m-r)/6})$.

######  Remark 5.2.

A puzzling feature of the argument is that we did not use any Diophantine
estimates for the equation
$\sum_{i\in\mathcal{I}(k)}F_{i}(e_{i}/F_{i})^{3}=0$. For example, if
$\mathcal{I}(k)=\\{i,j\\}$ and $F_{i}=F_{j}$, then $e_{i}+e_{j}=0$ and
$c_{i}=c_{j}$, so it would seem that the sum over $g_{k},e_{i},e_{j}$ is too
relaxed. But in fact, even only using the “first order” information that
$F_{i}c_{i},F_{j}c_{j}$ agree up to “second order” factors (here, squares), we
still get a near-optimal estimate!

It follows that $\sum_{n\asymp
N}\mathop{{\sum\nolimits^{\mathrlap{\prime}}}}_{\bm{c}\in\mathcal{C}}n^{-(m-1)/2}\lvert\widetilde{S}_{\bm{c}}(n)\rvert\lvert\widetilde{I}_{\bm{c}}(n)\rvert$
is at most $O_{\epsilon}$ of

$\displaystyle
X^{\epsilon}N^{1-m/2}(XC/N)^{1-(m-r)/4}\max(N,N^{1/3+(m-r)/6})\prod_{i\in\mathcal{I}}C_{i}^{1/2}(N/XC_{i})^{1/2}$
$\displaystyle\ll_{\epsilon}X^{\epsilon}N^{(m-r)/4-m/2}(XC)^{1-(m-r)/4}(N/X)^{r/2}\max(N,N^{1/3+(m-r)/6})$
$\displaystyle=X^{1-(r+m)/4+\epsilon}\max(N^{1-(m-r)/4},N^{1/3-(m-r)/12})C^{1-(m-r)/4}.$

Since $N^{1-(m-r)/4}=(N^{1/3-(m-r)/12})^{3}$, we analyze depending on the sign
of $1-(m-r)/4$. If $1-(m-r)/4\geq 0$, i.e. $r\geq m-4$, then we get

$X^{1-(r+m)/4+\epsilon}N^{1-(m-r)/4}C^{1-(m-r)/4}\ll
X^{1-(r+m)/4+\epsilon}Y^{1-(m-r)/4}Z^{1-(m-r)/4},$

which up to $X^{\epsilon}$ becomes
$X^{1-(r+m)/4}(X^{2})^{1-(m-r)/4}=X^{3+(r-3m)/4}$. On the other hand, if
$r\leq m-4$, i.e. $1-(m-r)/4\leq 0$, then we get

$X^{1-(r+m)/4+\epsilon}N^{1/3-(m-r)/12}C^{1-(m-r)/4}\ll
X^{1-(r+m)/4+\epsilon}.$

Note that $3+(r-3m)/4\leq 3/2-m/4$ for $r\leq 2m-6$, while $1-(r+m)/4\leq
3/2-m/4$ for $r\geq-2$. At least if $m\geq 6$, then it follows that

$X^{m-3}\sum_{n\ll
Y}\sideset{}{{}^{\prime}}{\sum}_{\bm{c}\in\mathbb{Z}^{m}}\bm{1}_{\bm{c}\neq\bm{0}}\cdot
n^{-(m-1)/2}\lvert\widetilde{S}_{\bm{c}}(n)\rvert\lvert\widetilde{I}_{\bm{c}}(n)\rvert\ll_{\epsilon}X^{m-3}X^{3/2-m/4+\epsilon}=X^{3m/4-3/2+\epsilon},$

which suffices for Theorem 1.28.

For $m\leq 5$, the cases $r\leq 2m-6$ and $r\leq m-4$ are likewise fully
resolved, so we are only left with the cases $r\geq\max(2m-5,m-3)$, i.e. $r=5$
for $m=5$, and $r=3,4$ for $m=4$. Since $r=\sum_{k}\lvert\mathcal{I}(k)\rvert$
and $\lvert\mathcal{I}(k)\rvert\geq 2$ for all $k$, we are in fact left with
the cases where the $\mathcal{I}(k)$ have sizes $(2,3)$ or $(5)$ when $m=5$,
or sizes $(2,2)$ or $(3)$ or $(4)$ when $m=4$. Since Heath-Brown already
covered $m=4$, it remains to improve the method above for $m=5$.

The key new input is that for $\mathcal{I}(k)$ odd,
$\sum_{i\in\mathcal{I}(k)}F_{i}(e_{i}/F_{i})^{3}=0$ has relatively fewer
solutions. By inspection of the table at the bottom of [heath1998circle]*p.
687, one easily shows that there are $O(Z^{2+\epsilon})$ possible $\bm{c}$’s
in the $(2,3)$ case, and $O(Z^{3/2+\epsilon})$ in the $(5)$ case. Either way,
the analog of [heath1998circle]*p. 688, Lemma 7.1 holds: there are
$O(Z^{m-3+\epsilon})$ solutions to $F^{\vee}(\bm{c})=0$ with
$\bm{c}\in[-Z,Z]^{m}$. The ensuing arguments in [heath1998circle]*pp. 688–689
for $m=4,6$ are equally valid for $m=5$, the key numerics being $2+m/2\geq
1+2m/3$ and $m-3\geq 1/2$, as well as $m-3\geq(m-2)/4$. The final result
[heath1998circle]*p. 690, Lemma 7.2 is that

$\sum_{n\geq
1}\sideset{}{{}^{\prime}}{\sum}_{\bm{c}\in\mathbb{Z}^{m}}\bm{1}_{\bm{c}\neq\bm{0}}\cdot
n^{-m}S_{\bm{c}}(n)I_{\bm{c}}(n)\ll_{\epsilon}X^{3/2+3m/4+\epsilon}.$

This readily implies (upon inspection of §2.1) that

$\displaystyle X^{m-3}\sum_{n\geq
1}\sideset{}{{}^{\prime}}{\sum}_{\lVert\bm{c}\rVert\leq
Z}\bm{1}_{\bm{c}\neq\bm{0}}\cdot
n^{-(m-1)/2}\widetilde{S}_{\bm{c}}(n)\widetilde{I}_{\bm{c}}(n)$
$\displaystyle=Y^{-2}\sum_{n\geq
1}\sideset{}{{}^{\prime}}{\sum}_{\lVert\bm{c}\rVert\leq
Z}\bm{1}_{\bm{c}\neq\bm{0}}\cdot n^{-m}S_{\bm{c}}(n)I_{\bm{c}}(n)$
$\displaystyle\ll_{\epsilon}Y^{-2}X^{3/2+3m/4+\epsilon}=X^{3m/4-3/2+\epsilon},$

as desired for Theorem 1.28.

######  Remark 5.3.

If there were no cube-full parts, the counting argument would be the only
ingredient needed. Heath-Brown’s main idea is that if $F^{\vee}(\bm{c})=0$ for
some primitive vector $\bm{c}$, then for any given prime $p\nmid 3\prod
F_{i}$, at least two of the $c_{i}$’s must be coprime to $p$. The end result
is that if $4\leq m\leq 6$, then the cube-free contribution appears to be
dominant.

######  Remark 5.4.

Still, the truth for $m=5$ is probably that the $F^{\vee}(\bm{c})=0$
contribution is $O_{\epsilon}(X^{2+\epsilon})$. It would be interesting to
make progress on this somehow, perhaps using ideas from [heath1998circle]*p.
690, §8 (which interprets the $(2,2)$ contribution for $m=4$ in terms of
rational lines on the cubic surface $V/\mathbb{Q}$, up to an error of
$O_{\epsilon}(X^{3m/4-3/2+\epsilon})=O_{\epsilon}(X^{3/2+\epsilon})$).

##  Appendix A General cubic forms: a tentative axiomatic framework

Let $F(\bm{x})$ be a smooth cubic form in $m$ variables. For the smooth
contribution to the delta method (i.e. $F^{\vee}(\bm{c})\neq 0$ portion), the
main missing ingredient towards generalizing Theorem 1.28 is a reasonably
sharp analog of Lemma 2.14 (ignoring the issue that the notion of a deleted
box $\mathcal{R}$ may or may not have an easy generalization).

Nonetheless, it seems reasonable to guess some second-moment bounds that would
allow us to conditionally (in the manner of Theorem 1.28) recover some results
for general cubic forms, such as [hooley2014octonary, hooley2015octonary,
hooley2016representation]. For simplicity, suppose $w(\bm{x})$ is supported on
the non-vanishing locus of the Hessian of $F$. Then we would like an analog of
Proposition 2.16.

######  Proposition A.1.

The statement (1) below is implied by (2). Also, (2)–(3) are equivalent.

  1. (1)

The second moment of $N^{-1/2}\sum_{n\ll N}\lvert
a^{\prime}_{\bm{c}}(n)\rvert$ over $\bm{c}\in[-C,C]^{m}$ with
$F^{\vee}(\bm{c})\neq 0$ is at most $C^{m}+N^{m/3}$, up to
$O_{\epsilon}(N^{\epsilon}C^{\epsilon})$.

  2. (2)

The sum
$\mathop{{\sum\nolimits^{\mathrlap{\prime}}}}_{n,\bm{c}}n^{-1}\lvert\widetilde{S}_{\bm{c}}(n)\rvert^{2}$,
taken over $n\ll N$ and $\bm{c}\in[-C,C]^{m}$ with $F^{\vee}(\bm{c})\neq 0$
and $\operatorname{rad}(n)\mid F^{\vee}(\bm{c})$, is at most $C^{m}+N^{m/3}$,
up to $O_{\epsilon}(N^{\epsilon}C^{\epsilon})$.

  3. (3)

The sum
$\mathop{{\sum\nolimits^{\mathrlap{\prime}}}}_{n,\bm{c}}n^{-1}\lvert\widetilde{S}_{\bm{c}}(n)\rvert^{2}$,
taken over $\bm{c}\in[-C,C]^{m}$ with $F^{\vee}(\bm{c})\neq 0$ and cube-full
$n\ll N$, is at most $C^{m}+N^{m/3}$, up to
$O_{\epsilon}(N^{\epsilon}C^{\epsilon})$.

###### Proof.

By the proof of Proposition 2.16, the second statement implies the first.
Furthermore, for $\bm{c}$ fixed in the second sum, the inner sum over $n$ is
sparse, so we are really bounding

$\sideset{}{{}^{\prime}}{\sum}_{\bm{c}}\sideset{}{{}^{\prime}}{\max}_{n\ll
N}n^{-1}\lvert\widetilde{S}_{\bm{c}}(n)\rvert^{2}.$

Factor $n=n_{1}n_{2}n_{3}$ with $n_{1}$ square-free, $n_{2}$ exactly square-
full, and $n_{3}$ cube-full. Then
$n_{1}^{-1/2}\lvert\widetilde{S}_{\bm{c}}(n_{1})\rvert\ll_{\epsilon}N^{\epsilon}$
by [hooley2014octonary]*p. 254, Lemma 8 and
$n_{2}^{-1}\lvert\widetilde{S}_{\bm{c}}(n_{2})\rvert\ll_{\epsilon}N^{\epsilon}$
by [hooley2014octonary]*p. 254, Lemma 9, so the second statement is really
about

$\sideset{}{{}^{\prime}}{\sum}_{\bm{c}}\sideset{}{{}^{\prime}}{\max}_{n_{3}\ll
N}n_{3}^{-1}\lvert\widetilde{S}_{\bm{c}}(n_{3})\rvert^{2},$

where $n_{3}$ is restricted to cube-full integers with
$\operatorname{rad}(n_{3})\mid F^{\vee}(\bm{c})$. However,
$\widetilde{S}_{\bm{c}}(p^{\geq 2})=0$ for $p\nmid F^{\vee}(\bm{c})$
[hooley2014octonary]*p. 255, Lemma 10, so the condition
$\operatorname{rad}(n_{3})\mid F^{\vee}(\bm{c})$ is redundant, whence the
equivalence of the second and third statements. ∎

######  Remark A.2.

One should be able to handle “exactly cube-full” integers $n=\kappa_{3}$ by
the “Hessian rank modulo $p$” analysis introduced in [hooley2014octonary], at
least for generic forms for which [hooley2014octonary]*p. 245, (6) holds for
all $r$ (see Hooley’s remarks at [hooley2014octonary]*p. 246, end of §3). It
is exponents $l\geq 4$ where the rank modulo $p$ does not appear to be fine
enough of an invariant (with the situation only getting worse as
$l\to\infty$).

Fix an approximation $\Psi_{1}=\sum_{n\geq 1}b_{\bm{c}}(n)n^{-s}$ of $\Phi$,
in the sense of Definition 1.12. To generalize Theorem 1.28, we need a
stronger second-moment hypothesis of the form

$\sideset{}{{}^{\prime}}{\sum}_{\bm{c}\in[-C,C]^{m}}\left\lvert\sum_{n\ll
N}b_{\bm{c}}(n)\right\rvert^{2}\ll_{\epsilon}(CN)^{\epsilon}\max(C^{m},N)\cdot
N\qquad\textnormal{for all $C,N>0$}.$

(Note that here we allow $\max(C,N^{1/m})$ to be as small as $N^{1/m}$,
whereas in Definition 1.18 we only allow $\max(Z,Y^{1/m})$ to be as small as
$Y^{1/3}$.)

######  Theorem A.3.

Fix $\Psi_{1}$ as above. Assume the cube-full second-moment bound stated
earlier, as well as the “stronger second-moment hypothesis” stated above. Then
as $X\to\infty$, the “$F^{\vee}(\bm{c})\neq 0$ contribution” to $N_{F,w}(X)$
is $O_{\epsilon}(X^{3m/4-3/2+\epsilon})$, provided $m\geq 4$ and $w(\bm{x})$
is supported on the non-vanishing locus of the Hessian of $F$.

###### Proof sketch.

By the Cauchy strategy of §3, it suffices to show that the quantity

$\displaystyle
N^{-(m-1)/2}(XC/N)^{1-m/2}[N_{0}(C^{m}+N_{0}^{m/3})]^{1/2}\cdot\max(C^{m/2},N_{1}^{1/2})N_{1}^{1/2}$
$\displaystyle\asymp(XC)^{1-m/2}\max(C^{m/2},N_{0}^{m/6})\max(C^{m/2},N_{1}^{1/2})$

is $O_{\epsilon}(X^{3/2-m/4+\epsilon})$. There are $2\times 2=4$ possible
outcomes for the pair of $\max(-,-)$’s. If neither is $C^{m/2}$, then we have

$(XC)^{1-m/2}N_{0}^{m/6}N_{1}^{1/2}\ll X^{1-m/2}\max(N^{m/6},N^{1/2})\ll
X^{1-m/2}Y^{m/6}=X^{1-m/4},$

which is satisfactory. If at least one $C^{m/2}$ appears, then the
$C$-exponent is positive, so we may replace $C$ with its upper bound $Z$ to
get

$(XZ)^{1-m/2}\max(Z^{m/2},N_{0}^{m/6})\max(Z^{m/2},N_{1}^{1/2})=X^{1-m/2}Z^{1+m/2},$

noting $Z\gg Y^{1/3}\gg N_{0}^{1/3}$ and $Z^{m}\gg Z^{3}\gg Y\gg N_{1}$. Up to
$X^{\epsilon}$, we end up with $X^{3/2-m/4}$, as desired. ∎

######  Remark A.4.

One can likely cover the range $C\ll X^{(m-1)/(2m+4)}$ using the unconditional
Deligne bound $C^{m}N_{1}^{2}$ instead of the large-sieve bound
$\max(C^{m},N_{1})N_{1}$. So the range required of the large sieve can be
weakened a bit. Furthermore, it appears that we can always handle the case
$C\ll N_{0}^{1/3}$ with Deligne (but still assuming the cube-full second-
moment bound).

######  Remark A.5.

For the $m=8$ result [hooley2014octonary, hooley2015octonary] specifically, in
which the $F^{\vee}(\bm{c})\neq 0$ sum is the only conditional ingredient, it
appears that the second-moment assumption for $N^{-1/2}\sum_{n\ll N}\lvert
a^{\prime}_{\bm{c}}(n)\rvert$ can be relaxed by a factor of $N^{2/3}$, to
$N^{2/3}\cdot(C^{m}+N^{m/3})$. Since there are $O(N^{1/3})$ cube-full numbers
$n\ll N$, it would then suffice to show that for all cube-full $n$, the sum
$\mathop{{\sum\nolimits^{\mathrlap{\prime}}}}_{\bm{c}}n^{-1}\lvert\widetilde{S}_{\bm{c}}(n)\rvert^{2}$,
taken over $\bm{c}\in[-C,C]^{m}$ with $F^{\vee}(\bm{c})\neq 0$, is roughly at
most $n^{1/3}\cdot(C^{m}+n^{m/3})$. For the analogous first moment, Heath-
Brown proved the bound $\sum_{\lVert\bm{c}\rVert\leq
C}n^{-1/2}\lvert\widetilde{S}_{\bm{c}}(n)\rvert\ll_{\epsilon}n^{\epsilon}(C^{m}+n^{m/3})$
[heath1983cubic]*p. 243, Lemma 14 for all square-full $n$.

######  Remark A.6.

The missing ingredients may or may not be easier for the special octonary
forms used in [hooley2016representation].

##  Appendix B Other open problems of a similar nature

###  B.1. Problems with a cubic flavor

####  B.1.1. The cubic convexity barrier

Beating the Hua bound (or “breaking convexity” as [wooley1995breaking] might
say) lies in the following family of (probably equivalent) problems:

  1. (1)

To show that the (smoothly weighted) count of solutions
$\bm{x}\in\mathbb{Z}^{6}$ to $x_{1}^{3}+\dots+x_{6}^{3}=0$, in the region
$\lVert\bm{x}\rVert\ll X$, is $O(X^{7/2-\delta_{3}})$ for some $\delta_{3}>0$.

  2. (2)

To show that the analogous smooth count for $x_{1}^{3}+\dots+x_{8}^{3}=0$
satisfies a Hardy–Littlewood asymptotic $cX^{5}+O(X^{5-\delta_{4}})$ with a
power saving $\delta_{4}>0$.

  3. (3)

In general, for $s\geq 4$, to show that the analogous count for
$x_{1}^{3}+\dots+x_{2s}^{3}=0$ satisfies a Hardy–Littlewood asymptotic of the
form $cX^{2s-3}+O(X^{3s/2-1-\delta_{s}})$ for some $\delta_{s}>0$.

These “even Fermat bounds” would all follow from an improvement of the current
generic cubic Weyl sum bound of $O_{\epsilon}(X^{3/4+\epsilon})$, which
appears quite difficult to beat; see e.g. [heath2009bounds] assuming $abc$. In
any case, the beauty of [hooley1986HasseWeil,
hooley_greaves_harman_huxley_1997, heath1998circle] lies in the averaging over
arcs (and the further averaging over moduli going beyond Kloosterman), which
is morally independent of the issue of pointwise Weyl bounds.

The equivalent family above would also imply similar bounds for odd numbers of
variables, e.g. an upper bound for $2s=5$ (a problem raised by
[bombieri2009problems]). At first glance, there does not appear to be an
_equivalent_ Diophantine problem involving $4$ variables, or any odd number
$2s$, although the general “restricted arcs” form of the $4$ variable problem
($s=2$) could in principle pare away most, if not all, but the most extreme
minor arcs for $s\geq 3$.

####  B.1.2. Other problems with a cubic flavor

Besides the “familiar” or “classical” Fermat cubic problems above, other
natural problems in the same vein include the following:

  1. (4)

To show that each smooth projective cubic surface $V(F)$ over $\mathbb{Q}$ has
$O_{F,\epsilon}(X^{3/2+\epsilon})$ solutions $\bm{x}\ll X$ away from its
rational lines—or at least $O_{F}(X^{12/7-\delta})$, to beat
[salberger_2015]’s bound.333Note however that one of the nice features of
[salberger_2015] is uniformity over $F$, which we do not consider here.

  2. (5)

To show that $x_{1}^{3}+\dots+x_{4}^{3}=0$ has $O_{\epsilon}(X^{2+\epsilon})$
solutions, in a way that generalizes to show that uniformly over $M\leq
X^{3/2}$, we have

$\int_{\theta\in\mathfrak{M}(M)}d\theta\,\lvert
T(\theta)\rvert^{4}\ll_{\epsilon}X^{\epsilon}\cdot(X+M^{2}/X),$

where $T(\theta)\colonequals\sum_{\lvert x\rvert\leq X}e(\theta x^{3})$ and
$\mathfrak{M}(M)\colonequals\bigcup_{q\leq
M}\\{\theta\in\mathbb{R}/\mathbb{Z}:\lvert q\theta-a\rvert\leq M/X^{3}\\}$.

(Unconditionally, [brudern1991ternary] obtained
$O_{\epsilon}(X^{\epsilon})\cdot(X+M^{2}/X+M^{7/2}/X^{3})$.)

  3. (6)

For $k\in\\{3,4\\}$, to show that the equation
$x_{1}^{2}+x_{2}^{3}+x_{3}^{k}=y_{1}^{2}+y_{2}^{3}+y_{3}^{k}$, with each
_monomial_ restricted to be of size at most $N$, satisfies a (smoothly
weighted) Hardy–Littlewood asymptotic of the form
$cN^{1+2\delta_{k}}+O_{\epsilon}(N^{1+\delta_{k}+\epsilon})$, where
$\delta_{k}\colonequals(1/2+1/3+1/k)-1$.

  4. (7)

For $k\in\\{3,4\\}$, to show that $x_{1}^{2}+x_{2}^{3}+x_{3}^{k}$ positively
represents all $n\leq N$ but an exceptional set of size
$O_{\epsilon}(N^{1-\delta_{k}+\epsilon})$—which [brudern1991ternary] proved
unconditionally for $k=5$, and obtained weaker bounds towards for $k=3,4$.

######  Remark B.1.

Progress on (4) would already be interesting for “somewhat general” (e.g.
diagonal) $F$’s. But for cubic surfaces $V(F)$ with $\mathbb{Q}$-lines,
unconditional bounds beyond [salberger_2015] are already known—e.g.
$O_{\epsilon}(X^{4/3+\epsilon})$ for the Fermat cubic.

(Note that general diagonal $V(F)$’s have no $\mathbb{Q}$-lines.)

######  Remark B.2.

The bound in (5), if true, may or may not be optimal (up to $\epsilon$). It is
optimal for $M=X^{3/2}$, at least.

######  Remark B.3.

The asymptotic error term $O_{\epsilon}(N^{1+\delta_{k}+\epsilon})$ in (6), if
valid, would likely be optimal up to $\epsilon$, due to “trivial loci” such as
$\bm{x}=\bm{y}$. It could be interesting to try to obtain a “secondary term”
asymptotic that precisely captures the influence of such loci; cf.
[vaughan2015squares]*Theorem 1.4 regarding the mixed ternary forms
$x_{1}^{2}+x_{2}^{2}+x_{3}^{k}$ for $k\geq 3$.

######  Remark B.4.

By mimicking [brudern1991ternary], one can show that the bound in (5) implies
the “exceptional set bound” in (7); see [restricted_cubic_moments]*Theorem
1.11 for details. It is also reasonable to expect “(5) to imply (6)” and “(6)
to imply (7)”—at least morally—but such expectations would need to be
carefully checked.

####  B.1.3. Conditional approaches

The bounds in (1)–(3), and the bound in (4) for diagonal $F$, can be
established conditionally under Hypothesis HW for smooth projective cubic
hypersurfaces of dimension $3,5,2s-3$, and $1$, respectively, in certain
families. See Theorem 1.28 for a unified treatment, which also clarifies how
one may “relax” Hypothesis HW.

Similarly, under Hypothesis HW for smooth hyperplane sections of
$V(x_{1}^{3}+\dots+x_{4}^{3})$, one can mimic Theorem 1.28 to prove the bound
in (5), and consequently the bound in (7). Again, some “relaxation” of
Hypothesis HW is possible; see [restricted_cubic_moments]*Theorem 1.6 for
details.444The situation for (6) may well be similar, but would need to be
carefully analyzed and written up. So at least in (5) and (7), a tiny
improvement over [brudern1991ternary]’s bounds should already be possible
using existing general automorphic large sieve inequalities; see
[restricted_cubic_moments]*Remark 1.9 for more details. But truly satisfying
progress on any of (1)–(7) will probably (and hopefully) require significant
new ideas.

######  Remark B.5.

In the absence of progress on GRH itself, the “GRH-on-average philosophy”
provides one of the most alluring approaches towards (1)–(7).

However, other ideas could also plausibly lead to interesting progress on
(1)–(7). For example, (4) might be susceptible to clever slicing methods (e.g.
those developed by Salberger or Heath-Brown), especially when combined with
advances in the theory of irrational varieties (including upper-bound sieves
and elliptic-curve statistics).

From another direction, Bourgain and Demeter (among others) have applied
decoupling theory on curves (for instance) to prove interesting point-counting
bounds, often even in the absence of translation invariance. Although the most
striking applications of decoupling to _classical_ Diophantine problems so far
seem to have been restricted to degree $\geq 5$ or codimension $\geq 2$, the
full power of decoupling remains to be understood, and one cannot yet rule out
the possibility of applying decoupling (or one of its non-archimedean allies,
as developed by Wooley and others) towards (1)–(7).

###  B.2. Counting on other varieties

For simplicity, we discuss only individual smooth projective complete
intersections $W\colonequals W(F_{1},\dots,F_{R})/\mathbb{Q}$ in
$\mathbb{P}^{m-1}_{\mathbb{Q}}$ with $m\geq 5$ and $R\colonequals 2$ and
$\deg{F_{1}}=\dots=\deg{F_{R}}=d\colonequals 2$, though there are certainly
many other interesting (affine or projective) varieties one could consider
(individually or on average).

Given $\bm{F}\colonequals(F_{1},\dots,F_{R})$, fix $w\in
C^{\infty}_{c}(\mathbb{R}^{m})$ supported away from $\bm{0}$. For $X\geq 1$,
let
$N_{\bm{F},w}(X)\colonequals\sum_{\bm{x}\in\mathbb{Z}^{m}}w(\bm{x}/X)\cdot\bm{1}_{\bm{F}(\bm{x})=\bm{0}}$,
and if $m=5$, let $N^{\prime}_{\bm{F},w}(X)$ denote the corresponding weighted
count restricted $\bm{x}\in\mathbb{Z}^{m}$ not lying on any $\mathbb{Q}$-line
of $W$.

For references on what is known about the Hasse principle for $W$, we refer
the reader to the introduction of [vishe2019rational]. When $m=5$, the “Hasse
principle up to Brauer–Manin” is unknown but conjectured to hold, and from a
quantitative point of view the best we know in general is Salberger’s
(currently unpublished) bound
$N^{\prime}_{\bm{F},w}(X)\ll_{\bm{F},\epsilon}X^{3/2+\epsilon}$.

Let $D\in\mathbb{Z}[\bm{c}]$ denote a suitable discriminant polynomial, so
that a given hyperplane section $W_{\bm{c}}/\mathbb{Q}$ of $W$ is singular if
and only if $D(\bm{c})=0$. Let

$S_{\bm{F},\bm{c}}(q)\colonequals\sideset{}{{}^{\prime}}{\sum}_{\bm{a}\in(\mathbb{Z}/q)^{r}}\sum_{\bm{x}\in(\mathbb{Z}/q)^{m}}e_{q}(\bm{a}\cdot\bm{F}(\bm{x})+\bm{c}\cdot\bm{x})$

(cf. the complete exponential sums defined in [heath2017simultaneous]), where
we restrict $\bm{a}$ to be _primitive_ modulo $q$. Then let
$\widetilde{S}_{\bm{F},\bm{c}}(q)\colonequals
q^{-(m+R)/2}S_{\bm{F},\bm{c}}(q)$, let
$\Phi(\bm{F},\bm{c},s)\colonequals\sum_{q\geq
1}\widetilde{S}_{\bm{F},\bm{c}}(q)q^{-s}$, and let
$\Psi_{1}(\bm{F},\bm{c},s)=\sum_{q\geq 1}b_{\bm{F},\bm{c}}(q)q^{-s}$ denote a
first-order approximation of $\Phi(\bm{F},\bm{c},s)$ in the sense suggested by
Definition 1.12.

Optimistic _preliminary calculations_ (subject to errors or unforeseen
difficulties) suggest the following conjecture:

######  Conjecture B.6.

Fix $\bm{F},w,\Psi_{1}$. Let $Y\colonequals X^{d/(1+1/R)}=X^{4/3}$. _Assume
that_ for all $X\geq 1$ and $Z\geq Y/X=X^{1/3}$, and for all $N\ll Y$ and
intervals $I\subseteq[N/2,2N]$, we have a uniform bound of the form

$\sideset{}{{}^{\prime}}{\sum}_{\bm{c}\in[-Z,Z]^{m}}\left\lvert\sum_{q\in
I}b_{\bm{F},\bm{c}}(q)\right\rvert^{2}\ll_{\epsilon}Z^{\epsilon}\max(Z^{m},Y)\cdot
N,$

where we restrict to $\bm{c}$ with $D(\bm{c})\neq 0$. _Also assume_ the
existence of a “fully satisfactory” two-dimensional Kloosterman method. Now
let $\theta\colonequals\frac{dR(m-1-R)}{2(R+1)}=\frac{2(m-3)}{3}$. Then

  1. (1)

$N^{\prime}_{\bm{F},w}(X)\ll_{\epsilon}X^{\theta+\epsilon}=X^{4/3+\epsilon}$
if $m=5$;

  2. (2)

$N_{\bm{F},w}(X)\ll_{\epsilon}X^{\theta+\epsilon}=X^{2+\epsilon}$ if $m=6$;
and

  3. (3)

$N_{\bm{F},w}(X)=cX^{m-dR}+O_{\epsilon}(X^{\theta+\epsilon})=cX^{m-4}+O_{\epsilon}(X^{2(m-3)/3+\epsilon})$
if $m\geq 7$ (with $c=c_{\bm{F},\bm{w}}$ being the usual Hardy–Littlewood
prediction if $m\geq 8$, and a more complicated constant that also
incorporates $\mathbb{Q}$-planes on $W$ if $m=7$).

######  Remark B.7.

The “elementary GRH on average” assumption above, and that in Theorem 1.28,
should belong to a wider framework of “average hypotheses” over “natural but
thin” geometric families. Hopefully at least one such family is sufficiently
rich yet tractable to inspire significant new ideas.

######  Remark B.8.

Over $D(\bm{c})\neq 0$, the Hasse–Weil $L$-functions $L(s,W_{\bm{c}})$ are
quite rich. Roughly speaking, each $W_{\bm{c}}$ (after passing to certain
standard $\ell$-adic Galois representations) corresponds (up to Tate twist) to
an abelian variety of dimension $(m-3)/2$ if $2\nmid m$, and to an Artin
representation of dimension $m$ if $2\mid m$. (See e.g.
[bernardara2016chow]*Theorem 2.1 or [reid1972complete].)

######  Remark B.9.

The $\bm{F}$’s considered in [heath2017simultaneous] define _singular_ $W$’s
(with singular loci of positive dimension, in fact), so are not included in
the conjecture above. But it would be interesting to see how much further one
can push [heath2017simultaneous] under automorphy and GRH for varieties
“related to” hyperplane sections of $W$. (The hyperplane sections $W_{\bm{c}}$
for [heath2017simultaneous] are all singular, but some desingularization might
be possible.)

######  Remark B.10.

The “two-dimensional Kloosterman method” in [heath2017simultaneous] is based
on “positivity” (cf. the works [hooley1986HasseWeil,
hooley_greaves_harman_huxley_1997] avoiding [duke1993bounds, heath1996new]),
and hence is not “fully satisfactory” (in that it does not readily generalize
to arbitrary $\bm{F}$’s). Finding a suitable “fully satisfactory”
generalization, even in the rational function field case (where the one-
dimensional Kloosterman method is “trivial to set up”), seems to be an
interesting but challenging problem; see [vishe2019rational] for some
fascinating progress555with fairly good cancellation, but not as much as one
might optimistically hope for in this direction.

######  Remark B.11.

At least for complete intersections, it might be interesting to try “lifting”
the Kloosterman method to the setting of universal torsors, but this could be
difficult (and maybe only directly relevant to $W$’s of small dimension, e.g.
projective surfaces).

##  Appendix C Natural modularity questions

In this section, we fix a smooth cubic form $F/\mathbb{Q}$ in $m\in\\{4,6\\}$
variables, and consider the Hasse–Weil $L$-functions $L(s,V_{\bm{c}})$
appearing in Example 1.13(4). Fix $\bm{c}$. (Here we restrict attention to
$\bm{c}\in\mathbb{Q}^{m}$ with $V_{\bm{c}}$ smooth of dimension $m_{\ast}$.)

###  C.1. A general discussion

If $m=4$, then $V_{\bm{c}}$ is a genus one curve over $\mathbb{Q}$, so
$L(s,V_{\bm{c}})$ is known to be the $L$-function of a weight $2$ modular cusp
form (since $L(s,V_{\bm{c}})=L(s,J(V_{\bm{c}}))$, and elliptic curves over
$\mathbb{Q}$ are known to be modular).

Now suppose $m=6$. Let $A_{\bm{c}}/\mathbb{Q}$ denote the Albanese variety of
$F(V_{\bm{c}})/\mathbb{Q}$, the Fano surface of lines on
$V_{\bm{c}}/\mathbb{Q}$. Now fix a prime $\ell$. Then it is known that the
_analytically normalized_ $L$-functions $L(s,V_{\bm{c}}),L(s,A_{\bm{c}})$
agree at all but finitely many places. So if we consider the $\ell$-adic
$G_{\mathbb{Q}}$-representations $M_{\bm{c}}\colonequals
H^{3}(V_{\bm{c}}\times\overline{\mathbb{Q}},\mathbb{Q}_{\ell})$ and
$V_{\ell}(A_{\bm{c}})^{\vee}$ defining $L(s,V_{\bm{c}}),L(s,A_{\bm{c}})$,
respectively, then $M_{\bm{c}}(1),V_{\ell}(A_{\bm{c}})^{\vee}$ agree up to
semi-simplification. (In fact $M_{\bm{c}}(1),V_{\ell}(A_{\bm{c}})^{\vee}$ are
probably known to be isomorphic. We do not know of a convenient reference for
this fact, but cf. [reid1972complete]*§4.3, Corollaries 4.29.)

Here $\dim A_{\bm{c}}=5$. But at least in general, abelian varieties
$A/\mathbb{Q}$ of dimension $5$ are not yet known to be automorphic. (In this
connection, the current state of the art is [boxer2018abelian], which
“potentially addresses” abelian _surfaces_ over certain fields like
$\mathbb{Q}$.)

However, one might ask if (at least) for certain $F$’s, like
$x_{1}^{3}+\dots+x_{6}^{3}$, the $A_{\bm{c}}$’s might have some special
structure that could “reduce” the complexity of $L(s,A_{\bm{c}})$. In this
vein, recall that the Hasse–Weil $L$-function of a diagonal projective
hypersurface over $\mathbb{Q}$ of degree $d\geq 2$ always factors into
$L$-functions of degree $\phi(d)$ associated to certain Hecke characters on
$\mathbb{Q}(\zeta_{d})$; at least up to bad factors, this observation is due
to Weil.

Now let $F\colonequals x_{1}^{3}+\dots+x_{6}^{3}$. Does some loose analog of
Weil’s observation hold for the “codimension-$2$ diagonal systems”
$V_{\bm{c}}$? In the most direct sense, the answer to this basic question is
probably negative in general (see Proposition C.1 below). So even when
$F=x_{1}^{3}+\dots+x_{6}^{3}$, the conjectured automorphy of $V_{\bm{c}}$ (or
$A_{\bm{c}}$) does not seem to easily follow from existing progress on the
Langlands program; one may well need interesting new ideas in order to probe
the family $\bm{c}\mapsto L(s,V_{\bm{c}})$ from an automorphic point of view.

###  C.2. Notes on an explicit diagonal system, and a conjecture

Armed with the theory of abelian varieties, we can prove the following
statement:

######  Proposition C.1.

Let $F\colonequals x_{1}^{3}+\dots+x_{6}^{3}$ and
$\bm{c}\colonequals(1,2,3,4,5,6)$. For each prime $\ell$, assume Tate’s global
semi-simplicity conjecture over $\mathbb{Q}$.666This assumption is convenient,
but it may well be removable. Then for each prime $\ell$ and number field
$K/\mathbb{Q}$, the $\ell$-adic $G_{\mathbb{Q}}$-representations
$M_{\bm{c}},V_{\ell}(A_{\bm{c}})$ are irreducible as $\ell$-adic
$G_{K}$-representations.

######  Remark C.2.

Proposition C.1 does not rule out the possibility that $L(s,V_{\bm{c}})$ could
be “deconstructed” in a subtler way, e.g. as the $9$th symmetric power
$L$-function of a $\operatorname{GL}_{2}$ representation.

######  Remark C.3.

It may or may not be natural to contrast Proposition C.1 with
[patterson1997asymptotic]*Theorem 3.1, a result showing that the “one-
dimensional constituents of $S_{\bm{c}}(q)$” (before taking mixed $6$th
moments over $a\bmod{q}$) are “nice” over $\mathbb{Q}(\zeta_{3})$ (and in
fact, amenable to statistical analysis via cubic metaplectic forms
[louvel2011distribution]).

The conclusion of Proposition C.1 is equivalent to the statement that
$\operatorname{End}(A_{\bm{c}}\times\overline{\mathbb{Q}})=\mathbb{Z}$. Thus
Proposition C.1 naturally suggests the following conjecture:

######  Conjecture C.4.

For _almost all_ $\bm{c}\in\mathbb{Z}^{m}$ with $F^{\vee}(\bm{c})\neq 0$, we
have $\operatorname{End}(A_{\bm{c}}\times\overline{\mathbb{Q}})=\mathbb{Z}$.

######  Remark C.5.

By [masser1996specialization], the conjecture above should be equivalent to a
question about the generic fiber of the family $A_{\bm{c}}$. Furthermore, a
“typical” special fiber should control the possible behavior at the generic
fiber. But Proposition C.1 does not seem to imply the conjecture “for free”
(without further calculation).

###### Proof of Proposition C.1.

For convenience, let $A\colonequals A_{\bm{c}}$. Now fix $\ell$. Then the
$\ell$-adic $G_{\mathbb{Q}}$-representations
$M_{\bm{c}}(1),V_{\ell}(A)^{\vee}$ are semi-simple by assumption, and thus
isomorphic. So by Faltings’ isogeny theorem (and the fact that endomorphism
rings of abelian varieties are torsion-free), our desired irreducibility
statement is equivalent to the statement that
$\operatorname{End}(A_{K})=\mathbb{Z}$ holds for all $K/\mathbb{Q}$. (In fact,
one can prove this equivalence unconditionally, using the theory of
composition series.)

We now prove the latter statement—which is independent of $\ell$. To begin, we
re-define $\ell\colonequals 2$ for convenience. By Appendix C.3.1,
$V_{\bm{c}}$ has good reduction at all primes $p\in[5,100]$, so by smooth
proper base change, $M_{\bm{c}}$ is unramified at all $p\in[5,100]$. Since
$M_{\bm{c}}(1)\cong V_{\ell}(A)^{\vee}$ as $\ell$-adic
$G_{\mathbb{Q}}$-representations, it follows that $V_{\ell}(A)$ is unramified
at all $p\in[5,100]$. So by the Néron–Ogg–Shafarevich criterion, $A$ has good
reduction at all $p\in[5,100]$.

Let
$\operatorname{End}^{0}(-)\colonequals\operatorname{End}(-)\otimes_{\mathbb{Z}}\mathbb{Q}$;
then $\operatorname{End}(-)$ is always a subring of
$\operatorname{End}^{0}(-)$. In particular, to achieve our original goal, it
suffices to prove that $\operatorname{End}^{0}(A_{K})=\mathbb{Q}$ holds for
all $K/\mathbb{Q}$.

Now fix $K/\mathbb{Q}$, and fix a Néron model
$\mathcal{A}_{K}/\mathcal{O}_{K}$ of $A_{K}$. Then for each prime $\wp$ of
$\mathcal{O}_{K}$, let $k_{\wp}\colonequals\mathcal{O}_{K}/\wp$ and
$f_{\wp}\colonequals[k_{\wp}:\mathbb{F}_{p}]$. Finally, for each _$A_{K}$
-good_ prime $\wp$, let $A_{\wp}\colonequals\mathcal{A}_{K}\times k_{\wp}$
denote the reduction of $A_{K}$ modulo $\wp$, let
$\pi_{\wp}\in\operatorname{End}(A_{\wp})$ denote the $k_{\wp}$-relative
Frobenius of $A_{\wp}$, and _if_ $\wp\nmid\ell$, let $P_{\wp}\in
1+t\mathbb{Z}[t]$ denote the “ _reverse_ characteristic polynomial” of
$\pi_{\wp}$. Then given $\wp\mid p\in[5,100]$, we have
$P_{\wp}(t)=\prod(1-\beta_{i}^{f_{\wp}}t)$, where
$P_{p\mathbb{Z}}(t)=\prod(1-\beta_{i}t)$, where for $p\in\\{5,7,11,13\\}$ we
know by Appendix C.3.1 that

  1. (1)

$P_{5\mathbb{Z}}(t)=(5t^{2}+1)\cdot(625t^{8}+100t^{6}+4t^{2}+1)$ at $p=5$,

  2. (2)

$P_{7\mathbb{Z}}(t)=(7t^{2}+t+1)\cdot(7t^{2}+4t+1)^{4}$ at $p=7$,

  3. (3)

$P_{11\mathbb{Z}}(t)=161051t^{10}+14641t^{9}+3993t^{8}+2420t^{7}+539t^{6}+14t^{5}+49t^{4}+20t^{3}+3t^{2}+t+1$
at $p=11$, and

  4. (4)

$P_{13\mathbb{Z}}(t)=(13t^{2}+1)\cdot(13t^{2}+7t+1)\cdot(13t^{2}+4t+1)^{3}$ at
$p=13$.

In order to put the data above to use, note that for each $A_{K}$-good prime
$\wp$, the theory of Néron models furnishes a canonical _reduction map_
$\operatorname{End}(A_{K})\to\operatorname{End}(A_{\wp})$, which is known to
be _injective_. On the other hand, for each $A_{K}$-good $\wp$, the results
[tate1966endomorphisms]*Theorems 1(a) and 2(a) (combined with the fact, noted
in [tate1966endomorphisms]*p. 141, par. 2, that over $k_{\wp}$, “isogeny-class
factorizations” correspond to “simple $E$-algebra factorizations”) imply that

$\operatorname{End}^{0}(A_{\wp})\cong E_{\wp,1}\times\cdots,$

where $E_{\wp,1},\dots$ are certain _central simple algebras_ (CSA’s) over the
_number field factors_ $F_{\wp,1},\dots$ of the commutative subalgebra
$\mathbb{Q}[\pi_{\wp}]\subseteq\operatorname{End}^{0}(A_{\wp})$, and where

$F_{\wp,j}\cong\mathbb{Q}[T]/(T^{10}P_{\wp,j}(1/T))\cong\mathbb{Q}[t]/(P_{\wp,j}(t))\quad\textnormal{and}\quad[E_{\wp,j}:F_{\wp,j}]=m_{\wp,j}^{2}$

correspond to an arbitrary factorization
$P_{\wp}(t)=P_{\wp,1}(t)^{m_{\wp,1}}\cdots$ of $P_{\wp}(t)$ into irreducibles
over $\mathbb{Q}$ of the form $1+t\mathbb{Z}[t]$.

Finally, fix $\wp\mid 11$ and $\wp^{\prime}\mid 5$. Then by Lemma C.7(1),
$P_{\wp}$ is irreducible over $\mathbb{Q}$. So
$\operatorname{End}^{0}(A_{\wp})=\mathbb{Q}[\pi_{\wp}]\cong F_{\wp,1}\cong
L_{11}$. The algebra $\operatorname{End}^{0}(A_{K})$ is thus isomorphic to a
number field $F\subseteq L_{11}$. Now on the one hand, $F\subseteq L_{11}$
automatically implies $[F:\mathbb{Q}]\in\\{1,5,10\\}$, by Lemma C.7(1). On the
other hand, by Lemma C.7(2)–(3) (applying (2) if $2\nmid f_{\wp^{\prime}}$,
and (3) if $2\mid f_{\wp^{\prime}}$), we must have
$\operatorname{End}^{0}(A_{\wp^{\prime}})\cong E_{\wp^{\prime},1}\times
E_{\wp^{\prime},2}$, where each $E_{\wp^{\prime},j}$ is a CSA of dimension
$m_{\wp^{\prime},j}^{2}\in\\{1,4\\}$ over a number field $F_{\wp^{\prime},j}$
of degree $[F_{\wp^{\prime},j}:\mathbb{Q}]\in\\{1,2,4,8\\}$. Now fix
$j\in[2]$, and let $F^{\prime}_{j}$ denote the image of the composition
$F\cong\operatorname{End}^{0}(A_{K})\to\operatorname{End}^{0}(A_{\wp^{\prime}})\operatorname{\twoheadrightarrow}E_{\wp^{\prime},j}$.
Then certainly
$[F_{\wp^{\prime},j}F^{\prime}_{j}:F_{\wp^{\prime},j}]\leq[E_{\wp^{\prime},j}:F_{\wp^{\prime},j}]=m_{\wp^{\prime},j}^{2}\leq
4$; but $F_{\wp^{\prime},j}$ is _central_ in $E_{\wp^{\prime},j}$, so
$F_{\wp^{\prime},j}F^{\prime}_{j}$ is a _field_ , and hence
$[F^{\prime}_{j}:\mathbb{Q}]\mid[F_{\wp^{\prime},j}F^{\prime}_{j}:\mathbb{Q}]\perp
5$. But the surjection $F\operatorname{\twoheadrightarrow}F^{\prime}_{j}$ is
_nonzero_ , because $E_{\wp^{\prime},j}\neq 0$ (and all of our ring maps,
including $F\to E_{\wp^{\prime},j}$, are _unital_). Also, $F$ is a _field_.
Therefore, $F^{\prime}_{j}\cong F$. It follows that
$[F^{\prime}_{j}:\mathbb{Q}]=[F:\mathbb{Q}]$, whence
$[F:\mathbb{Q}]\in\\{1,5,10\\}\cap\mathbb{Z}_{5}^{\times}=1$. So
$[\operatorname{End}^{0}(A_{K}):\mathbb{Q}]=1$, as desired. ∎

######  Remark C.6.

One could slightly shorten the proof, by replacing $K$ with a suitable
quadratic extension $K^{\prime}$ (if necessary) to reduce to the case where
there exists $\wp\mid 5$ with $2\mid f_{\wp}$.

######  Lemma C.7.

Let $Q_{5}(t)\colonequals P_{5\mathbb{Z}}(t)$ and $Q_{11}(t)\colonequals
P_{11\mathbb{Z}}(t)$ be the specific polynomials at $p\in\\{5,11\\}$ written
above. Given $p\in\\{5,11\\}$ and $f\geq 1$, write
$Q_{p}(t)=\prod(1-\beta_{i}t)$, let

$Q_{p^{f}}(t)\colonequals\prod(1-\beta_{i}^{f}t),$

let $Q_{p^{f},1}(t)^{m(p^{f},1)}\cdots$ denote a factorization of
$Q_{p^{f}}(t)$ into irreducibles over $\mathbb{Q}$ of the form
$1+t\mathbb{Z}[t]$, and let $\sqrt{Q_{p^{f}}}\colonequals Q_{p^{f},1}\cdots$
denote a “radical” of $Q_{p^{f}}$.

  1. (1)

The polynomial $Q_{11}(t)$ is irreducible, and the field
$L_{11}\colonequals\mathbb{Q}[t]/Q_{11}$ contains exactly $1$ nontrivial
subfield $L^{\prime}_{11}$. Here $[L_{11}:\mathbb{Q}]=10$ and
$[L^{\prime}_{11}:\mathbb{Q}]=5$. Furthermore, for each $f\geq 1$, the
polynomial $Q_{11^{f}}(t)$ is irreducible, and $\mathbb{Q}[t]/Q_{11^{f}}\cong
L_{11}$.

  2. (2)

We can write $Q_{5}=Q_{5,1}Q_{5,2}$ with $\deg Q_{5,1}=2$ and $\deg
Q_{5,2}=8$. Under these conventions, the fields
$F_{5,j}\colonequals\mathbb{Q}[t]/Q_{5,j}$ have degree $\deg Q_{5,j}$.
Furthermore, for each _odd_ $f\geq 1$, we have $Q_{5^{f}}=\sqrt{Q_{5^{f}}}$
and $\mathbb{Q}[t]/Q_{5^{f}}\cong F_{5,1}\times F_{5,2}$.

  3. (3)

We can write $\sqrt{Q_{5^{2}}}=Q_{5^{2},1}Q_{5^{2},2}$ with $\deg
Q_{5^{2},1}=1$ and $\deg Q_{5^{2},2}=4$. Under these conventions, the fields
$F_{5^{2},j}\colonequals\mathbb{Q}[t]/Q_{5^{2},j}$ have degree $\deg
Q_{5^{2},j}$. Furthermore, for each _even_ $f\geq 2$, we have
$Q_{5^{f}}=(\sqrt{Q_{5^{f}}})^{2}$ and $\mathbb{Q}[t]/\sqrt{Q_{5^{f}}}\cong
F_{5^{2},1}\times F_{5^{2},2}$.

######  Remark C.8.

It is important to view $L_{11},F_{5,2}$ as “abstract” (rather than
“embedded”) number fields, since they are not Galois (even though
$F_{5,1},F_{5^{2},1},F_{5^{2},2}$ happen to be Galois).

Before proceeding, note in general that $Q_{q}(t)\mid Q_{q^{f}}(t^{f})$, so
the algebra map $\mathbb{Q}[t]\to\mathbb{Q}[t]/Q_{q}$ given by $t\mapsto
t^{f}$ factors through a “canonical” map

$\phi_{q,f}\colon\mathbb{Q}[t]/Q_{q^{f}}\to\mathbb{Q}[t]/Q_{q}$

between two $10$-dimensional algebras. In general, $\phi_{q,f}$ need not be
bijective (or equivalently, injective or surjective), but it is always nonzero
(because it sends $1$ to $1$). But we can still analyze $Q_{q^{f}}(t)$
reasonably well, using the Galois-theoretic correspondence between irreducible
polynomials over $\mathbb{Q}$ on the one hand, and $G_{\mathbb{Q}}$-orbits in
$\overline{\mathbb{Q}}$ on the other.

###### Proof of (1).

Fix $f\geq 1$. Let $\phi\colonequals\phi_{11,f}$. By Appendix C.3.1, we know
that $Q_{11}$ is irreducible, i.e. $L_{11}$ is a field. So $G_{\mathbb{Q}}$
acts _transitively_ on the roots of $Q_{11}$ in $\overline{\mathbb{Q}}$, and
hence also on the roots of $Q_{11^{f}}$. Thus $\sqrt{Q_{11^{f}}}$ is
irreducible—i.e. $\mathbb{Q}[t]/Q_{11^{f}}$ is a _local_ Artinian ring. Since
$\phi\neq 0$, it follows that $\mathbb{Q}[t]/Q_{11^{f}}\cong L_{11}$ holds if
and only if $Q_{11^{f}}=\sqrt{Q_{11^{f}}}$.

So to prove (1), it remains to show that $Q_{11^{f}}(t)$ is square-free (and
thus irreducible), and compute the subfields of $L_{11}$. (We can do these
tasks in either order.)

For the first task, write $Q_{11}(t)=\prod(1-\beta_{i}t)$ with
$\beta_{1},\dots,\beta_{10}\in\overline{\mathbb{Q}}$; then we need to show
that $\beta_{1}^{f},\dots,\beta_{10}^{f}$ are pairwise distinct. Assume for
contradiction that $\beta_{i}^{f}=\beta_{j}^{f}$ for some distinct $i,j$. Then
$\beta_{i}/\beta_{j}$ is a _primitive_ $n$th root of unity for some $n\mid f$.
In particular, $\zeta_{n}\in\mathbb{Q}(\beta_{i},\beta_{j})$. Since
$[L_{11}:\mathbb{Q}]=10$, it follows that
$\phi(n)=[\mathbb{Q}(\zeta_{n}):\mathbb{Q}]\mid 10d$ for some
$d\in\\{1,2,\dots,9\\}$. But in general, $\phi(n)\geq\sqrt{n/2}$, so
$90\geq\sqrt{n/2}$, whence $n<2\cdot 100^{2}$. By Appendix C.3.2, we conclude
that $n\leq 330$. Hence by Appendix C.3.2,
$[\mathbb{Q}(\beta_{i}^{n}):\mathbb{Q}]=10$, so the Galois conjugates
$\beta_{1}^{n},\dots,\beta_{10}^{n}$ must be pairwise distinct—contradicting
the fact that $(\beta_{i}/\beta_{j})^{n}=1$. This contradiction completes the
first task.

For the second task, see Appendix C.3.2. ∎

###### Proof of (2).

By Appendix C.3.1, we can (and must) take $Q_{5,1}\colonequals 5t^{2}+1$ and
$Q_{5,2}\colonequals 625t^{8}+100t^{6}+4t^{2}+1$. (These are irreducible, i.e.
$F_{5,1}$ and $F_{5,2}$ are fields.)

Now fix $f\geq 1$ with $2\nmid f$. Let $\phi\colonequals\phi_{5,f}$. For
convenience, write $Q_{5,1}=\prod_{i\leq 2}(1-\beta_{i}t)$ and
$Q_{5,2}=\prod_{i\geq 3}(1-\beta_{i}t)$ with
$\beta_{1},\dots,\beta_{10}\in\overline{\mathbb{Q}}$. Then let
$Q_{5,1,f}\colonequals\prod_{i\leq 2}(1-\beta_{i}^{f}t)$ and
$Q_{5,2,f}\colonequals\prod_{i\geq 3}(1-\beta_{i}^{f}t)$. For each $j$, Galois
theory implies that $\sqrt{Q_{5,j,f}}$ is irreducible; cf. the proof of (1).
Furthermore, $Q_{5,1,f}$ is clearly square-free, since
$\beta_{1}^{f}=-\beta_{2}^{f}\neq\beta_{2}^{f}$.

In fact, $Q_{5,2,f}$ is also square-free. Assume not; then arguing as in (1),
we obtain distinct $i,j\geq 3$, and an integer $n\leq 330$, such that $n\mid
f$ and $(\beta_{i}/\beta_{j})^{n}=1$. But here $2\nmid f$, so $2\nmid n$. So
by Appendix C.3.2, $[\mathbb{Q}(\beta_{i}^{n}):\mathbb{Q}]=8$, so the Galois
conjugates $\beta_{3}^{n},\dots,\beta_{10}^{n}$ must be pairwise distinct—a
contradiction.

From the above information, we conclude that $Q_{5,j,f}$ is irreducible for
each $j$. Since $\deg Q_{5,1,f}\neq\deg Q_{5,2,f}$, it then follows that the
polynomial $Q_{5^{f}}=Q_{5,1,f}Q_{5,2,f}$ is square-free. Consequently, by the
Chinese remainder theorem modulo $Q_{5^{f}},Q_{5}$, we can identify $\phi$
with the product of the canonical maps
$\psi_{j}\colon\mathbb{Q}[t]/Q_{5,j,f}\to\mathbb{Q}[t]/Q_{5,j}$. But
$\psi_{1},\psi_{2}$ are nonzero maps between _fields_ —and thus isomorphisms.
So $\phi$ is an isomorphism itself, whence
$\mathbb{Q}[t]/Q_{5^{f}}\cong\mathbb{Q}[t]/Q_{5}\cong F_{5,1}\times F_{5,2}$,
as desired. ∎

###### Proof of (3).

Following (2) and its proof, write $Q_{5,1}=\prod_{i\leq 2}(1-\beta_{i}t)$ and
$Q_{5,2}=\prod_{i\geq 3}(1-\beta_{i}t)$ with
$\beta_{1},\dots,\beta_{10}\in\overline{\mathbb{Q}}$. The polynomials
$Q_{5,1},Q_{5,2}$ are _even_ , so we may assume that
$\beta_{1}+\beta_{2}=\beta_{3}+\beta_{4}=\cdots=0$. Then in particular,
$Q_{5^{2}}(t)=Q_{5^{2},1}(t)^{2}Q_{5^{2},2}(t)^{2}$, where
$Q_{5^{2},1}\colonequals\prod_{2\mid i\leq
2}(1-\beta_{i}^{2}t)=Q_{5,1}(t^{1/2})=5t+1$ and
$Q_{5^{2},2}\colonequals\prod_{2\mid i\geq
3}(1-\beta_{i}^{2}t)=Q_{5,2}(t^{1/2})=625t^{4}+100t^{3}+4t^{1}+1$. (Note that
$Q_{5^{2},j}(t^{2})=Q_{5,j}(t)$, and each $Q_{5,j}$ is irreducible, so each
$Q_{5^{2},j}$ is certainly irreducible.)

Now fix $f\geq 1$ with $2\mid f$. Let $\phi\colonequals\phi_{5^{2},f/2}$. Let
$Q_{5^{2},1,f/2}\colonequals\prod_{2\mid i\leq 2}(1-\beta_{i}^{f}t)$ and
$Q_{5^{2},2,f/2}\colonequals\prod_{2\mid i\geq 3}(1-\beta_{i}^{f}t)$, and let
$Q_{5,1,f}\colonequals\prod_{i\leq 2}(1-\beta_{i}^{f}t)=Q_{5^{2},1,f/2}^{2}$
and $Q_{5,2,f}\colonequals\prod_{i\geq
3}(1-\beta_{i}^{f}t)=Q_{5^{2},2,f/2}^{2}$. For each $j$, Galois theory implies
that $\sqrt{Q_{5^{2},j,f/2}}$ is irreducible; cf. the proof of (1).
Furthermore, $Q_{5^{2},1,f/2}$ is clearly square-free, since $\deg
Q_{5^{2},1,f/2}=1$.

In fact, $Q_{5^{2},2,f/2}$ is also square-free. Assume not; then arguing as in
(1), we obtain distinct _even_ indices $i,j\geq 3$, and an integer $n\leq
330$, such that $n\mid f/2$ and $(\beta_{i}^{2}/\beta_{j}^{2})^{n}=1$. So by
Appendix C.3.2, $[\mathbb{Q}((\beta_{i}^{2})^{n}):\mathbb{Q}]=4$, so the
Galois conjugates $(\beta_{4}^{2})^{n},\dots,(\beta_{10}^{2})^{n}$ must be
pairwise distinct—a contradiction.

We conclude that $Q_{5^{2},j,f/2}$ is irreducible for each $j$. Since $\deg
Q_{5^{2},1,f/2}\neq\deg Q_{5^{2},2,f/2}$, it then follows that
$\sqrt{Q_{5^{f}}}=Q_{5^{2},1,f/2}Q_{5^{2},2,f/2}$ and
$Q_{5^{f}}=(\sqrt{Q_{5^{f}}})^{2}$. By studying the “canonical” maps
$\phi^{\prime}\colon\mathbb{Q}[t]/\sqrt{Q_{5^{f}}}\to\mathbb{Q}[t]/\sqrt{Q_{5^{2}}}$
and
$\psi^{\prime}_{j}\colon\mathbb{Q}[t]/Q_{5^{2},j,f/2}\to\mathbb{Q}[t]/Q_{5^{2},j}$,
we obtain an isomorphism
$\mathbb{Q}[t]/\sqrt{Q_{5^{f}}}\cong\mathbb{Q}[t]/\sqrt{Q_{5^{2}}}\cong
F_{5^{2},1}\times F_{5^{2},2}$, as desired; cf. the proof of (2). ∎

###  C.3. Supporting code

####  C.3.1. Local zeta polynomials

Running (in SageMath)

    def Delta(c_1,c_2,c_3,c_4,c_5,c_6):
        return Integer(expand(
            3*prod(sqrt(c_1)^3 + i[0]*sqrt(c_2)^3 + i[1]*sqrt(c_3)^3
            + i[2]*sqrt(c_4)^3 + i[3]*sqrt(c_5)^3 + i[4]*sqrt(c_6)^3
        for i in cartesian_product([[-1,1],[-1,1],[-1,1],[-1,1],[-1,1]]))));
    factor(Delta(1,2,3,4,5,6))

yields `3^13 * 996001 * 1898591 * 107541241 * 1722583559`. So in particular,
$V_{\bm{c}}$ in Proposition C.1 has good reduction at all primes
$p\in[5,100]$. Now in [debarre2021lines]’s supplementary code [github-alaface-
CubLin-2016]*algorithms.txt, re-define the function ZetaG as follows:

    // Input: (c,q) with q an odd prime power,
    //              c[6] ne 0 in Fq, and V_c/Fq smooth of dimension 3
    // Output: the characteristic polynomial P_1(F(V_c))
    ZetaG := function(c,q)
     Fq := FiniteField(q);
     P4<[x]> := ProjectiveSpace(Fq,4);
     X := Scheme(P4,c[6]^3*(&+[x[i]^3 : i in [1..5]])
                    - (&+[c[i]*x[i] : i in [1..5]])^3);
     L := Scheme(P4,[x[2]+x[4], x[3]+x[5],
                                c[6]*x[1] - &+[c[i]*x[i] : i in [1..5]]]);
     R<t> := PolynomialRing(Rationals());
     n := [Mr(X,L,r) : r in [1..5]];
     u := -&+[n[r]/r*t^r : r in [1..5]];
     ll := Coefficients(R!&+[u^k/Factorial(k) : k in [0..5]])[1..6];
     cf := ll cat [q^i*ll[6-i] : i in [1..5]];
     return &+[cf[i]*t^(i-1) : i in [1..11]];
    end function;

Then from the Magma session

    > load "algorithms.txt";
    > ZetaG([1,2,3,4,5,6],5);
    3125*t^10 + 1125*t^8 + 100*t^6 + 20*t^4 + 9*t^2 + 1
    > ZetaG([1,2,3,4,5,6],7);
    16807*t^10 + 40817*t^9 + 50421*t^8 + 40572*t^7 + 23478*t^6
    + 10182*t^5 + 3354*t^4 + 828*t^3 + 147*t^2 + 17*t + 1
    > ZetaG([1,2,3,4,5,6],11);
    161051*t^10 + 14641*t^9 + 3993*t^8 + 2420*t^7 + 539*t^6
    + 14*t^5 + 49*t^4 + 20*t^3 + 3*t^2 + t + 1
    > ZetaG([1,2,3,4,5,6],13);
    371293*t^10 + 542659*t^9 + 432809*t^8 + 234572*t^7 + 94718*t^6
    + 29666*t^5 + 7286*t^4 + 1388*t^3 + 197*t^2 + 19*t + 1

and the (SageMath) factorizations (over $\mathbb{Z}[t]$)

  1. (1)

`(5*t^2 + 1) * (625*t^8 + 100*t^6 + 4*t^2 + 1)` at $p=5$,

  2. (2)

`(7*t^2 + t + 1) * (7*t^2 + 4*t + 1)^4` at $p=7$,

  3. (3)

`irreducible` at $p=11$, and

  4. (4)

`(13*t^2 + 1) * (13*t^2 + 7*t + 1) * (13*t^2 + 4*t + 1)^3` at $p=13$,

we obtain the polynomials
$P_{5\mathbb{Z}},P_{7\mathbb{Z}},P_{11\mathbb{Z}},P_{13\mathbb{Z}}$ as quoted
in the proof of Proposition C.1.

####  C.3.2. Number field subfield computations

The SageMath code

    max_tested = 0; max_discovered = 0
    for n in range(1,2*100^2):
        max_tested = n
        for d in range(1,10):
            if (10*d) % euler_phi(n) == 0:
                max_discovered = n; break
    print(max_tested,max_discovered)

yields `19999 330`, showing that

$\max\\{n\in[1,2\cdot
100^{2}-1]:\exists\;d\in\\{1,2,\dots,9\\}\;\textnormal{such that}\;\phi(n)\mid
10d\\}=330.$

We now study $Q_{11}$. The SageMath code

    max_tested = 0; R.<t> = QQ[]
    Q_11 = ZetaG([1,2,3,4,5,6],11); K.<a> = NumberField(Q_11)
    for n in range(1,500):
        max_tested = n; L = K.subfield(a^n)[0]
        if L.absolute_degree() != 10:
            print(n)
    print("Tested up to",max_tested)

yields `Tested up to 499`, showing that for each positive integer $n<500$, the
field $\mathbb{Q}[t]/Q_{11}$ is generated by $\overline{t}^{n}$ (i.e.
$\overline{t}^{n}$ is a _primitive element_ for $\mathbb{Q}[t]/Q_{11}$).
Meanwhile,

    R.<t> = QQ[]; Q_11 = ZetaG([1,2,3,4,5,6],11); K.<a> = NumberField(Q_11)
    print(K.is_galois(),[Z[0].absolute_degree() for Z in K.subfields()])

yields `False [1, 5, 10]`, showing that the field
$L_{11}\colonequals\mathbb{Q}[t]/Q_{11}$ is non-Galois, with exactly $1$
nontrivial subfield $L^{\prime}_{11}$—and furthermore,
$[L^{\prime}_{11}:\mathbb{Q}]=5$.

We now study $Q_{5,2}$. The SageMath code

    max_tested = 0; R.<t> = QQ[]
    Q_52 = 625*t^8 + 100*t^6 + 4*t^2 + 1; K.<a> = NumberField(Q_52)
    for n in range(1,500,2):
        max_tested = n; L = K.subfield(a^n)[0]
        if L.absolute_degree() != 8:
            print(n)
    print("Tested odds up to",max_tested)

yields `Tested odds up to 499`, showing that for each positive _odd_ integer
$n<500$, the field $\mathbb{Q}[t]/Q_{5,2}$ is generated by $\overline{t}^{n}$.

Finally, we study $Q_{5^{2},2}$. The SageMath code

    max_tested = 0; R.<t> = QQ[]
    Q_522 = 625*t^4 + 100*t^3 + 4*t^1 + 1; K.<a> = NumberField(Q_522)
    for n in range(1,500):
        max_tested = n; L = K.subfield(a^n)[0]
        if L.absolute_degree() != 4:
            print(n)
    print("Tested up to",max_tested)

yields `Tested up to 499`, showing that for each positive integer $n<500$, the
field $\mathbb{Q}[t]/Q_{5^{2},2}$ is generated by $\overline{t}^{n}$.

##  Appendix D “Correctly” defining discriminants of hyperplane sections

Fix $m\geq 3$. Our goal in Appendix D is to “correctly” define the
discriminant of a hyperplane section $\mathcal{V}_{\bm{c}}/\mathbb{Z}$; we
will justify Definition D.6 (below) as a reasonable candidate.

Throughout Appendix D, we will use the adjective “generic” in the “symbolic”
sense (i.e. we will work with “indeterminate coefficients”).

###  D.1. Definitions for generic arbitrary $F$

######  Definition D.1 (Cf. [saito2012discriminant]*§2.1).

Let $G$ denote a generic cubic form in $x_{1},\dots,x_{m-1}$. Let
$\operatorname{disc}(G)\in\mathbb{Z}[G_{ijk}]$ be a _primitive_ form in
$G_{ijk}$, square-free over $\mathbb{Q}$, such that given
$G/\overline{\mathbb{Q}}$, we have $\operatorname{disc}(G)\neq 0$ if and only
if $V(G)$ is smooth of codimension $1$.

######  Remark D.2.

Here $\operatorname{disc}(G)$ is $\pm 1$-unique, absolutely irreducible, and
_homogeneous_ of degree $(m-1)2^{m-2}$. Also, $V(G)/\mathbb{F}_{p}$ is smooth
of codimension $1$ if and only if $p\nmid\operatorname{disc}(G)$. For the
Fermat cubic $G=x_{1}^{3}+\dots+x_{m-1}^{3}$, we have
$\operatorname{disc}(G)=\pm 3^{\mathfrak{e}_{m}}$, where

$\mathfrak{e}_{m}\colonequals\frac{(-1)^{m-1}-2^{m-1}}{3}+(m-1)2^{m-2}=3\cdot\bm{1}_{m=3}+9\cdot\bm{1}_{m=4}+27\cdot\bm{1}_{m=5}+69\cdot\bm{1}_{m=6}+\cdots.$

(For the Fermat calculation in general, see [gelfand2008discriminants]*p. 434,
(1.8) and Proposition 1.7; for $m=4$, see [saito2012discriminant]*§5.3; for
$m=3$, see [saito2012discriminant]*§5.1.)

######  Definition D.3 (Cf. [terakado2018determinant]*§1.1).

Let $F$ denote a generic cubic form in $x_{1},\dots,x_{m}$. Let
$\operatorname{disc}(F,\bm{c})\in\mathbb{Z}[F_{ijk},\bm{c}]$ be a _primitive_
form in $F_{ijk},\bm{c}$, square-free over $\mathbb{Q}$, such that given