# How often does a cubic hypersurface have a rational point?

Lea Beneish1 1Department of Mathematics, University of North Texas, Denton,
TX, USA<EMAIL_ADDRESS>and  Christopher Keyes2 2Department of
Mathematics, King’s College London, London, UK
and Heilbronn Institute for Mathematical Research, Bristol, UK
<EMAIL_ADDRESS>
###### Abstract.

A cubic hypersurface in $\mathbb{P}^{n}$ defined over $\mathbb{Q}$ is given by
the vanishing locus of a cubic form $f$ in $n+1$ variables. It is conjectured
that when $n\geq 4$, such cubic hypersurfaces satisfy the Hasse principle.
This is now known to hold on average due to recent work of Browning, Le
Boudec, and Sawin. Using this result, we determine the proportion of cubic
hypersurfaces in $\mathbb{P}^{n}$, ordered by the height of $f$, with a
rational point for $n\geq 4$ explicitly as a product over primes $p$ of
rational functions in $p$. In particular, this proportion is equal to 1 for
cubic hypersurfaces in $\mathbb{P}^{n}$ for $n\geq 9$; for $100\%$ of cubic
hypersurfaces, this recovers a celebrated result of Heath-Brown that non-
singular cubic forms in at least 10 variables have rational zeros. In the
$n=3$ case, we give a precise conjecture for the proportion of cubic surfaces
in $\mathbb{P}^{3}$ with a rational point.

##  1\. Introduction

00footnotetext: License: for the purposes of open access, the authors have
applied a Creative Commons Attribution (CC BY) license to any Author Accepted
Manuscript version arising from this submission.

A cubic hypersurface in $\mathbb{P}^{n}$ defined over $\mathbb{Q}$ is given by
the vanishing locus of an integral cubic form $f$ in $n+1$ variables,

(1.1) $X_{f}\colon f(x_{0},\ldots,x_{n})=\sum\limits_{0\leq i\leq j\leq k\leq
n}a_{ijk}x_{i}x_{j}x_{k}=0.$

We are interested in the proportion of $X_{f}$ that possess a rational point.
In studying the rational points of such hypersurfaces, it is often useful to
study their local points. If a variety $X/\mathbb{Q}$ possesses
$\mathbb{Q}_{v}$-points for all places $v$ of $\mathbb{Q}$, we say $X$ is
everywhere locally soluble. This property is a necessary but not sufficient
condition for $X$ to possess rational points. If $X$ is everywhere locally
soluble, but still fails to have a rational point, we say there is an
obstruction to the Hasse principle (i.e., the Hasse principle does not hold).

More generally, a degree $d$ hypersurface in $\mathbb{P}^{n}$ is Fano if
$d\leq n$. Over any number field, it is known that smooth Fano hypersurfaces
of dimension at least $3$ do not have a Brauer–Manin obstruction to the Hasse
principle [PV04, Appx. A]. It has further been conjectured by Colliot-Thélène
[CT03] that this is the only possible obstruction to the Hasse principle,
hence the Hasse principle should hold. Browning, Le Boudec, and Sawin [BLBS23]
prove that this is almost always true over the rationals in the following
precise sense: in the limit as $A$ tends to infinity, the proportion of Fano
hypersurfaces defined over $\mathbb{Q}$ of fixed degree $d$ in
$\mathbb{P}^{n}$ (except in the case $n=d=3$) with height at most $A$ is equal
to the proportion of such hypersurfaces that are everywhere locally soluble.

Explicitly, counting hypersurfaces by the Euclidean height of the vector
$\boldsymbol{a}\in\mathbb{Z}^{\binom{n+d}{d}}$ of coefficients of the defining
degree $d$ form $f\in\mathbb{Z}[x_{0},\ldots,x_{n}]$,

$\operatorname{ht}(f):=||\boldsymbol{a}||_{2}=\left(\sum_{0\leq
i_{1}\leq\ldots\leq i_{d}\leq n}a_{i_{1}\ldots i_{d}}^{2}\right)^{1/2},$

we define the natural densities

(1.2) $\displaystyle\rho_{d,n}$
$\displaystyle=\lim_{A\to\infty}\frac{\\#\\{f:\operatorname{ht}(f)\leq A,\
X_{f}(\mathbb{Q})\neq\emptyset\\}}{\\#\\{f:\operatorname{ht}(f)\leq A\\}}$
(1.3) $\displaystyle\rho^{\mathrm{ELS}}_{d,n}$
$\displaystyle=\lim_{A\to\infty}\frac{\\#\\{f:\operatorname{ht}(f)\leq A,\
X_{f}(\mathbb{Q}_{v})\neq\emptyset\text{ for all
}v\\}}{\\#\\{f:\operatorname{ht}(f)\leq A\\}}$

where $X_{f}$ denotes the degree $d$ hypersurface in $\mathbb{P}^{n}$ cut out
by $f=0$ and $v$ runs through all places of $\mathbb{Q}$. For fixed
$(d,n)\neq(3,3)$ satisfying $2\leq d\leq n$, Browning, Le Boudec, and Sawin
show that $\rho_{d,n}=\rho_{d,n}^{\mathrm{ELS}}$ [BLBS23, Theorem 1.1]. They
also deduce that for $(d,n)\neq(2,2)$ satisfying $2\leq d\leq n$, we have
$\rho_{d,n}>0$ [BLBS23, Corollary 1.2]; that is, for a positive proportion of
degree $d$ Fano hypersurfaces $X_{f}\subset\mathbb{P}^{n}$ — now excluding
plane conics ($d=n=2$) but including cubic surfaces ($d=n=3$) — we have
$X_{f}(\mathbb{Q})\neq\emptyset$.

In this paper, for each $n\geq 4$, we determine the proportion $\rho_{3,n}$
explicitly as a product of local factors and compute it numerically to high
precision. A striking feature is that the local factors are uniform in $p$,
given by rational functions.

######  Theorem 1.1.

Let $n\geq 4$. There exist polynomials $g_{n}(t),h_{n}(t)\in\mathbb{Z}[t]$
such that the proportion of cubic hypersurfaces in $\mathbb{P}^{n}$ ordered by
Euclidean height that possess a rational point is given by

$\rho_{3,n}=\begin{cases}\displaystyle\prod_{p\text{
prime}}\left(1-\frac{g_{n}(p)}{h_{n}(p)}\right)&4\leq n\leq 8\\\ 1&n\geq
9.\end{cases}$

The polynomials $g_{n},h_{n}$ are given explicitly in (6.7) – (6.16).

In the case of cubic surfaces ($d=n=3$),
$\rho_{3,3}=\rho^{\mathrm{ELS}}_{3,3}$ has been conjectured but is not known
to hold [PV04, Conjecture 2.2(ii)]. We can, however, give a similarly explicit
description of $\rho^{\mathrm{ELS}}_{3,3}$, leading to a conjectural formula
for $\rho_{3,3}$.

######  Conjecture 1.2 (Cubic surfaces).

Let $n=3$. The proportion of cubic surfaces in $\mathbb{P}^{3}$ ordered by
Euclidean height that possess a rational point is given by111Note that the
conjectured formula for $\frac{g_{3}(p)}{h_{3}(p)}$ is not presented in lowest
form, to save space.

$\rho_{3,3}=\prod_{p}\textstyle\left(1-\frac{\left(3p^{26}+p^{24}+p^{23}+4p^{22}-3p^{21}+3p^{20}+2p^{19}+2p^{18}-p^{17}+p^{14}+p^{13}-2p^{12}+3p^{11}+3p^{7}\right)\left(p^{2}+1\right)\left(p+1\right)^{2}\left(p-1\right)^{4}}{9\left(p^{13}-1\right)\left(p^{7}+1\right)\left(p^{7}-1\right)\left(p^{6}+1\right)\left(p^{5}-1\right)\left(p^{3}+1\right)\left(p^{3}-1\right)}\right).$

Numerically, $\rho_{3,3}\approx 0.999927$.

######  Remark 1.3 (asymptotics and numerics).

We record below in Table 1.1 how quickly each local factor approaches 1,
deduced from the explicit descriptions given in Section 6. We also record the
approximate numerical values of $\rho_{3,n}$ (including the conjectural value
of $\rho_{3,3}$). The details of these calculations and their precision are
discussed in Section 7.

Table 1.1. Asymptotics and numerics for $\rho_{3,n}$ $n$ | $\frac{g_{n}(t)}{h_{n}(t)}\sim$ | $\rho_{3,n}\approx$
---|---|---
3 | $1/3t^{10}$ |  $0.999927$ (conj.)
4 | $1/9t^{22}$ | $1-5.022\cdot 10^{-9}$
5 | $1/9t^{43}$ | $1-1.343\cdot 10^{-15}$
6 | $1/9t^{78}$ | $1-3.502\cdot 10^{-26}$
7 | $1/27t^{129}$ | $1-5.152\cdot 10^{-42}$
8 | $1/27t^{201}$ | $1-6.222\cdot 10^{-64}$

For $n\geq 9$, Theorem 1.1 recovers a consequence of Heath-Brown’s celebrated
result that non-singular cubic forms in $n+1\geq 10$ variables have rational
zeros [HB83]. By combining [BLBS23] with a complete description of the
probability that $X_{f}(\mathbb{Q}_{p})\neq\emptyset$, we are able to further
determine $\rho_{3,n}$ for $n\geq 4$ (see Theorem 2.3). The existence of
nontrivial $p$-adic zeros for cubic forms in 10 or more variables was
established by Dem’yanov for $p\neq 3$ [Dem50] and later by Lewis for all
primes $p$ [Lew52]. When $n=8$, work of Hooley shows that the Hasse principle
holds for nonsingular cubic forms [Hoo88]. Heath-Brown’s result is sharp in
that there exist cubic forms in $9$ variables which fail to be everywhere
locally soluble; this is reflected in Theorem 1.1 as we compute
$\rho_{3,8}<1$, and extended to all $n\geq 4$ (conjecturally $n\geq 3$).

In light of [BLBS23], our strategy is to determine
$\rho^{\mathrm{ELS}}_{3,n}$, the proportion of everywhere locally soluble
cubic hypersurfaces in $\mathbb{P}^{n}$. We are able to do this by applying
work of Poonen and Voloch [PV04] and Bright, Browning, and Loughran [BBL16] to
describe $\rho_{d,n}^{\mathrm{ELS}}$ as a product of local factors. These
local factors are essentially the probabilities that
$X_{f}(\mathbb{Q}_{v})\neq\emptyset$ for a randomly chosen cubic form $f$ over
$\mathbb{Q}_{v}$, which we determine explicitly and uniformly for finite
places $v$. There have been other recent works studying everywhere local
solubility in various families of hypersurfaces (see, for example, Browning,
Fisher–Ho–Park, Hirakawa–Kanamura [Bro17, FHP21, HK21]).

Most notably, Theorem 1.1 may be regarded as a cubic analogue of work of
Bhargava, Cremona, Fisher, Jones, and Keating on the density of integral
quadratic forms in $n+1$ variables with a nontrivial integral zero; in
particular, they give an explicit description of $\rho_{2,n}$ as a product of
local probabilities which are rational functions in $p$ [BCF+16b].

Our methods also recover $\rho^{\mathrm{ELS}}_{3,2}$, the density of
everywhere local soluble plane cubic curves, first computed by Bhargava,
Cremona, and Fisher as a product of explicit local densities [BCF16a]. Less is
known about the density of plane cubics with a global point, $\rho_{3,2}$.
Bhargava showed both that a positive proportion of plane cubics fail the Hasse
principle and that a positive proportion have a rational point [Bha14,
Theorems 1, 2], i.e. $0<\rho_{3,2}<\rho^{\mathrm{ELS}}_{3,2}$. Combining his
methods with conjectures on the distribution of ranks of elliptic curves over
$\mathbb{Q}$, Bhargava further conjectured
$\rho_{3,2}=\frac{1}{3}\rho^{\mathrm{ELS}}_{3,2}$ [Bha14, Conjecture 6].

This paper is organized as follows. In Section 2 we give an overview of the
strategy of the paper. In Section 3 we analyze the possible ways in which a
cubic form can factor over a finite field and determine their probabilities of
occurrence in cases of interest. In Sections 4 and 5, we compute the
probability that a $p$-adic cubic hypersurface has a $\mathbb{Q}_{p}$-point
given certain conditions on its reduction $\overline{X_{f}}$; the former
handles the case where the reduction is not a configuration of conjugate
hyperplanes over $\mathbb{F}_{p}$, while the latter handles precisely these
cases. In Section 6 we prove Theorem 1.1 and give the explicit formulas for
the local factors, and in Section 7 we describe how we obtain precise
numerical values.

### Acknowledgments

The authors are grateful to Jackson Morrow for bringing [BLBS23] to their
attention and would like to thank Tim Browning, Tom Fisher, Rachel Newton, and
Bjorn Poonen for helpful comments on an earlier draft. CK was supported by the
Additional Funding Programme for Mathematical Sciences, delivered by EPSRC
(EP/V521917/1) and the Heilbronn Institute for Mathematical Research.

##  2\. Everywhere local solubility

Let $X/\mathbb{Q}$ be a variety and $v$ denote a place of $\mathbb{Q}$.

######  Definition 2.1.

$X$ is locally soluble at $\boldsymbol{v}$ if $X(\mathbb{Q}_{v})\neq\emptyset$
and everywhere locally soluble if $X(\mathbb{Q}_{v})\neq\emptyset$ for all
places $v$.

It follows from the inclusions $X(\mathbb{Q})\hookrightarrow
X(\mathbb{Q}_{v})$ that everywhere local solubility is necessary for
$X(\mathbb{Q})\neq\emptyset$. Thus in studying how often a degree $d$
hypersurface $X_{f}$ has a rational point, it will be useful to keep track of
how often $X_{f}$ is everywhere locally soluble. We denote by
$\rho^{\mathrm{ELS}}_{d,n}$ the natural density of integral degree $d$ forms
$f$ (with respect to the Euclidean height
$\operatorname{ht}(f)=||\boldsymbol{a}||_{2}$) for which the hypersurface
$X_{f}\subset\mathbb{P}^{n}$ is everywhere locally soluble, given in (1.3).

The limit definition of $\rho^{\mathrm{ELS}}_{d,n}$ is unwieldy to compute
with. However, it follows from work of Bright, Browning, and Loughran [BBL16,
Theorem 1.4] that when $n\geq 2$ and $(d,n)\neq(2,2)$,

(2.1) $\rho^{\mathrm{ELS}}_{d,n}=\rho_{d,n}(\infty)\prod_{p}\rho_{d,n}(p),$

where $\rho_{d,n}(v)$ is the density of $v$-adic degree $d$ hypersurfaces with
a $v$-adic point. This is made precise for the finite places by letting
$\mu_{p}$ to be the normalized Haar measure on
$\mathbb{Z}_{p}^{\binom{n+d}{d}}$, the space of integral $p$-adic degree $d$
forms, and taking

$\rho_{d,n}(p)=\mu_{p}\left(\left\\{f\in\mathbb{Z}_{p}[x_{0},\ldots,x_{n}]:X_{f}(\mathbb{Q}_{p})\neq\emptyset\right\\}\right),$

where $f$ is identified with the tuple corresponding to its coefficients in
$\mathbb{Z}_{p}^{\binom{n+d}{d}}$.

In their original paper on random hypersurfaces, Poonen and Voloch proved
(2.1) with a different choice of height function when defining
$\rho^{\mathrm{ELS}}_{d,n}$ and $\rho_{d,n}(\infty)$ [PV04, Theorem 3.6]; see
Remark 2.2. Similar product formulae for the densities of everywhere local
solubility hold in other families of varieties. Poonen and Stoll showed that a
version of (2.1) holds for the density of everywhere locally soluble
hyperelliptic curves [PS99], in work which predates [PV04]. Bright, Browning,
and Loughran generalized their approach to show the analogue of (2.1) holds
for families coming from fibers of maps to affine or projective space, subject
to certain geometric conditions [BBL16]. This was employed by the authors to
prove an analogue of (2.1) for families of superelliptic curves [BK23]. In
another direction, Fisher, Ho, and Park show that the analogue of (2.1) holds
for families of hypersurfaces in products of projective space [FHP21, Theorem
1.1].

######  Remark 2.2 (choices when defining densities).

Let $\Psi\subset\mathbb{R}^{\binom{n+d}{d}}$ denote a bounded subset of
positive measure with boundary measure zero, and define

$\rho^{\mathrm{ELS}}_{d,n,\Psi}=\lim_{A\to\infty}\frac{\\#\\{f\in
A\Psi\cap\mathbb{Z}^{\binom{n+d}{d}}:X_{f}\text{ ELS}\\}}{\\#\\{f\in
A\Psi\cap\mathbb{Z}^{\binom{n+d}{d}}\\}}.$

Let $\Psi^{\prime}\subseteq\Psi$ denote the subset of $f$ for which
$X_{f}(\mathbb{R})\neq\emptyset$ and $\mu_{\infty}$ be the Lebesgue measure on
$\mathbb{R}^{\binom{n+d}{d}}$. It follows from [BBL16, Proposition 3.2] that

$\rho^{\mathrm{ELS}}_{d,n,\Psi}=\frac{\mu_{\infty}(\Psi^{\prime})}{\mu_{\infty}(\Psi)}\prod_{p}\rho_{d,n}(p).$

We take
$\rho_{d,n,\Psi}(\infty)=\frac{\mu_{\infty}(\Psi^{\prime})}{\mu_{\infty}(\Psi)}$
to get (2.1) and note that the product over finite places does not depend on
$\Psi$.

Different choices of $\operatorname{ht}(f)$ correspond to different choices of
$\Psi$. In their original paper, Poonen and Voloch used
$\operatorname{ht}(f)=||\boldsymbol{a}||_{\infty}$, i.e. the maximum absolute
value of the coefficients of $f$; this corresponds to taking
$\Psi=[-1,1]^{\binom{n+d}{d}}$. The Euclidean height
$\operatorname{ht}(f)=||\boldsymbol{a}||_{2}$, used here and in [BLBS23],
corresponds to taking $\Psi$ to be a sphere of radius 1 in
$\mathbb{R}^{\binom{n+d}{d}}$. In either case,
$\rho^{\mathrm{ELS}}_{d,n,\Psi}/\rho^{\mathrm{ELS}}_{d,n,\Psi}(\infty)$
coincide. We also point out that in [BLBS23], Browning, Le Boudec, and Sawin
count only primitive $f$, i.e. those whose coefficients have no common
divisor. By a standard Möbius inversion argument, the densities $\rho_{d,n}$
and $\rho^{\mathrm{ELS}}_{d,n}$ are unchanged whether we choose to count all
integral $f$ or just primitive $f$.

Returning to families of cubic hypersurfaces, the local factors
$\rho_{3,n}(v)$ can be computed explicitly. Since real cubic forms possess
real zeros, we have $\rho_{3,n}(\infty)=1$. For the finite places $v=p$, we
prove that these local probabilities are given in terms of rational functions,
uniformly in $p$.

######  Theorem 2.3.

Let $n\geq 1$. There exist $g_{n}(t),h_{n}(t)\in\mathbb{Z}[t]$ such that for
all primes $p$ we have

$\rho_{3,n}(p)=\begin{cases}1-\frac{g_{n}(p)}{h_{n}(p)}&1\leq n\leq 8\\\
1&n\geq 9,\end{cases}$

with $g_{n},h_{n}$ given explicitly in (6.1) – (6.16).

In the following sections we set out to compute $\rho_{3,n}(p)$ for $n\geq 1$,
thereby proving Theorem 2.3. Theorem 1.1 then follows by applying (2.1) and
[BLBS23]. The strategy is to reduce modulo $p$ and wherever possible find
points on $\overline{X_{f}}(\mathbb{F}_{p})$ which may be lifted to $p$-adic
points via Hensel’s lemma. If $\overline{X_{f}}(\mathbb{F}_{p})$ consists only
of singular points, we apply a transformation and repeat this process of
reducing and lifting. Along the way, we need to keep track of various
factorization and lifting probabilities.

We begin in Section 3 by measuring how often a cubic form $f$ over
$\mathbb{F}_{p}$ factors so that the associated hypersurface over
$\mathbb{F}_{p}$ decomposes into a configuration of hyperplanes conjugate over
$\mathbb{F}_{p^{3}}$; these are the aforementioned factorization
probabilities. A key insight, made in Section 4, is that when the reduction
$\overline{X_{f}}$ is not one of these distinguished configurations,
$X_{f}(\mathbb{F}_{p})$ contains a point which lifts via Hensel’s lemma to a
point in $X_{f}(\mathbb{Q}_{p})$. In these cases, the lifting probability is
simply 1. When $\overline{X_{f}}$ is a configuration of conjugate hyperplanes,
it possesses only singular $\mathbb{F}_{p}$-points, and the lifting
probabilities present a challenge. In Section 5, we write $\rho_{3,n}(p)$ as a
linear combination of the appropriate factorization and lifting probabilities
and investigate relations between them. This culminates in a large system of
relations which can be solved symbolically by the computer algebra system Sage
[Sag21], completing the proof of Theorem 2.3 by obtaining the expressions
(6.1) – (6.16) for $n\leq 8$ and verifying $\rho_{3,n}(p)=1$ when $n\geq 9$.

The case of binary cubic forms ($n=1$) was already known; see e.g. [BCFG22,
§1.2.3]. In the case of plane cubic curves ($n=2$), the conclusion of Theorem
2.3 was shown by Bhargava, Cremona, and Fisher [BCF16a]. Indeed, for $n=1,2$,
the $g_{n}(t)$ and $h_{n}(t)$ that we produce agree with these known results.
While our overall strategy is similar to theirs in spirit, in dimensions $n>2$
certain factorization cases require extra attention, and our approach requires
three “phases” rather than the two needed for the planar situation.
Furthermore, we choose to block the variables $x_{0},\ldots,x_{n}$ in a way
that allows us to work essentially uniformly in $n$. For that reason, in what
follows we allow $n=1,2$ and make note of similarities and differences for the
reader familiar with [BCF16a].

A feature of interest in this approach is that it is entirely uniform in $p$.
The same cannot be said for local solubility problems in other families,
including genus one hyperelliptic curves [BCF21] and genus four trigonal
superelliptic curves [BK23], where the local densities are given by rational
functions only for sufficiently large $p$. In the latter case, those rational
functions also depend on the residue class of $p$ modulo 3.

######  Remark 2.4 (finite extensions).

Theorem 2.3 can be extended to finite extensions $K/\mathbb{Q}$ as follows.
For a completion $K_{v}$ at a finite place $v$, let $\mathcal{O}_{v}$ denote
the valuation ring and $\mathbb{F}_{v}$ the residue field with
$q=\\#\mathbb{F}_{v}$. At each finite place $v$, the $v$-adic Haar measure of
the set of cubic forms $f\in\mathcal{O}_{v}[x_{0},\ldots,x_{n}]$ with
$X_{f}(\mathcal{O}_{v})\neq\emptyset$ is denoted $\rho_{3,n}(v)$ and given
explicitly by the same formula

$\rho_{3,n}(v)=1-\frac{g_{n}(q)}{h_{n}(q)}.$

Aside from the factorization probabilities presented in Section 3 for
arbitrary finite fields $\mathbb{F}_{q}$, we elect to restrict to the setting
of $K=\mathbb{Q}$, since this is where we have
$\rho_{3,n}=\rho^{\mathrm{ELS}}_{3,n}=\prod_{p}\rho_{3,n}(p)$ for $n\geq 4$
due to [BLBS23]. However, the proof of Proposition 4.1 and the strategy in
Section 5 generalize readily to $K_{v}$.

##  3\. Factorization probabilities

Fix a prime power $q$ and let $\mathbb{F}_{q}$ denote the finite field with
$q$ elements. Let $f\in\mathbb{F}_{q}[x_{0},\ldots,x_{n}]$ be a nonzero cubic
form. In Table 3.1 below, we record the possible factorizations of $f$ over
the algebraic closure $\overline{\mathbb{F}_{q}}$ along with their geometric
descriptions. Here each (decorated) $\ell$ denotes a distinct linear form, $g$
denotes a quadratic form, and $\sigma$ denotes the Frobenius action generating
the Galois group of an extension $\mathbb{F}_{q^{r}}/\mathbb{F}_{q}$.

Table 3.1. Possible geometric factorizations of a cubic hypersurface over $\mathbb{F}_{q}$ Symbol | Factorization | Description
---|---|---
$(1^{3})$ | $f=\ell^{3}$ | triple hyperplane over $\mathbb{F}_{q}$
$(1^{2}1)$ | $f=\ell^{2}\ell^{\prime}$ | double hyperplane and hyperplane both over $\mathbb{F}_{q}$
$(111)$ | $f=\ell_{1}\ell_{2}\ell_{3}$ | three distinct hyperplanes over $\mathbb{F}_{q}$
$(111)_{2}$ | $f=\ell(\sigma\ell)\ell^{\prime}$ | two conjugate hyperplanes over $\mathbb{F}_{q^{2}}$ and hyperplane over $\mathbb{F}_{q}$
$(111)_{3}$ | $f=\ell(\sigma\ell)(\sigma^{2}\ell)$ | three conjugate hyperplanes over $\mathbb{F}_{q^{3}}$
$(21)$ | $f=g\ell$ | quadric hypersurface and hyperplane both over $\mathbb{F}_{q}$
$(3)$ | $f$ | geometrically irreducible cubic hypersurface over $\mathbb{F}_{q}$

We are especially interested in the case where $f$ factors over
$\overline{\mathbb{F}_{q}}$ as the product of conjugates of a linear form over
$\mathbb{F}_{q^{3}}$, i.e. those of the form $(1^{3})$ or $(111)_{3}$ in Table
3.1. When $n=2$, $(1^{3})$ corresponds to a triple line, and $(111)_{3}$ can
be either a star, or a triangle as described in [BCF16a]. We can make these
precise for general $n$ as follows.

######  Definition 3.1.

Let $f\in\mathbb{F}_{q}[x_{0},\ldots,x_{n}]$ be a nonzero cubic form. Suppose
there exists a linear form
$\ell=b_{0}x_{0}+\ldots+b_{n}x_{n}\in\mathbb{F}_{q^{3}}[x_{0},\ldots,x_{n}]$
such that

$f=\prod_{\sigma\in\operatorname{Gal}(\mathbb{F}_{q^{3}}/\mathbb{F}_{q})}\sigma\ell$

and whose coefficients span a dimension $i$ $\mathbb{F}_{q}$-subspace of
$\mathbb{F}_{q^{3}}$,

$i=\dim_{\mathbb{F}_{q}}\operatorname{span}\\{b_{0},\ldots,b_{n}\\}.$

Then we say $f$ has factorization type $\boldsymbol{i}$.

The factorization type of $f$ is invariant under linear change of coordinates
over $\mathbb{F}_{q}$ and scaling $f$ by an element of
$\mathbb{F}_{q}^{\times}$. Type 1 is equivalent to $f=\ell^{3}$, or $X_{f}$ a
hyperplane with multiplicity 3. Types 2 and 3 correspond to the $(111)_{3}$
factorization of Table 3.1, or $X_{f}$ composed of three conjugate
hyperplanes; the pairwise intersections coincide for type 2, while they do not
for type 3 (the generic case).

######  Remark 3.2.

Suppose $f\in\mathbb{F}_{q}[x_{0},\ldots,x_{n}]$ is a nonzero cubic form with
factorization type $i\in\\{1,2,3\\}$. There is a bijection between
$X_{f}(\mathbb{F}_{q})$ and $\mathbb{P}^{n-i}(\mathbb{F}_{q})$. This can be
seen by making a linear change of variables resulting in
$i=\dim_{\mathbb{F}_{q}}\operatorname{span}\\{b_{0},\ldots,b_{i-1}\\}$ and
$b_{j}=0$ for $j\geq i$ and seeing that

$X_{f}(\mathbb{F}_{q})=\\{[0:\ldots:0:x_{i}:\ldots:x_{n}]:[x_{i}:\ldots:x_{n}]\in\mathbb{P}^{n-i}(\mathbb{F}_{q})\\}.$

Let $N_{n}$ denote the number of nonzero cubic forms
$f\in\mathbb{F}_{q}[x_{0},\ldots,x_{n}]$ and $N_{n,i}$ the number of such
forms having factorization type $i$ for $i=1,2,3$. We take
$\xi_{n,i}=N_{n,i}/N_{n}$ to be the probability that a nonzero cubic form has
type $i$; this is equal to the probability (in the sense of Haar measure) that
a primitive cubic form $f\in\mathbb{Z}_{p}[x_{0},\ldots,x_{n}]$ has reduction
$\overline{f}$ with type $i$. Let us further define

$\xi_{n,0}=1-\xi_{n,1}-\xi_{n,2}-\xi_{n,3},$

the probability of not being one of these three types.

######  Lemma 3.3.

Let $n\geq 0$. We have the following values for $\xi_{n,i}$.

$\displaystyle\xi_{n,0}$
$\displaystyle=1-\frac{q^{3n-3}+2q^{n+3}+2q^{n+2}+2q^{n+1}-2q^{2}-2q-3}{3\left(q^{2}+q+1\right)\left(q^{\binom{n+3}{3}}-1\right)}$
$\displaystyle\xi_{n,1}$
$\displaystyle=\frac{q^{n+1}-1}{q^{\binom{n+3}{3}}-1}$
$\displaystyle\xi_{n,2}$
$\displaystyle=\frac{{\left(q^{2n+1}-q^{n+1}-q^{n}+1\right)}q}{3\left(q^{\binom{n+3}{3}}-1\right)}$
$\displaystyle\xi_{n,3}$
$\displaystyle=\frac{{\left(q^{3n}-q^{2n}-q^{2n+1}-q^{2n-1}+q^{n+1}+q^{n-1}+q^{n}-1\right)}q^{3}}{3{\left(q^{2}+q+1\right)}{\left(q^{\binom{n+3}{3}}-1\right)}}$

###### Proof.

When $n=0$, we have $f=ax_{0}^{3}$ which has type 1. Thus $\xi_{0,1}=1$ and
$\xi_{0,0}=\xi_{0,2}=\xi_{0,3}=0$, which are in agreement with the stated
formulas.

From here on we assume $n\geq 1$. We have $N_{n}=q^{\binom{n+3}{3}}-1$. The
key idea is to count the linear forms into which $f$ factors appropriately,
and account for multiplicity due to scaling and conjugation. To compute
$N_{n,1}$, the number of cubic forms in $n+1$ variables that factor as a
triple hyperplane, we count linear forms over $\mathbb{F}_{q}$. Since there
are $\frac{q^{n+1}-1}{q-1}$ such hyperplanes, after accounting for possible
scaling, we obtain

$N_{n,1}=q^{n+1}-1.$

The value of $N_{1,2}$ was computed in [BCF16a, Lemma 5]. For general $n$, we
compute $N_{n,2}$ by again counting linear forms
$\ell=\sum_{k=0}^{n}b_{k}x_{k}$, but this time they are defined over
$\mathbb{F}_{q^{3}}$ and their coefficients span a 2-dimensional
$\mathbb{F}_{q}$-subspace. After scaling, we may assume that the first nonzero
coefficient occurs at index $s$ and is $b_{s}=1$, and the first coefficient
$b_{k}\notin\mathbb{F}_{q}$ occurs at index $t$, for which there are $q^{3}-q$
possibilities. Thus we have, after accounting for conjugating $\ell$ and
scaling at the end,

$\displaystyle N_{n,2}$ $\displaystyle=\frac{q-1}{3}\sum_{0\leq
s<n}\sum_{s<t\leq n}q^{t-s-1}(q^{3}-q)q^{2(n-t)}$
$\displaystyle=\frac{(q-1)(q^{3}-q)}{3}\sum_{0\leq
s<n}q^{n-s-1}\left(\frac{q^{n-s}-1}{q-1}\right)$
$\displaystyle=\frac{q\left(q^{2n+1}-q^{n+1}-q^{n}+1\right)}{3}.$

The value of $N_{2,3}$ also appeared implicitly in [BCF16a, §2.2.3]. For
general $n$, the calculation of $N_{n,3}$ is quite similar to that of
$N_{n,2}$, with indices $s,t$ as above, and also a first coefficient $b_{u}$
that is not contained in $\operatorname{span}_{\mathbb{F}_{q}}\\{1,b_{t}\\}$.
Thus we compute

$\displaystyle N_{n,3}$ $\displaystyle=\frac{q-1}{3}\sum_{0\leq
s<n-1}\sum_{s<t<n}\sum_{t<u\leq
n}q^{t-s-1}(q^{3}-q)q^{2(u-t-1)}(q^{3}-q^{2})q^{3(n-u)}$
$\displaystyle=\frac{(q-1)(q^{3}-q)(q^{3}-q^{2})}{3}\sum_{0\leq
s<n-1}\sum_{s<t<n}q^{2n-s-t-3}\left(\frac{q^{n-t}-1}{q-1}\right)$
$\displaystyle=\frac{(q^{3}-q)(q^{3}-q^{2})}{3}\sum_{0\leq
s<n-1}\left(\frac{q^{n-s-1}(q^{2n-2s-2}-1)}{q^{2}-1}-\frac{q^{n-s-2}(q^{n-s-1}-1)}{q-1}\right)$
$\displaystyle=\frac{q(q^{3}-q^{2})}{3}\left(\frac{q^{3}(q^{3n-3}-1)}{q^{3}-1}-\frac{q(q^{2n-2}-1)}{q-1}+\frac{q^{n-1}-1}{q-1}\right).$

Finally, we have $N_{n,0}=N_{n}-N_{n,1}-N_{n,2}-N_{n,3}$ and the stated values
of $\xi_{n,i}$ follow from $\xi_{n,i}=N_{n,i}/N_{n}$. ∎

######  Remark 3.4.

We can also interpret the counts $N_{n,i}$ for $i=1,2,3$ above in terms of the
number of $\mathbb{F}_{q}$-points on certain Grassmannians. Recall
$\operatorname{Gr}(r,k)$ is a variety whose $\mathbb{F}_{q}$-points
parametrize $r$-dimensional subspaces of $\mathbb{F}_{q}^{k}$. For $n\geq i$,
we have

$N_{n,i}=\\#\operatorname{Gr}(n+1-i,n+1)(\mathbb{F}_{q})\cdot N_{i-1,i}.$

To see this, recall from Remark 3.2 that a cubic form
$f\in\mathbb{F}_{q}[x_{0},\ldots,x_{n}]$ with factorization type $i$ has its
geometric components intersect over a codimension $i$ linear subspace of
$\mathbb{P}^{n}$ over $\mathbb{F}_{q}$. There are
$\\#\operatorname{Gr}(n+1-i,n+1)(\mathbb{F}_{q})$ choices for this subspace.
Changing coordinates, we may assume this intersection is at
$x_{0}=\cdots=x_{i-1}=0$. In order for $f=\prod\sigma(\ell)$ to vanish on this
subspace, it must involve only the coordinates $x_{0},\ldots,x_{i-1}$, and by
definition there are $N_{i-1,i}$ such $f$ with factorization type $i$.

###  3.1. Additional conditions

We continue with $f\in\mathbb{F}_{q}[x_{0},\ldots,x_{n}]$ a nonzero cubic form
and $X_{f}/\mathbb{F}_{q}$ the associated cubic hypersurface. Recalling our
notation (1.1), we write $f=\sum_{0\leq i\leq j\leq k\leq
n}a_{ijk}x_{i}x_{j}x_{k}$.

######  Definition 3.5.

We say $f$ satisfies condition (1), or the point condition, if $a_{000}\neq
0$, i.e. $X_{f}$ does not contain the $\mathbb{F}_{q}$-point $[1:0:\ldots:0]$.

######  Definition 3.6.

We say $f$ satisfies condition (2), or the line condition, if
$f(x_{0},x_{1},0,\ldots,0)$ is an irreducible binary cubic form over
$\mathbb{F}_{q}$, i.e. the intersection of $X_{f}$ with the line
$x_{2}=\ldots=x_{n}=0$ contains no $\mathbb{F}_{q}$-points.

In [BCF16a, BCF+16b], similar point and line conditions are used in recursive
computations of the probability of $p$-adic solubility for plane cubic curves
and quadratic forms. To deal with cubic hypersurfaces of higher dimension, we
find we will need an additional condition.

######  Definition 3.7.

We say $f$ satisfies condition (3), or the plane condition, if
$f(x_{0},x_{1},x_{2},0,\ldots,0)$ has factorization type 3 (the triangle
configuration), i.e. the intersection of $X_{f}$ with the plane
$x_{3}=\ldots=x_{n}=0$ contains no $\mathbb{F}_{q}$-points.

Let $\xi^{(j)}_{n,i}$ denote the probability that a nonzero cubic form
satisfying condition $(j)$ has factorization type $i$ for $1\leq i,j\leq 3$.
This is again equal to the probability (in the sense of Haar measure) that a
primitive cubic form $f\in\mathbb{Z}_{p}[x_{0},\ldots,x_{n}]$ with reduction
$\overline{f}$ satisfying condition $j$ has factorization type $i$. As before,
we further define

$\xi^{(j)}_{n,0}=1-\xi^{(j)}_{n,1}-\xi^{(j)}_{n,2}-\xi^{(j)}_{n,3},$

the probability of not being one of these three types. For $n=1,2$,
$\xi^{(j)}_{n,i}$ were computed in [BCF16a, Proposition 7].

######  Lemma 3.8.

For $n\geq 1$ and $j\leq n+1$ we have the following values for
$\xi^{(j)}_{n,i}$.

$i\backslash j$ | 1 | 2 | 3
---|---|---|---
0 | $\displaystyle 1-\frac{q^{2n}+q^{n+2}-q^{n}+2}{3q^{\binom{n+3}{3}-n-1}}$ | $\displaystyle 1-\frac{1}{q^{\binom{n+3}{3}-3n-1}}$ | $\displaystyle 1-\frac{1}{q^{\binom{n+3}{3}-3n-8}}$
1 | $\displaystyle\frac{1}{q^{\binom{n+3}{3}-n-1}}$ | 0 | 0
2 | $\displaystyle\frac{{\left(q+1\right)}{\left(q^{n+1}-1\right)}}{3q^{\binom{n+3}{3}-n-1}}$ | $\displaystyle\frac{1}{q^{\binom{n+3}{3}-2n-2}}$ | 0
3 | $\displaystyle\frac{q^{2n-1}-q^{n-1}-q^{n}+1}{3q^{\binom{n+3}{3}-n-2}}$ | $\displaystyle\frac{q^{n-1}-1}{q^{\binom{n+3}{3}-2n-2}}$ | $\displaystyle\frac{1}{q^{\binom{n+3}{3}-3n-4}}$

###### Proof.

The proof is analogous to that of Lemma 3.3. Let $N_{n}^{(j)}$ denote the
number of cubic forms $f$ in $n+1$ variables subject to condition $(j)$ and
$N_{n,i}^{(j)}$ denote the number of such forms with factorization type $i$.

Now allowing the point, line, and plane conditions to be applied, we compute
$N_{n}^{(j)}$ by recognizing that the condition imposes a constraint on the
coefficients in the first $j$ variables, with the rest free to be chosen from
$\mathbb{F}_{q}$. Explicitly, we find

$\displaystyle N_{n}^{(1)}$ $\displaystyle=(q-1)q^{\binom{n+3}{3}-1},$
$\displaystyle N_{n}^{(2)}$
$\displaystyle=N_{1,2}q^{\binom{n+3}{3}-4}=\frac{1}{3}(q-1)^{2}(q+1)q^{\binom{n+3}{3}-3},$
$\displaystyle N_{n}^{(3)}$
$\displaystyle=N_{2,3}q^{\binom{n+3}{3}-10}=\frac{1}{3}(q-1)^{3}(q+1)q^{\binom{n+3}{3}-7}.$

To compute $N_{n,i}^{(j)}$, we essentially repeat the calculations in Lemma
3.3. The condition $(j)$ has the effect of simplifying several steps, much in
the same way that it is easier to count monic polynomials than all
polynomials. When $0<i<j$ we have $N_{n,i}^{(j)}=\xi^{(j)}_{n,i}=0$, since
condition $(j)$ imposes that the coefficients of the linear form into which
$f$ factors (possibly over $\mathbb{F}_{q^{3}}$) span a $j$-dimensional
$\mathbb{F}_{q}$-vector space, ruling out type $i$. Here are the values of
$N_{n,i}^{(j)}$, for completeness.

$\displaystyle N_{n,1}^{(1)}$ $\displaystyle=(q-1)q^{\binom{n+3}{3}-1}$
$\displaystyle N_{n,2}^{(1)}$ $\displaystyle=\frac{q-1}{3}\sum_{0<t\leq
n}q^{t-1}(q^{3}-q)q^{2(n-t)}$
$\displaystyle=\frac{1}{3}q^{n}(q^{2}-1)(q^{n}-1)$ $\displaystyle
N_{n,2}^{(2)}$ $\displaystyle=\frac{1}{3}(q-1)(q^{3}-q)q^{2n-2}$
$\displaystyle N_{n,3}^{(1)}$ $\displaystyle=\frac{q-1}{3}\sum_{0<t\leq
n-1}\sum_{t<u\leq n}q^{t-1}(q^{3}-q)q^{2(u-t-1)}(q^{3}-q^{2})q^{3(n-u)}$
$\displaystyle=\frac{1}{3}q^{n+1}(q-1)(q^{2n-1}-q^{n}-q^{n-1}+1)$
$\displaystyle N_{n,3}^{(2)}$ $\displaystyle=\frac{q-1}{3}\sum_{1<u\leq
n}(q^{3}-q)q^{2(u-2)}(q^{3}-q^{2})q^{3(n-u)}$
$\displaystyle=\frac{1}{3}q^{2n-1}(q-1)(q^{2}-1)(q^{n-1}-1)$ $\displaystyle
N_{n,3}^{(3)}$ $\displaystyle=\frac{1}{3}q^{3n-3}(q-1)^{3}(q+1)$

We then compute $N_{n,0}^{(j)}$ as before and take the ratios with
$N_{n}^{(j)}$ to obtain the stated values. ∎

######  Remark 3.9.

When $n=0$, we have $\xi_{0,1}^{(1)}=1$ and $\xi_{0,i}^{(1)}=0$ for $i=0,2,3$.
When $i=2$ this is not evident from the value presented in Lemma 3.8 due to
extra cancellation. From the proof, however, we do see $N_{0,2}^{(1)}=0$.

##  4\. Lifting probabilities that are 1

Let $f\in\mathbb{Z}_{p}[x_{0},\ldots,x_{n}]$ be a primitive cubic form. We can
characterize those $f$ for which $X_{f}$ has no $p$-adic points as follows.

######  Proposition 4.1.

Let $n\geq 1$ and suppose $X_{f}(\mathbb{Q}_{p})=\emptyset$. Then
$\overline{f}$ has factorization type 1, 2, or 3 described above in Definition
3.1

The proof will proceed by induction on $n$. We will need an intermediate
result for the inductive step.

######  Lemma 4.2.

Let $n\geq 3$ and suppose $f\in\mathbb{F}_{q}[x_{0},\ldots,x_{n}]$ is a
nonzero cubic form. Suppose that for all hyperplanes $H\subset\mathbb{P}^{n}$
defined over $\mathbb{F}_{q}$, we have either that $H$ is an irreducible
component of $X_{f}$ or $X_{f}\cap H\subset\mathbb{P}^{n-1}$ is the union of
conjugate hyperplanes over $\mathbb{F}_{q^{3}}$. Then $f$ has type 1, 2, or 3.

###### Proof.

By Chevalley’s theorem, there exists a point $P\in X_{f}(\mathbb{F}_{q})$.
After a linear change of coordinates on $\mathbb{P}^{n}$, we may assume
$P=[0:\ldots:0:1]$. Such a change of coordinates may change whether a given
hyperplane $H$ is a component of $X_{f}$, or the type of $X_{f}\cap H$, but
the hypothesis still holds, and $f$ has type $i=1,2,3$ if and only if it does
after the change of coordinates.

Let $H_{i}\colon x_{i}=0$ denote a coordinate hyperplane and let
$f_{i}=f(x_{0},\ldots,x_{i-1},0,x_{i+1},\ldots,x_{n})$ be the (possibly zero)
cubic form cutting.out the intersection $X_{f_{i}}=X_{f}\cap H_{i}\subseteq
H_{i}\simeq\mathbb{P}^{n-1}$. Note that $H_{i}$ is an irreducible component of
$X_{f}$ if and only if $f_{i}=0$, or equivalently $x_{i}\mid f$. If $f_{i}\neq
0$ then $X_{f_{i}}$ is a configuration of conjugate hyperplanes, i.e. $f_{i}$
has factorization type 1, 2, or 3 from Definition 3.1.

We first consider the case where $f_{i}=0$ for some $i$. Since $n\geq 3$,
there is at least one $j$ such that $f_{j}\neq 0$, hence we have $x_{i}\mid
f_{j}$. Since $f_{j}$ is the product of conjugate linear forms, we must have
$f_{j}=a_{iii}x_{i}^{3}$ (recall $a_{ijk}$ is the coefficient of
$x_{i}x_{j}x_{k}$ in $f$ as in (1.1)). In particular, this implies
$x_{j^{\prime}}\nmid f$ for all $j^{\prime}\neq i,j$, and for any such
$j^{\prime}$ we have $f_{j^{\prime}}=a_{iii}x_{i}^{3}$. From this we deduce

$f=a_{iii}x_{i}^{3}+x_{i}x_{j}g_{j}=a_{iii}x_{i}^{3}+x_{i}x_{j^{\prime}}g_{j^{\prime}}$

for linear forms $g_{j},g_{j^{\prime}}\in\mathbb{F}_{q}[x_{0},\ldots,x_{n}]$.
This forces $f=a_{iii}x_{i}^{3}+a_{ijj^{\prime}}x_{i}x_{j}x_{j^{\prime}}$.
Then there exists $k\notin\\{i,j,j^{\prime}\\}$ such that $f=f_{k}\neq 0$ has
factorization type 1, 2, or 3.

Suppose now that $f_{i}\neq 0$ for all $i$. For $0\leq i<n$ we have
$f_{i}=\prod_{\sigma\in\operatorname{Gal}(\mathbb{F}_{q^{3}}/\mathbb{F}_{q})}\sigma(\ell_{i})$
for a nonzero linear form $\ell_{i}\in\mathbb{F}_{q^{3}}[x_{0},\ldots,x_{n}]$.
Since $P\in X_{f_{i}}$ we must have $\ell_{i}(0,\ldots,0,1)=0$ and thus the
$x_{n}$-coefficient of $\ell_{i}$ vanishes. Using this, we claim that
$a_{ijn}=0$ for all $i,j\in\\{0,\ldots,n\\}$. For any $i,j$, there exists
$k\in\\{0,\ldots,n\\}-\\{i,j,n\\}$ and the $x_{i}x_{j}x_{n}$ terms of $f$ and
$f_{k}$ coincide. But
$f_{k}=\prod_{\sigma\in\operatorname{Gal}(\mathbb{F}_{q^{3}}/\mathbb{F}_{q})}\sigma(\ell_{k})$
for $\ell_{k}$ with zero $x_{n}$-coefficient, and thus $a_{ijn}=0$. We
conclude that no terms containing $x_{n}$ show up in $f$, and therefore
$f=f_{n}$ has type 1, 2, or 3, as desired. ∎

######  Remark 4.3.

Lemma 4.2 fails to hold when $n=2$, as $f=x_{0}x_{1}x_{2}$ is readily seen to
be a counterexample. There also exist smooth and geometrically irreducible
plane curves over $\mathbb{F}_{q}$ whose coordinate cubic forms are all type 1
or 2 (type 3 cannot occur for binary cubic forms). For example, when $3\nmid
q$ this is the case for diagonal plane cubics of the form $X\colon
c_{0}x_{0}^{3}+c_{1}x_{1}^{3}+c_{2}x_{2}^{3}=0$ if $c_{0},c_{1},c_{2}$
represent distinct classes in
$\mathbb{F}_{q}^{\times}/(\mathbb{F}_{q}^{\times})^{3}$.

###### Proof of Proposition 4.1.

For $n=1$, the case of primitive binary cubic forms
$f\in\mathbb{Z}_{p}[x_{0},x_{1}]$, we observe that factorization types 1 and 2
correspond to to $\overline{f}$ having a linear factor of multiplicity 3 and
$\overline{f}$ being irreducible over $\mathbb{F}_{p}$, respectively. If
$\overline{f}$ has neither of these factorization types, then it has a linear
factor of multiplicity 1 and so must $f$ by Hensel’s lemma.

For $n\geq 2$ we proceed by induction on $n$. When $n=2$, the case of plane
cubic curves, if $\overline{X_{f}}$ is smooth and geometrically irreducible
then by an application of the Hasse–Weil bound on $\overline{X_{f}}$ we have

$\\#\overline{X_{f}}(\mathbb{F}_{p})\geq p+1-2\sqrt{p}>0.$

In particular, there exists a smooth $\mathbb{F}_{p}$-point to lift via
Hensel’s lemma to a point in $X_{f}(\mathbb{Q}_{p})$. If $\overline{X_{f}}$ is
geometrically irreducible but not smooth, its normalization
$\widetilde{\overline{X_{f}}}$ is a smooth genus zero curve over
$\mathbb{F}_{p}$ with $p+1$ $\mathbb{F}_{p}$-points. At most 2 of those points
map to singular points of $\overline{X_{f}}$, so there are at least $p-1>0$
smooth $\mathbb{F}_{p}$-points on $\overline{X_{f}}$ which may be lifted to
$X_{f}(\mathbb{Q}_{p})$ via Hensel’s Lemma.

This leaves only the geometrically reducible cases. When $\overline{X_{f}}$
has a reduced line defined over $\mathbb{F}_{p}$ as a component, it contains
$p+1$ $\mathbb{F}_{p}$-points; again, at most two of these intersect another
component, leaving at least $p-1>0$ smooth points which may be lifted via
Hensel’s lemma. The only remaining possibilities lacking a reduced line
defined over $\mathbb{F}_{p}$ as an irreducible component are precisely types
1, 2, and 3 of Definition 3.1, concluding the proof for $X_{f}$ a plane curve.

Suppose now that $n>2$ and assume the conclusion holds for $n-1$. Since
$X_{f}(\mathbb{Q}_{p})=\emptyset$ then for all hyperplanes
$H\subset\mathbb{P}^{n}$ defined over $\mathbb{Q}_{p}$ we have $(X_{f}\cap
H)(\mathbb{Q}_{p})=\emptyset$. Fixing a model for $H$ over $\mathbb{Z}_{p}$
and reducing modulo $p$, we have either $\overline{(X_{f}\cap
H)}=\overline{H}$ or $\overline{(X_{f}\cap
H)}\subsetneq\overline{H}\simeq\mathbb{P}^{n-1}_{\mathbb{F}_{q}}$ is a union
of conjugate hyperplanes defined over $\mathbb{F}_{p^{3}}$ by the inductive
hypothesis. Invoking Lemma 4.2, $\overline{f}$ must have type 1, 2, or 3. ∎

######  Remark 4.4.

Kopparty and Yekhanin [KY08, Lemma 3.2] show that for $d$ prime,
$n\geq\frac{d}{2}+1$, and $q\geq 32n^{4}$, if a degree $d$ form
$f\in\mathbb{F}_{q}[x_{0},\ldots,x_{n}]$ has no $\mathbb{F}_{q}$-solutions
(other than $x_{0}=\ldots=x_{n}=0$), then there exists a linear form $\ell$
such that
$f=\prod_{\sigma\in\operatorname{Gal}(\mathbb{F}_{q^{d}}/\mathbb{F}_{q})}\sigma\ell$.
Their proof invokes the Chevalley–Warning theorem as well as the Weil
conjectures.

An immediate consequence of Proposition 4.1 is that if the reduction
$\overline{f}$ of a primitive cubic form
$f\in\mathbb{Z}_{p}[x_{0},\ldots,x_{n}]$ does not have type 1, 2, or 3, then
$X_{f}(\mathbb{Q}_{p})\neq\emptyset$. In other words, the probability of
$X_{f}$ having a $\mathbb{Q}_{p}$ point, given this condition on
$\overline{f}$, is 1:

$\frac{\mu_{p}\left(\left\\{f\in\mathbb{Z}_{p}[x_{0},\ldots,x_{n}]:f\text{
primitive cubic form},\ \overline{f}\text{ not type }1,\ 2,\ 3,\ \text{and
}X_{f}(\mathbb{Q}_{p})\neq\emptyset\right\\}\right)}{\mu_{p}\left(\left\\{f\in\mathbb{Z}_{p}[x_{0},\ldots,x_{n}]:f\text{
primitive cubic form},\ \overline{f}\text{ not type }1,\ 2,\
3\right\\}\right)}=1,$

where $\mu_{p}$ denotes the normalized $p$-adic Haar measure as usual.

##  5\. Lifting probabilities that are not 1

Let us return to the challenge of computing $\rho_{3,n}(p)$, the probability
that a nonzero cubic form over $\mathbb{Z}_{p}$ in $n+1$ variables has a
solution, which we denote hereafter by $\rho_{n}(p)$, suppressing the degree,
to streamline the notation. In this section we drop the dependence on $p$ for
our notation, writing $\rho_{n}$ for the local probability $\rho_{n}(p)$.

In the previous section, we determined that when $\overline{f}$ does not have
type 1, 2, or 3, $X_{f}(\mathbb{Q}_{p})$ is nonempty. The goal of this section
is to complete the computation of $\rho_{n}(p)$, beginning with defining
lifting probabilities conditional on $\overline{f}$ having type $i=1,2,3$.

######  Definition 5.1.

Let $\sigma_{n,i}$ denote the probability (in the sense of normalized Haar
measure, as usual) that a primitive cubic form $f$ has a
$\mathbb{Q}_{p}$-solution, given that $\overline{f}$ has factorization type
$i$,

$\sigma_{n,i}=\frac{\mu_{p}\left(\\{f\in\mathbb{Z}_{p}[x_{0},\ldots,x_{n}]:f\text{
primitive cubic form},\ \overline{f}\text{ has type }i,\
X_{f}(\mathbb{Q}_{p})\neq\emptyset\\}\right)}{\mu_{p}\left(\\{f\in\mathbb{Z}_{p}[x_{0},\ldots,x_{n}]:f\text{
primitive cubic form},\ \overline{f}\text{ has type }i\\}\right)}.$

By Proposition 4.1 and the definition of $\xi_{n,i}$, we have

(5.1)
$\rho_{n}=\xi_{n,0}+\xi_{n,1}\sigma_{n,1}+\xi_{n,2}\sigma_{n,2}+\xi_{n,3}\sigma_{n,3}.$

By reducing the problem to computing $\sigma_{n,i}$, we obtain some additional
structure in being able to write our cubic forms as the product of conjugate
linear forms. Namely, we will make key changes of variables that allow
$\sigma_{n,i}$ to be related to new conditional lifting probabilities of cubic
forms in fewer variables. This process will then be iterated until we have
enough relations to solve for all the lifting probabilities.

Before embarking on this journey, let us define a few additional
probabilities. Let $\rho_{n}^{(j)}$ denote the probability that a nonzero
cubic form over $\mathbb{Z}_{p}$ in $n+1$ variables satisfying condition $(j)$
has a $\mathbb{Q}_{p}$-solution. We then have the analogue of (5.1),

(5.2)
$\rho_{n}^{(j)}=\xi^{(j)}_{n,0}+\sum_{i=1,2,3}\xi^{(j)}_{n,i}\sigma_{n,i}.$

######  Remark 5.2.

In the $n=2$ case, Bhargava, Cremona, and Fisher [BCF16a] determine

$\rho_{2}=1-\frac{p^{9}-p^{8}+p^{6}-p^{4}+p^{3}+p^{2}-2p+1}{3(p^{2}+1)(p^{4}+1)(p^{6}+p^{3}+1)},\text{
so }1-\rho_{2}\sim\frac{1}{3p^{3}}.$

Their approach takes advantage of the fact that plane curves reducing to a
“triangle” configuration, i.e. when $\overline{f}$ has factorization type 3,
have no solutions, i.e. $\sigma_{2,3}=0$. While they employ similar changes of
variables and overall strategy, this allows for a considerably simpler web of
relations. Our approach is not hindered by allowing $n=2$, so we will not
exclude this case; the reader may view our results in the $n=2$ case as a
repackaging of their result.

There are some differences between the two papers worth briefly highlighting.
One minor difference comes from our decision for $\xi_{n,i}$ to be the
probability that a nonzero form up to scaling possesses factorization type
$i$. Others appear to be merely notational (e.g. they denote with
$\beta_{1}^{\prime\prime}$ what we denote by $\xi_{2,2}^{(1)}$), but we find
substantial utility in being able to easily index both the factorization types
$i=1,2,3$ and the conditions $(j)$.

### Valuation tables

Here we introduce a bookkeeping tool that will be employed extensively in the
remainder of this section, which we refer to as a valuation table for a
$p$-adic cubic form $f\in\mathbb{Z}_{p}[x_{0},\ldots,x_{n}]$. This table
records known information about the $p$-adic valuations of the coefficients of
$f$, streamlining how we keep track of the recursive process through which we
are able to compute $\rho_{n},\rho_{n}^{(j)},\sigma_{n,i},\sigma_{n,i}^{(j)}$,
and other lifting probabilities we will soon define.

Similar constructions appeared in the calculation of $\rho_{2}$ [BCF16a, e.g.
Lemma 12], as well as in that of the density of soluble quadratic forms
[BCF+16b, Lemma 3.3]. To handle cubics in more than three variables while
keeping things from getting (too) out of hand, we employ a blocking strategy
which allows our arguments to be essentially uniform in $n$.

Suppose we partition the set of variables $\\{x_{0},\ldots,x_{n}\\}$ into $r$
(nonempty) subsets. Since for our purposes we only need $r\leq 4$, we label
them by $S,T,U,W$ to avoid excessive decoration. Generally, each subset will
consist of consecutively indexed variables, e.g.
$S=\\{x_{0},\ldots,x_{i-1}\\}$ and $T=\\{x_{i},\ldots,x_{n}\\}$.

This induces a partition of the set of cubic monomials
$\\{x_{i}x_{j}x_{k}:0\leq i\leq j\leq k\leq n\\}$ into subsets of the form

$S^{s}T^{t}U^{u}W^{w}=S^{s}\times T^{t}\times U^{u}\times W^{w}\text{ such
that }s+t+u+w=3,$

that is, those monomials $x_{i}x_{j}x_{k}$ for which exactly $s$ of
$x_{i},x_{j},x_{k}$ lies in $S$, exactly $t$ lie in $T$, etc. When $r<4$, we
drop the extraneous empty subset $W$, and possibly also $U$, from the notation
altogether.

######  Definition 5.3.

Let

$v_{stuw}=\min\\{v_{p}(a_{ijk}):x_{i}x_{j}x_{k}\in S^{s}T^{t}U^{u}W^{w}\\}.$

A valuation table for $\boldsymbol{f}$ is a collection of arrays, indexed by
$w$, that records information about $v_{stuw}$ in the $u$-th row and $t$-th
column.

This is perhaps best seen by illustration for $r=2,3,4$ separately.

######  Example 5.4.

Suppose $r=2$, so $\\{x_{0},\ldots,x_{n}\\}=S\coprod T$. Let

$v_{st}=\min\\{v_{p}(a_{ijk}):x_{i}x_{j}x_{k}\in S^{s}T^{t}\\}.$

In this case, the valuation table is a single row, with columns indexed by the
value of $t$ starting with $t=0$ on the left and $t=3$ on the right.

Table 5.1. The shape of a valuation table for a partition with $r=2$ $*$ | $*$ | $*$ | $*$
---|---|---|---

######  Example 5.5.

Suppose $r=3$, so $\\{x_{0},\ldots,x_{n}\\}=S\coprod T\coprod U$. Let

$v_{stu}=\min\\{v_{p}(a_{ijk}):x_{i}x_{j}x_{k}\in S^{s}T^{t}U^{u}\\}.$

Now the valuation table forms a triangle, with the entry in the $u$-th row and
$t$-th column (indexing by zero, so the top left entry is $u=t=0$, the bottom
left entry is $(u,t)=(3,0)$, the top right is $(u,t)=(0,3)$, etc.) corresponds
to information known about $v_{stu}$ with $s=3-t-u$.

Table 5.2. The shape of a valuation table for a partition with $r=3$ $*$ | $*$ | $*$ | $*$
---|---|---|---
$*$ | $*$ | $*$ |
$*$ | $*$ |  |
$*$ |  |  |

######  Example 5.6.

Suppose $r=4$, so $\\{x_{0},\ldots,x_{n}\\}=S\coprod T\coprod U\coprod W$. Let

$v_{stuw}=\min\\{v_{p}(a_{ijk}):x_{i}x_{j}x_{k}\in S^{s}T^{t}U^{u}W^{w}\\}.$

We would like to represent information about $v_{stuw}$ along lattice points
forming a tetrahedron, but we will settle for “slicing” the tetrahedron at
each value of $w$.

That is, the information about $v_{stuw}$ is recorded in the $u$-th row and
$t$-th column of the $w$-th slice (again indexing by zero as in the previous
example), with $s=3-t-u-w$, of the table below.

Table 5.3. The shape of a valuation table for a partition with $r=4$ $w=0$ | $w=1$ | $w=2$ | $w=3$
---|---|---|---
$*$ | $*$ | $*$ | $*$ | $*$ | $*$ | $*$ | $*$ | $*$ | $*$
$*$ | $*$ | $*$ |  | $*$ | $*$ |  | $*$ |  |
$*$ | $*$ |  |  | $*$ |  |  |  |  |
$*$ |  |  |  |  |  |  |  |  |

A corner entry in a valuation table corresponds to (exactly) one of $s,t,u,w$
equal to 3. For example, if $s=3$, the polynomial
$h_{S}=\sum_{x_{i},x_{j},x_{k}\in S}a_{ijk}x_{i}x_{j}x_{k}$ corresponds to the
cubic form in $\\#S$ variables obtained from $f$ by setting $x_{i}=0$ for all
$x_{i}\in T,U,W$. The other corner entries similarly represent cubic forms in
only the variables in $T,U,W$, etc. Note that when $r=4$, the single entry in
the $w=3$ slice is a corner, but none of the entries in the $w=1,2$ slices are
corners.

We will occasionally decorate the corner entries to keep track of additional
information. A subscript $*_{i}$ for $i=1,2,3$ indicates reduction type $i$:
that is, if the $s=3$ corner entry is decorated as such, then the cubic form
$h_{S}$ described above has reduction type $i$ after dividing out by any
common factors of $p$.

A superscript $*^{(j)}$ for $j=1,2,3$ indicates condition $(j)$: again using
$s=3$, this indicates that $h_{S}$ above satisfies the point, line, or plane
condition ($j=1,2,3$ respectively). Note that we are assuming an ordering on
the variables within $S$ (or $T,U,W$) in declaring that $h_{S}$ satisfies
condition $(j)$. We will always mean this to be with the variables in
increasing order. For example, if $S=\\{x_{0},\ldots,x_{i-1}\\},\
T=\\{x_{i},\ldots,x_{n}\\}$, and the $t=3$ corner entry in a valuation table
reads $=0^{(j)}$, it indicates that the cubic form

$h_{T}(x_{i},\ldots,x_{n})=f(0,\ldots,0,x_{i},\ldots,x_{n})$

is nonvanishing modulo $p$ and satisfies condition $(j)$. That is, after
setting all but its first $j$ variables $x_{i},\ldots,x_{i+j-1}$ to zero, we
only have the trivial solution modulo $p$,

$\overline{h_{T}}(x_{i},\ldots,x_{i+j-1},0,\ldots,0)=0\implies
x_{i}=\ldots=x_{i+j-1}=0.$

###  5.1. Phase I

We first set out to compute $\sigma_{n,i}$. In this first phase of
computation, we will employ key changes of variables which will become a
recurring theme in later phases. The idea is that when $f$ has factorization
type $i$, we can produce another $p$-adic cubic form $f_{\mathrm{I}}$ such
that $X_{f}(\mathbb{Q}_{p})$ and $X_{f_{\mathrm{I}}}(\mathbb{Q}_{p})$ are in
bijection, with $f_{\mathrm{I}}$ taking a shape particularly amenable to
further analysis.

Suppose $f\in\mathbb{Z}_{p}[x_{0},\ldots,x_{n}]$ is a cubic form with
factorization type $i$. After scaling, we may assume $f$ is primitive, i.e.
$p$ does not divide all coefficients. Recall that
$\overline{f}=\prod_{\sigma\in\operatorname{Gal}(\mathbb{F}_{p^{3}}/\mathbb{F}_{p})}\sigma(b_{0}x_{0}+\ldots+b_{n}x_{n})$
where $\dim\operatorname{span}\\{b_{j}\\}_{j=0}^{n}=i$. After a linear change
of the coordinates $x_{j}$, we may assume that $\\{b_{j}\\}_{j=0}^{n}$ is
spanned by its first $i$ elements $b_{0},\ldots,b_{i-1}$, and thus also that
$b_{j}=0$ for all $j\geq i$.

Translating things back to $f$, after the aforementioned change of coordinates
we have

$f=g(x_{0},\ldots,x_{i-1})+p\sum_{0\leq j\leq
k<i}\ell_{jk}(x_{i},\ldots,x_{n})x_{j}x_{k}+p\sum_{0\leq
j<i}q_{j}(x_{i},\ldots,x_{n})x_{j}+ph(x_{i},\ldots,x_{n})$

where $\ell_{jk}$, $q_{j}$, and $h$ are linear, quadratic, and cubic forms in
$n+1-i$ variables over $\mathbb{Z}_{p}$. This situation can be compactly
described by taking $S=\\{x_{0},\ldots,x_{i-1}\\}$ and
$T=\\{x_{i},\ldots,x_{n}\\}$ and forming the valuation table of $f$ below in
Table 5.4.

Table 5.4. Valuation table for $f$ $=0_{i}$ | $\geq 1$ | $\geq 1$ | $\geq 1$
---|---|---|---

Suppose $[x_{0}:\ldots:x_{n}]\in X_{f}(\mathbb{Q}_{p})$. Throughout, after
clearing denominators, we may assume $x_{0},\ldots,x_{n}\in\mathbb{Z}_{p}$ and
share no common factor of $p$. We claim that $p\mid x_{0},\ldots,x_{i-1}$.
Reducing modulo $p$, we have that $\overline{g}(x_{0},\ldots,x_{i-1})=0$ is
necessary. However, since $\overline{g}=\overline{f}$ has factorization type
$i$, the only such solution is $x_{0}=\ldots=x_{i-1}=0\pmod{p}$. Indeed,

  * •

when $i=1$ we have $g(x_{0})=ax_{0}^{3}$ for some
$a\in\mathbb{Z}_{p}^{\times}$;

  * •

when $i=2$, $\overline{g}$ is an irreducible binary cubic form over
$\mathbb{F}_{p}$ and hence has no nontrivial solutions;

  * •

when $i=3$ we have $\overline{g}$ is a ternary cubic form in the “triangle”
configuration and hence has no nontrivial solutions.

This motivates another change of coordinates, this time replacing
$x_{0},\ldots,x_{i-1}$ by $px_{0},\ldots,px_{i-1}$ and dividing by $p$,

(5.3) $\displaystyle f_{\mathrm{I}}$
$\displaystyle=\frac{1}{p}f(px_{0},\ldots,px_{i-1},x_{i},\ldots,x_{n})$
$\displaystyle=p^{2}g(x_{0},\ldots,x_{i-1})+p^{2}\sum_{0\leq j\leq
k<i}\ell_{jk}(x_{i},\ldots,x_{n})x_{j}x_{k}+p\sum_{0\leq
j<i}q_{j}(x_{i},\ldots,x_{n})x_{j}+h_{\mathrm{I}}(x_{i},\ldots,x_{n}),$

and we have a bijection between $X_{f}(\mathbb{Q}_{p})$ and
$X_{f_{\mathrm{I}}}(\mathbb{Q}_{p})$. We pause to record how this change of
variables affects the valuations of coefficients of $f_{\mathrm{I}}$ below in
Table 5.5, opting to lean on this notation as things get more involved.

Table 5.5. Valuation table for $f_{\mathrm{I}}$ $=2_{i}$ | $\geq 2$ | $\geq 1$ | $\geq 0$
---|---|---|---

We are now ready to define additional lifting probabilities and relate them to
$\sigma_{n,i}$.

######  Definition 5.7.

Let $\tau_{n,ij}$ denote the probability that $f$ has a $p$-adic solution,
given that $f$ has type $i$ and after the transformations described above,
$f_{\mathrm{I}}$ is primitive and has type $j$.

######  Definition 5.8.

Let $\sigma_{n,i}^{\prime}$ denote the probability that $f$ has a $p$-adic
solution, given that $f$ has type $i$ and after the transformations described
above, the coefficients of $f_{\mathrm{I}}$ have minimal $p$-adic valuation at
least 1.

######  Lemma 5.9.

Let $n\geq 1$ and $i\in\\{1,2,3\\}$. If $i=n+1$ we have

$\sigma_{1,2}=\sigma_{2,3}=0.$

For $i<n+1$, $\sigma_{n,i}$ satisfies the relation

(5.4)
$\sigma_{n,i}=\left(1-\frac{1}{p^{\binom{n-i+3}{3}}}\right)\left(\xi_{n-i,0}+\sum_{1\leq
j\leq
3}\xi_{n-i,j}\tau_{n,ij}\right)+\frac{1}{p^{\binom{n-i+3}{3}}}\sigma_{n,i}^{\prime}.$

###### Proof.

That $\sigma_{1,2}=\sigma_{2,3}=0$ is merely the observation that if a binary
(resp. ternary) cubic form $f$ has type 2 (resp. type 3), then its reduction
contains no $\mathbb{F}_{p}$-points, and hence $X_{f}$ can have no $p$-adic
points.

Excluding $(n,i)=(1,2),(2,3)$ and given a polynomial $f$ with factorization
type $i$, we perform the transformations above to obtain $f_{\mathrm{I}}$ with
valuation table given by Table 5.5. Note that
$\overline{f_{\mathrm{I}}}=\overline{h_{\mathrm{I}}}$, so (provided
$\overline{h_{\mathrm{I}}}\neq 0$) if
$X_{f}(\mathbb{Q}_{p})=X_{f_{\mathrm{I}}}(\mathbb{Q}_{p})=\emptyset$, then by
Proposition 4.1 we have $\overline{h_{\mathrm{I}}}$ has factorization type 1,
2, or 3.

Since $h_{\mathrm{I}}$ is a general cubic form in $n-i+1$ variables, it has
$\binom{n-i+3}{3}$ coefficients, hence the probability that $h_{\mathrm{I}}$
is primitive is given by $1-p^{\binom{n-i+3}{3}}$. If $h_{\mathrm{I}}$ is not
primitive, then by definition the lifting probability is given by
$\sigma_{n,i}^{\prime}$. If $h_{\mathrm{I}}$ is primitive, then $\xi_{n-i,j}$
represents the probability that $h_{\mathrm{I}}$ has factorization type $j$
(recall $j=0$ means that $h_{\mathrm{I}}$ does not have types 1, 2, or 3). For
$j\neq 0$, we again find by definition that $\tau_{n,ij}$ is the probability
of a lift.

In the $j=0$ case, $\overline{h_{\mathrm{I}}}$ does not have type 1, 2, or 3,
so it has a nontrivial solution $h_{\mathrm{I}}(x_{i},\ldots,x_{n})\equiv
0\pmod{p}$ which lifts to a $p$-adic solution. From this it follows that
$f_{\mathrm{I}}(0,\ldots,0,x_{i},\ldots,x_{n})\equiv 0\pmod{p}$ lifts to a
$p$-adic point on $X_{f_{\mathrm{I}}}$. Thus in this case the lifting
probability is 1. Putting this together with our previous observations yields
(5.4). ∎

######  Remark 5.10.

We should point out that $\tau_{n,ij}$ is not well defined when $i+j>n+1$,
since the polynomial $h_{\mathrm{I}}$ in $n-i+1$ variables cannot have
factorization type $j>n-i+1$. However, in these cases we have $\xi_{n-i,j}=0$,
so (5.4) is still well defined. We will therefore sweep this abuse of notation
under the rug, both here and in similar future instances.

It will be useful to define additional probabilities contingent on whether or
not $h_{\mathrm{I}}$ satisfies the point, line, or plane condition.

######  Definition 5.11.

Let $\sigma_{n,i}^{(k)}$ denote the probability that $f$ has a $p$-adic
solution, given that $f$ has type $i$ and after the transformations described
above, $f_{\mathrm{I}}$ satisfies condition $(k)$, i.e.
$\overline{f_{\mathrm{I}}}(x_{i},\ldots,x_{i+k-1},0,\ldots,0)$ has no
nontrivial solutions.

Thus $\sigma_{n,i}^{(k)}$ describes the lifting probability when $f$ has type
$i$ and $f_{\mathrm{I}}$ has the valuation table given in Table 5.6 where
$h_{\mathrm{I}}$ satisfies condition $(k)$.

Table 5.6. Valuation table for $f_{\mathrm{I}}$ where $\overline{f_{\mathrm{I}}}$ satisfies condition $(k)$ $=2_{i}$ | $\geq 2$ | $\geq 1$ | $=0^{(k)}$
---|---|---|---

Next we give an analogue of Lemma 5.9. The proof is entirely analogous — in
fact it is slightly simpler in that $f_{\mathrm{I}}$ is known to be primitive
since it satisfies condition $(k)$ — so we omit it.

######  Lemma 5.12.

Let $n\geq 2$ and $1\leq i,k\leq 3$. Then $\sigma_{n,i}^{(k)}$ satisfies the
relation

(5.5) $\sigma_{n,i}^{(k)}=\xi_{n-i,0}^{(k)}+\sum_{1\leq j\leq
3}\xi_{n-i,j}^{(k)}\tau_{n,ij}.$

We conclude this phase by computing a relation for $\sigma_{n,i}^{\prime}$.
This offers a first glimpse into the interplay between the various lifting
probabilities we have defined so far, and a glimmer of hope that in continuing
on to phases II and III, we will actually accumulate enough relations to solve
for $\rho_{n}$.

######  Lemma 5.13.

Let $n\geq 2$ and $1\leq i\leq 3$ and $(n,i)\neq(2,3)$. Then
$\sigma_{n,i}^{\prime}$ satisfies the relation

(5.6)
$\sigma_{n,i}^{\prime}=1-\frac{1}{p^{i\binom{n-i+2}{2}}}+\frac{1}{p^{i\binom{n-i+2}{2}}}\left(\left(1-\frac{1}{p^{\binom{n-i+3}{3}}}\right)\left(\xi_{n-i,0}+\sum_{1\leq
j\leq
3}\xi_{n-i,j}\sigma_{n,j}^{(i)}\right)+\frac{1}{p^{\binom{n-i+3}{3}}}\rho_{n}^{(i)}\right).$

###### Proof.

The starting point for $\sigma_{n,i}^{\prime}$ is a cubic form $f$ from which
$f_{\mathrm{I}}$ is produced via the phase I transformations (5.3). If
$h_{\mathrm{I}}$ is not primitive, then the valuation table of
$\frac{1}{p}f_{\mathrm{I}}$ is given below.

Table 5.7. Valuation table for $\frac{1}{p}f_{\mathrm{I}}$ $=1_{i}$ | $\geq 1$ | $\geq 0$ | $\geq 0$
---|---|---|---

Reducing modulo $p$, we have

$\overline{\frac{1}{p}f_{\mathrm{I}}}=\sum_{0\leq
j<i}\overline{q_{j}}(x_{i},\ldots,x_{n})x_{j}+\overline{\frac{1}{p}h_{\mathrm{I}}}(x_{i},\ldots,x_{n}).$

Suppose for some $0\leq j<i$ the quadratic form $q_{j}$ is nonvanishing mod
$p$. Then there exists some (nontrivial) input $(x_{i},\ldots,x_{n})$ for
which $q_{j}(x_{i},\ldots,x_{n})\not\equiv 0\pmod{p}$. Then
$\overline{\frac{1}{p}f_{\mathrm{I}}}$ is linear in $x_{j}$, so it has a
solution which lifts to a $p$-adic solution, producing a point in
$X_{f}(\mathbb{Q}_{p})$.

Suppose instead that all $q_{j}$ fail to be primitive. Counting coefficients
of each $q_{j}$, this happens with probability $1/p^{i\binom{n-i+2}{2}}$. Now
we turn our attention to the factorization type of $h_{\mathrm{I}}/p$. As a
cubic form in $n-i+3$ variables, it fails to be primitive with probability
$1/p^{\binom{n-i+3}{3}}$, in which case the valuation table for
$f_{\mathrm{I}}/p$ is given by Table 5.8 below.

Table 5.8. Valuation table for $\frac{1}{p}f_{\mathrm{I}}$ $=1_{i}$ | $\geq 1$ | $\geq 1$ | $\geq 1$
---|---|---|---

Here we can divide again by $p$, returning us to the case of a general cubic
form in $n+1$ variables satisfying condition $(i)$, since
$\overline{\frac{1}{p^{2}}f_{\mathrm{I}}}(x_{0},\ldots,x_{i-1},0,\ldots,0)$
has no solutions. Thus in the case that $h_{\mathrm{I}}/p$ fails to be
primitive, the lifting probability is given by $\rho_{n}^{(i)}$.

If $h_{\mathrm{I}}/p$ is primitive (which occurs with probability
$1-1/p^{\binom{n-i+3}{3}}$, then its reduction has factorization type
$j=1,2,3$ with probability $\xi_{n-i,j}$, and none of those with probability
$\xi_{n-i,0}$. In the latter case, a liftable solution exists by the same
argument as in Lemma 5.9 using Proposition 4.1.

In the former case, we claim the probability of a liftable solution is given
by $\sigma_{n,j}^{(i)}$. To realize this, we first carry out a change of
coordinates in $x_{i},\ldots,x_{n}$ isolating the first $j$ variables of
$h_{\mathrm{I}}/p$. Repartitioning the set of variables into
$S=\\{x_{0},\ldots,x_{i-1}\\}$, $T=\\{x_{i},\ldots,x_{j-1}\\}$, and
$U=\\{x_{j},\ldots,x_{n}\\}$, we obtain the left side of the valuation table
below suggesting a natural reindexing of the variables to form the table on
the right side.

Table 5.9. Valuation tables for $\frac{1}{p}f_{\mathrm{I}}$ after isolating $x_{i},\ldots,x_{i+j-1}$ in $h_{\mathrm{I}}/p$ and after renumbering $S$ |  |  |  |  | $T$ |  | $T$ |  |  |  |  | $S\cup U$
---|---|---|---|---|---|---|---|---|---|---|---|---
| $=1_{i}$ | $\geq 1$ | $\geq 1$ | $=0_{j}$ |  |  |  | $=0_{j}$ | $\geq 1$ | $\geq 1$ | $\geq 1^{(i)}$ |
| $\geq 1$ | $\geq 1$ | $\geq 1$ |  |  | $\xrightarrow{\text{reindex}}$ |  |  |  |  |  |
| $\geq 1$ | $\geq 1$ |  |  |  |  |  |  |  |  |  |
| $\geq 1$ |  |  |  |  |  |  |  |  |  |  |
$U$ |  |  |  |  |  |  |  |  |  |  |  |

From Table 5.9 we see that after reindexing and performing the usual phase I
transformations (5.3) on the resulting cubic form, we land precisely in the
situation of $\sigma_{n,j}^{(i)}$. Putting together these observations yields
(5.6), completing the proof of the lemma. ∎

######  Example 5.14 (binary cubic forms).

We can illustrate the approach of phase I by computing $\rho_{1}$, the density
of $p$-adic binary cubic forms with at least one root, which can be deduced
from the existing literature; see e.g. [BCFG22, §1.2.3].

We have $\xi_{1,3}=\xi^{(1)}_{1,3}=0$ by Lemmas 3.3 and 3.8 and
$\sigma_{1,2}=0$ by Lemma 5.9. Thus the equations (5.1) for $\rho_{n}$ and
(5.2) for $\rho_{n}^{(1)}$ specialize to

$\displaystyle\rho_{1}$ $\displaystyle=\xi_{1,0}+\xi_{1,1}\sigma_{1,1}$
$\displaystyle=\frac{2p(p+1)}{3(p^{2}+1)}+\frac{1}{p^{2}+1}\sigma_{1,1}$
$\displaystyle\rho_{1}^{(1)}$
$\displaystyle=\xi^{(1)}_{1,0}+\xi^{(1)}_{1,1}\sigma_{1,1}$
$\displaystyle=\frac{2(p^{2}-1)}{3p^{2}}+\frac{1}{p^{2}}\sigma_{1,1}.$

By Lemmas 5.9, 5.12, 5.13, specializing (5.4), (5.5), (5.6) to $n=1$, $i=1$
reveals

$\displaystyle\sigma_{1,1}$
$\displaystyle=\left(1-\frac{1}{p}\right)\tau_{1,11}+\frac{1}{p}\sigma_{1,1}^{\prime}$
$\displaystyle=\frac{1}{p}\sigma_{1,1}^{\prime}$
$\displaystyle\sigma_{1,1}^{(1)}$ $\displaystyle=\tau_{1,11}$
$\displaystyle=0$ $\displaystyle\sigma_{1,1}^{\prime}$
$\displaystyle=1-\frac{1}{p}+\frac{1}{p}\left(\left(1-\frac{1}{p}\right)\sigma_{1,1}^{(1)}+\frac{1}{p}\rho_{1}^{(1)}\right)$
$\displaystyle=1-\frac{1}{p}+\frac{1}{p^{2}}\rho_{1}^{(1)},$

with the right hand equality following once we observe that $\tau_{1,11}=0$.

We can see this directly by unwinding Definition 5.7 as follows. Suppose $f$
is a binary form with type 1, so after a change of variables we may assume its
valuation table is given by Table 5.4 with $S=\\{x_{0}\\}$ and
$T=\\{x_{1}\\}$. Any primitive solution $[x_{0}:x_{1}]$ therefore has $p\mid
x_{0}$. After the transformation (5.3), we see $f_{\mathrm{I}}$ has valuation
table given below, and thus $p\mid x_{1}$.

Table 5.10. Valuation table for $f_{\mathrm{I}}$ when $n=1$ and $i=1$ $=2_{1}$ | $\geq 2$ | $\geq 1$ | $=0_{1}$
---|---|---|---

This contradicts the original $[x_{0}:x_{1}]$ being primitive, so
$\tau_{1,11}=0$. Solving the relations above among $\rho_{1}$,
$\rho_{1}^{(1)}$, $\sigma_{1,1}$, and $\sigma_{1,1}^{\prime}$, we obtain

(5.7)
$\rho_{1}=\frac{2p^{4}+3p^{3}+p^{2}+3p+2}{3(p^{4}+p^{3}+p^{2}+p+1)}=1-\frac{(p^{2}+1)^{2}}{3(p^{4}+p^{3}+p^{2}+p+1)}.$

###  5.2. Phase II

The goal of phase II is to compute $\tau_{n,ij}$: the lifting probability
given that $f$ has type $i$ and the $f_{\mathrm{I}}$ produced in phase I has
type $j$. Here $i,j\in\\{1,2,3\\}$ and $i+j\leq n+1$ (see Remark 5.10). Our
approach is essentially the same as in phase I, in that we first isolate the
first $j$ variables of $h_{\mathrm{I}}$, then produce $f_{\mathrm{II}}$ such
that
$X_{f_{\mathrm{II}}}(\mathbb{Q}_{p}),X_{f_{\mathrm{I}}}(\mathbb{Q}_{p}),X_{f}(\mathbb{Q}_{p})$
are in bijection, and analyze the reduction $\overline{f_{\mathrm{II}}}$.

Recall from phase I that $\overline{f_{\mathrm{I}}}=\overline{h_{\mathrm{I}}}$
for $h_{\mathrm{I}}(x_{i},\ldots,x_{n})$ with factorization type $j$. After an
invertible linear change of coordinates in only $x_{i},\ldots,x_{n}$, we may
assume $f_{\mathrm{I}}$ has the valuation table below for the partition
$S=\\{x_{0},\ldots,x_{i-1}\\}$, $T=\\{x_{i},\ldots,x_{i+j-1}\\}$, and
$U=\\{x_{i+j},\ldots,x_{n}\\}$ (note that $U=\emptyset$ if $i+j=n+1$).

Table 5.11. Valuation table for $f_{\mathrm{I}}$ $=2_{i}$ | $\geq 2$ | $\geq 1$ | $=0_{j}$
---|---|---|---
$\geq 2$ | $\geq 1$ | $\geq 1$ |
$\geq 1$ | $\geq 1$ |  |
$\geq 1$ |  |  |

Note that $[x_{0}:\ldots:x_{n}]\in X_{f_{\mathrm{I}}}(\mathbb{Q}_{p})$ implies
$p\mid x_{i},\ldots,x_{i+j-1}$. This leads us to define

$f_{\mathrm{II}}=\frac{1}{p}f_{\mathrm{I}}(x_{0},\ldots,x_{i-1},px_{i},\ldots,px_{i+j-1},x_{i+j},\ldots,x_{n})$

with valuation table below such that $X_{f_{\mathrm{II}}}(\mathbb{Q}_{p})$ and
$X_{f_{\mathrm{I}}}(\mathbb{Q}_{p})$ are in bijection.

Table 5.12. Valuation table for $f_{\mathrm{II}}$ $=1_{i}$ | $\geq 2$ | $\geq 2$ | $=2_{j}$
---|---|---|---
$\geq 1$ | $\geq 1$ | $\geq 2$ |
$\geq 0$ | $\geq 1$ |  |
$\geq 0$ |  |  |

Note the $SU^{2}$ entry in Table 5.12 above. If these coefficients are
nonvanishing modulo $p$, then $\overline{f_{\mathrm{II}}}$ is linear in at
least one of the variables $x_{0},\ldots,x_{i-1}$, so an
$\mathbb{F}_{p}$-solution can be found and lifted to a $\mathbb{Q}_{p}$-point
on $X_{f_{\mathrm{II}}}$. If the $SU^{2}$ entry has valuation at least 1, then
we must analyze
$\overline{f_{\mathrm{II}}}=h_{\mathrm{II}}(x_{i+j},\ldots,x_{n})$.

######  Definition 5.15.

Let $\theta_{n,ijk}$ denote the probability that $f$ has a $p$-adic solution
given that

  * •

$f$ has type $i$,

  * •

$f_{\mathrm{I}}$ has type $j$,

  * •

the $SU^{2}$ coefficients of $f_{\mathrm{II}}$ have minimal valuation at least
1, and

  * •

$f_{\mathrm{II}}$ is primitive and reduces to type $k$.

######  Definition 5.16.

Let $\tau_{n,ij}^{\prime}$ denote the probability that $f$ has a $p$-adic
solution given that $f$ has type $i$, $f_{\mathrm{I}}$ has type $j$, and the
resulting $\overline{f_{\mathrm{II}}}$ vanishes modulo $p$.

######  Lemma 5.17.

Let $n\geq 1$ and $i,j\in\\{1,2,3\\}$. If $i+j=n+1$ then

$\tau_{n,ij}=0.$

If $i+j<n+1$ then

(5.8)
$\tau_{n,ij}=\scalebox{0.85}{$\displaystyle\left(1-\frac{1}{p^{i\binom{n-i-j+2}{2}}}\right)+\frac{1}{p^{i\binom{n-i-j+2}{2}}}\left(\left(1-\frac{1}{p^{\binom{n-i-j+3}{3}}}\right)\left(\xi_{n-i-j,0}+\sum_{1\leq
k\leq
3}\xi_{n-i-j,k}\theta_{n,ijk}\right)+\frac{1}{p^{\binom{n-i-j+3}{3}}}\tau_{n,ij}^{\prime}\right)$}.$

###### Proof.

Suppose $i+j=n+1$, i.e. that $U$ is empty. We know from phase I that if
$[x_{0}:\ldots:x_{n}]\in X_{f}(\mathbb{Q}_{p})$ then $p\mid
x_{0},\ldots,x_{i-1}$. The reduction $\overline{f_{\mathrm{II}}}$ has type $j$
in exactly $j$ variables, and hence $p\mid x_{i},\ldots,x_{n}$ as well, a
contradiction. Assume moving forward that $i+j<n+1$

The probability that the coefficients of monomials in $SU^{2}$ are all
divisible by $p$ is equal to that of $i$ independent quadratic forms in
$n-i-j$ variables vanishing modulo $p$, which is $1/p^{i\binom{n-i-j+2}{2}}$.
If any of the $SU^{2}$ coefficients are $p$-adic units, then
$\overline{f_{\mathrm{II}}}$ is linear in $x_{m}$ for some $0\leq m<i$, and
there exists a point in $\overline{X_{f_{\mathrm{II}}}}(\mathbb{F}_{p})$ which
lifts to $X_{f_{\mathrm{II}}}(\mathbb{Q}_{p})$ by Hensel’s Lemma (this is the
same argument used in Lemma 5.13 with the $ST^{2}$ coefficients).

If all the coefficients of $SU^{2}$ monomials are divisible by $p$, then the
probability that $\overline{f_{\mathrm{II}}}$ vanishes altogether is that of a
cubic form in $n-i-j$ variables vanishing, $1/p^{\binom{n-i-j+3}{3}}$. In this
case the lifting probability is $\tau_{n,ij}^{\prime}$ by definition.

Assuming $\overline{f_{\mathrm{II}}}\neq 0$, we stratify by reduction type.
For $k=1,2,3$, type $k$ occurs with probability $\xi_{n-i-j,k}$ and the
associated lifting probability is $\theta_{n,ijk}$ by definition. If
$\overline{f_{\mathrm{II}}}$ has none of these types, which occurs with
probability $\xi_{n-i-j,0}$ then
$X_{f_{\mathrm{II}}}(\mathbb{Q}_{p})\neq\emptyset$ by Proposition 4.1. ∎

######  Lemma 5.18.

Let $n\geq 2$ and $i,j\in\\{1,2,3\\}$ such that $i+j<n+1$. Then we have

(5.9)
$\tau_{n,ij}^{\prime}=1-\frac{1}{p^{ij(n-i-j+1)+j\binom{n-i-j+2}{2}}}+\frac{1}{p^{ij(n-i-j+1)+j\binom{n-i-j+2}{2}}}\left(\xi_{n-j,0}^{(i)}+\sum_{0\leq
k\leq 3}\xi_{n-j,k}^{(i)}\sigma_{n,k}^{(j)}\right)$

###### Proof.

We begin by dividing through by $p$, producing the valuation table below.

Table 5.13. Valuation table for initial step in computing $\tau_{n,ij}^{\prime}$ (colors added for emphasis) $=0_{i}$ | $\geq 1$ | $\geq 1$ | $=1_{j}$
---|---|---|---
$\geq 0$ | $\geq 0$ | $\geq 1$ |
$\geq 0$ | $\geq 0$ |  |
$\geq 0$ |  |  |

Looking at the center left column of Table 5.13, we argue that if the
coefficients of monomials in $STU$ (shown in blue) or $TU^{2}$ (shown in red)
are units in $\mathbb{Z}_{p}$, then we can find a lift. Let us see why
explicitly.

Suppose one of the $TU^{2}$ coefficients is nonzero and momentarily specialize
$x_{0}=\ldots=x_{i-1}=0$. The resulting cubic form is linear in one of the
variables in $T$, so there exists a Hensel-liftable solution to a point in
$X_{f_{\mathrm{II}}}(\mathbb{Q}_{p})$ (we saw this previously in Lemmas 5.13
and 5.17).

Suppose instead that all $TU^{2}$ coefficients are divisible by $p$, but some
$STU$ coefficient is a $p$-adic unit; for concreteness, say
$a_{0in}\in\mathbb{Z}_{p}^{\times}$. Again we find
$\overline{\frac{1}{p}f_{\mathrm{II}}}$ is linear in $x_{i}$, so there exists
an $\mathbb{F}_{p}$ point with, say $x_{0}=1,\ x_{n}=1$, which can be lifted
via Hensel’s lemma in $x_{i}$ to a point in
$X_{f_{\mathrm{II}}}(\mathbb{Q}_{p})$.

Thus we have found a lift unless all $STU,TU^{2}$ coefficients vanish, which
occurs with the same probability as $ij$ linear forms in $n-i-j$ variables and
$j$ quadratic forms in $n-i-j$ variables simultaneously vanishing modulo $p$,
$1/p^{ij(n-i-j+1)+j\binom{n-i-j+2}{2}}$. If this occurs, we repartition our
set of variables into $(S\cup U)\coprod T$, as shown in Table 5.14.

Table 5.14. Repartitioning step in computing $\tau_{n,ij}^{\prime}$ $S$ |  |  |  |  | $T$ |  | $S\cup U$ |  |  |  |  | $T$
---|---|---|---|---|---|---|---|---|---|---|---|---
| $=0_{i}$ | $\geq 1$ | $\geq 1$ | $=1_{j}$ |  |  |  | $=0^{(i)}$ | $\geq 1$ | $\geq 1$ | $=1_{j}$ |
| $\geq 0$ | $\geq 1$ | $\geq 1$ |  |  | $\xrightarrow{\text{reindex}}$ |  |  |  |  |  |
| $\geq 0$ | $\geq 1$ |  |  |  |  |  |  |  |  |  |
| $\geq 0$ |  |  |  |  |  |  |  |  |  |  |
$U$ |  |  |  |  |  |  |  |  |  |  |  |

The factorization type is now $k$ with probability $\xi_{n-j,k}^{(i)}$. If
$k=0$ we have a lift by Proposition 4.1; otherwise, performing the usual phase
I operations (5.3) to isolate the first $k$ variables of $S\cup U$ and relabel
once more, we find ourselves in the situation of $\sigma_{n,k}^{(j)}$. Putting
everything together yields (5.9). ∎

###  5.3. Phase III

Our final goal is to compute $\theta_{n,ijk}$ in terms of the various lifting
probabilities already defined. We recall our initial situation from the
definition: $f,f_{\mathrm{I}}$, had reduction types $i,j$ and in phase II we
found $f_{\mathrm{II}}$ had reduction type $k$ with its $SU^{2}$ coefficients
in $p\mathbb{Z}_{p}$; note this implies $i+j+k\leq n+1$. We record the
valuation table below for this situation with
$\\{x_{0},\ldots,x_{n}\\}=S\coprod T\coprod U$ as before with
$S=\\{x_{0},\ldots,x_{i-1}\\},T=\\{x_{i},\ldots,x_{i+j-1}\\}$ as before and
$U=\\{x_{i+j},\ldots,x_{n}\\}$.

Table 5.15. Valuation table for $f_{\mathrm{II}}$ $=1_{i}$ | $\geq 2$ | $\geq 2$ | $=2_{j}$
---|---|---|---
$\geq 1$ | $\geq 1$ | $\geq 2$ |
$\geq 1$ | $\geq 1$ |  |
$=0_{k}$ |  |  |

After possibly an invertible linear transformation over $\mathbb{Z}_{p}$
involving only $x_{i+j},\ldots,x_{n}$, we may isolate the $k$ variables
$x_{i+j},\ldots,x_{i+j+k-1}$ in $\overline{f_{\mathrm{II}}}$. To make this
precise, we instead modify our partition by
$U=\\{x_{i+j},\ldots,x_{i+j+k-1}\\}$ and add the block
$W=\\{x_{i+j+k},\ldots,x_{n}\\}$. This is neatly represented in the valuation
table below.

Table 5.16. Valuation table for initial situation of $\theta_{n,ijk}$ $w=0$ | $w=1$ | $w=2$ | $w=3$
---|---|---|---
$=1_{i}$ | $\geq 2$ | $\geq 2$ | $=2_{j}$ | $\geq 1$ | $\geq 1$ | $\geq 2$ | $\geq 1$ | $\geq 1$ | $\geq 1$
$\geq 1$ | $\geq 1$ | $\geq 2$ |  | $\geq 1$ | $\geq 1$ |  | $\geq 1$ |  |
$\geq 1$ | $\geq 1$ |  |  | $\geq 1$ |  |  |  |  |
$=0_{k}$ |  |  |  |  |  |  |  |  |

As in phases I, II, we see that for any $[x_{0}:\ldots:x_{n}]\in
X_{f_{\mathrm{II}}}(\mathbb{Q}_{p})$, we must have $p\mid
x_{i+j},\ldots,x_{i+j+k-1}$, motivating yet another change of variables

$f_{\mathrm{III}}=\frac{1}{p}f_{\mathrm{II}}(x_{0},\ldots,x_{i+j-1},px_{i+j},\ldots,px_{i+j+k-1},x_{i+j+k},\ldots,x_{n})$

with valuation table given below.

Table 5.17. Valuation table for $f_{\mathrm{III}}$ (color added for emphasis) $w=0$ | $w=1$ | $w=2$ | $w=3$
---|---|---|---
$=0_{i}$ | $\geq 1$ | $\geq 1$ | $=1_{j}$ | $\geq 0$ | $\geq 0$ | $\geq 1$ | $\geq 0$ | $\geq 0$ | $\geq 0$
$\geq 1$ | $\geq 1$ | $\geq 2$ |  | $\geq 1$ | $\geq 1$ |  | $\geq 1$ |  |
$\geq 2$ | $\geq 2$ |  |  | $\geq 2$ |  |  |  |  |
$=2_{k}$ |  |  |  |  |  |  |  |  |

If the $STW$ (shown in blue) and $TW^{2}$ (shown in red) entries in Table 5.17
were both divisible by $p$ at least once, then this table would “collapse”
into one resembling Table 5.15. In making this precise, we prove the
following.

######  Lemma 5.19.

Let $n\geq 2$ and $i,j,k\in\\{1,2,3\\}$. If $i+j+k=n+1$ then

$\theta_{n,ijk}=0.$

If $i+j+k<n+1$ then we have

(5.10) $\theta_{n,ijk}=\scalebox{0.9}{$\displaystyle 1-\frac{1}{p^{ij(n-i-
j-k+1)+j\binom{n-i-j-k+2}{2}}}+\frac{1}{p^{ij(n-i-j-k+1)+j\binom{n-i-
j-k+2}{2}}}\left(\xi_{n-j-k,0}^{(i)}+\sum_{1\leq\ell\leq
3}\xi_{n-j-k,\ell}^{(i)}\theta_{n,jk\ell}\right)$}.$

###### Proof.

That $\theta_{n,ijk}=0$ when $i+j+k=n+1$ follows from assuming
$[x_{0}:\ldots:x_{n}]\in X_{f_{\mathrm{III}}}(\mathbb{Q}_{p})$ and tracing it
back to a point on $X_{f}(\mathbb{Q}_{p})$ with all coordinates divisible by
$p$, producing a contradiction as in Lemmas 5.9 and 5.17. Assume then that
$i+j+k<n$.

If the $STW$ or $TW^{2}$ coefficients (the blue and red entries, respectively
in Table 5.17) are in $\mathbb{Z}_{p}^{\times}$ then there exists an
$\mathbb{F}_{p}$-point on $X_{f_{\mathrm{III}}}$ that lifts via Hensel’s
lemma. The argument follows exactly the same as that in Lemma 5.18. These
coefficients are all in $p\mathbb{Z}_{p}$ with probability equal to that of
$ij$ linear forms in $n-i-j-k$ variables and $j$ quadratic forms in $n-k-j-k$
variables over $\mathbb{F}_{p}$ vanishing, given by $1/p^{ij(n-i-
j-k+1)+j\binom{n-i-j-k+2}{2}}$.

Assuming we are in this situation, the reduction is a cubic form in $n-j-k+1$
variables,

$\overline{f_{\mathrm{III}}}=\overline{f_{\mathrm{III}}}(x_{0},\ldots,x_{i-1},x_{i+j+k},\ldots,x_{n}),$

due to all other monomials vanishing. Note that $f_{\mathrm{III}}$ is known to
satisfy condition $(i)$. As is our custom, we analyze the lifting probability
when $f_{\mathrm{III}}$ has type $\ell=1,2,3$, which each occur with
probability $\xi_{n-j-k,\ell}^{(i)}$; the lifting probability is 1 otherwise
by Proposition 4.1.

Repartitioning by setting $S^{\prime}=S\cup W$, $T^{\prime}=T$, and
$U^{\prime}=U$, we obtain a rotated version of Table 5.15, so the lifting
probability is given by $\theta_{n,jk\ell}$. This final step is illustrated
below in Table 5.18. Putting everything together yields (5.10). ∎

Table 5.18. Rotating and collapsing the valuation tetrahedron for
$f_{\mathrm{III}}$ (color added for emphasis)

$w=0$ | $w=1$ | $w=2$ | $w=3$
---|---|---|---
$=0_{i}$ | $\geq 1$ | $\geq 1$ | $=1_{j}$ | $\geq 0$ | $\geq 1$ | $\geq 1$ | $\geq 0$ | $\geq 1$ | $\geq 0$
$\geq 1$ | $\geq 1$ | $\geq 2$ |  | $\geq 1$ | $\geq 1$ |  | $\geq 1$ |  |
$\geq 2$ | $\geq 2$ |  |  | $\geq 2$ |  |  |  |  |
$=2_{k}$ |  |  |  |  |  |  |  |  |

| $S\cup W$ |  |  |  |  | $T$
---|---|---|---|---|---|---
|  | $=0^{(i)}$ | $\geq 1$ | $\geq 1$ | $=1_{j}$ |
$\xrightarrow{\text{reindex}}$ |  | $\geq 1$ | $\geq 1$ | $\geq 2$ |  |
|  | $\geq 2$ | $\geq 2$ |  |  |
|  | $=2_{k}$ |  |  |  |
| $U$ |  |  |  |  |

##  6\. Obtaining explicit rational functions

We are now ready to prove Theorem 2.3, that for $n\geq 1$ and all primes $p$,
we have

$\rho_{n}(p)=1-\frac{g_{n}(p)}{h_{n}(p)}$

for the explicit polynomials below in (6.1) – (6.16) when $1\leq n\leq 8$, and
$\rho_{n}(p)=1$ for $n\geq 9$. Note that $\frac{g_{n}}{h_{n}}$ as presented
here are not all in lowest terms, in order to make the expressions more
compact.

(6.1) $g_{1}(p)=(p^{2}+1)^{2}$ (6.2) $h_{1}(p)=3(p^{4}+p^{3}+p^{2}+p+1)$ (6.3)
$g_{2}(p)=p^{9}-p^{8}+p^{6}-p^{4}+p^{3}+p^{2}-2p+1$ (6.4)
$h_{2}(p)=3(p^{6}+p^{3}+1)(p^{4}+1)(p^{2}+1)$ (6.5)
$g_{3}(p)=\left(3p^{26}+p^{24}+p^{23}+4p^{22}-3p^{21}+3p^{20}+2p^{19}+2p^{18}-p^{17}+p^{14}+p^{13}-2p^{12}+3p^{11}+3p^{7}\right)\left(p^{2}+1\right)\left(p+1\right)^{2}\left(p-1\right)^{4}$
(6.6)
$h_{3}(p)=9\left(p^{13}-1\right)\left(p^{7}+1\right)\left(p^{7}-1\right)\left(p^{6}+1\right)\left(p^{5}-1\right)\left(p^{3}+1\right)\left(p^{3}-1\right)$
(6.7)
$g_{4}(p)=\left(p^{46}+3p^{41}+p^{40}-p^{39}+p^{37}+p^{36}+p^{35}-3p^{34}+3p^{27}-p^{26}+p^{25}+p^{19}\right)\left(p^{2}+1\right)\left(p+1\right)^{2}\left(p-1\right)^{4}$
(6.8)
$h_{4}(p)=9\left(p^{19}-1\right)\left(p^{17}-1\right)\left(p^{10}+1\right)\left(p^{9}+1\right)\left(p^{9}-1\right)\left(p^{7}-1\right)\left(p^{5}+1\right)$
(6.9)
$g_{5}(p)=\left(3p^{91}-3p^{90}+3p^{88}+3p^{85}-3p^{84}+3p^{82}-3p^{81}+3p^{79}+3p^{78}+3p^{76}-3p^{75}+3p^{73}-2p^{72}+p^{71}+4p^{70}-3p^{69}+3p^{67}-3p^{66}+3p^{64}-3p^{62}+3p^{61}+3p^{59}+3p^{58}-3p^{56}+3p^{55}-3p^{53}+3p^{52}+3p^{49}-3p^{47}+3p^{46}-3p^{44}+3p^{43}-3p^{41}+3p^{40}-3p^{38}+3p^{37}\right)\left(p^{5}-1\right)\left(p^{2}+1\right)\left(p+1\right)^{2}\left(p-1\right)^{4}$
(6.10)
$h_{5}(p)=27\left(p^{27}-1\right)\left(p^{25}-1\right)\left(p^{23}-1\right)\left(p^{14}+1\right)\left(p^{13}+1\right)\left(p^{13}-1\right)\left(p^{12}+1\right)\left(p^{7}+1\right)\left(p^{7}-1\right)\left(p^{6}+1\right)$
(6.11)
$g_{6}(p)=\left(3p^{105}+p^{97}+p^{96}+p^{95}-3p^{93}+3p^{81}\right)\left(p+1\right)^{2}\left(p-1\right)^{7}$
(6.12)
$h_{6}(p)=27\left(p^{31}-1\right)\left(p^{24}-p^{23}+p^{19}-p^{18}+p^{17}-p^{16}+p^{14}-p^{13}+p^{12}-p^{11}+p^{10}-p^{8}+p^{7}-p^{6}+p^{5}-p+1\right)\left(p^{20}-p^{19}+p^{17}-p^{16}+p^{14}-p^{13}+p^{11}-p^{10}+p^{9}-p^{7}+p^{6}-p^{4}+p^{3}-p+1\right)\left(p^{17}+1\right)\left(p^{17}-1\right)\left(p^{16}+1\right)\left(p^{11}-1\right)\left(p^{8}+p^{7}-p^{5}-p^{4}-p^{3}+p+1\right)\left(p^{8}-p^{7}+p^{5}-p^{4}+p^{3}-p+1\right)\left(p^{8}+1\right)\left(p^{6}+1\right)\left(p^{5}+1\right)\left(p^{5}-1\right)\left(p^{4}+1\right)\left(p^{3}+1\right)\left(p^{3}-1\right)^{3}$
(6.13)
$g_{7}(p)=\left(p^{4}+1\right)\left(p^{2}+1\right)^{2}\left(p+1\right)^{4}\left(p-1\right)^{9}p^{141}$
(6.14)
$h_{7}(p)=27\left(p^{43}-1\right)\left(p^{41}-1\right)\left(p^{24}-p^{23}+p^{21}-p^{20}+p^{18}-p^{17}+p^{15}-p^{14}+p^{12}-p^{10}+p^{9}-p^{7}+p^{6}-p^{4}+p^{3}-p+1\right)\left(p^{22}+1\right)\left(p^{20}+1\right)\left(p^{19}+1\right)\left(p^{19}-1\right)\left(p^{13}-1\right)\left(p^{12}+p^{11}-p^{9}-p^{8}+p^{6}-p^{4}-p^{3}+p+1\right)\left(p^{12}-p^{11}+p^{9}-p^{8}+p^{6}-p^{4}+p^{3}-p+1\right)\left(p^{11}+1\right)\left(p^{11}-1\right)\left(p^{10}+1\right)\left(p^{8}-p^{7}+p^{5}-p^{4}+p^{3}-p+1\right)\left(p^{7}+1\right)\left(p^{5}+1\right)\left(p^{5}-1\right)\left(p^{3}-1\right)^{3}$
(6.15)
$g_{8}(p)=\left(p^{9}-1\right)\left(p^{7}-1\right)\left(p^{4}+1\right)\left(p^{2}+1\right)^{2}\left(p+1\right)^{3}\left(p-1\right)^{9}p^{219}$
(6.16)
$h_{8}(p)=27\left(p^{53}-1\right)\left(p^{49}-1\right)\left(p^{47}-1\right)\left(p^{40}-p^{39}+p^{35}-p^{34}+p^{30}-p^{28}+p^{25}-p^{23}+p^{20}-p^{17}+p^{15}-p^{12}+p^{10}-p^{6}+p^{5}-p+1\right)\left(p^{32}-p^{31}+p^{29}-p^{28}+p^{26}-p^{25}+p^{23}-p^{22}+p^{20}-p^{19}+p^{17}-p^{16}+p^{15}-p^{13}+p^{12}-p^{10}+p^{9}-p^{7}+p^{6}-p^{4}+p^{3}-p+1\right)\left(p^{27}+1\right)\left(p^{27}-1\right)\left(p^{26}+1\right)\left(p^{25}+1\right)\left(p^{25}-1\right)\left(p^{24}+1\right)\left(p^{17}-1\right)\left(p^{13}+1\right)\left(p^{13}-1\right)\left(p^{12}+1\right)\left(p^{11}-1\right)\left(p^{6}+1\right)\left(p^{3}-1\right)^{3}$

###### Proof of Theorem 2.3.

To calculate $\rho_{n}(p)$, we assemble the relations (5.1), (5.2), (5.4),
(5.5), (5.6), (5.8), (5.9), and (5.10) into a system of 64 linear equations in
the 64 unknown lifting probabilities $\rho$, $\rho^{(i)}$, $\sigma_{i}$,
$\sigma_{i}^{(k)}$, $\sigma_{i}^{\prime}$, $\tau_{ij}$, $\tau_{ij}^{\prime}$,
$\theta_{ijk}$, where $i,j,k$ take values in $\\{1,2,3\\}$. The unique
solutions in the field of rational functions in the transcendental $p$ yield
the values for $\rho_{n}(p)$ described explicitly above in (6.1) – (6.16) in
terms of $g_{n}(p),h_{n}(p)$ for $1\leq n\leq 8$. For $n=9$ the unique
solution is $\rho_{9}=1$ (and all other lifting probabilities are equal to 1).
For $n>9$, the conclusion is deduced by specializing, i.e. to a nonzero cubic
form in only 10 of the $n+1$ variables, setting the rest to zero.

In practice, this was accomplished via a symbolic solve in Sage. While all 64
probabilities could be solved for at once, it is was significantly faster to
observe sub-linear systems in fewer unknowns and exploit this by solving them
first, thereby reducing the size of subsequent solves:

  1. (1)

Observe that the 27 equations (5.10) in $\theta_{n,ijk}$ involve no other
lifting probabilities.

  2. (2)

Once $\theta_{n,ijk}$ have been determined, the 27 equations
$\eqref{eq:sigma_cond},\eqref{eq:tau},\eqref{eq:tau_prime}$ depend only on
$\sigma_{n,i}^{(k)}$, $\tau_{n,ij}$, $\tau_{n,ij}^{\prime}$.

  3. (3)

Solve for the remaining 10 probabilities $\rho_{n}$, $\rho_{n}^{(i)}$,
$\sigma_{n,i}$, $\sigma_{n,i}^{\prime}$ in the system (5.1), (5.2), (5.4),
(5.6).

An implementation of this approach is available in the file
compute_rho_p.ipynb available in our GitHub repository [BK24]. ∎

##  7\. Numerical approximations

We now seek to obtain precise numerical approximations for
$\rho^{\mathrm{ELS}}_{n}$, the density of everywhere locally soluble cubic
hypersurfaces when $2\leq n\leq 8$ (and hence $\rho_{n}$ for $n\geq 4$ by
[BLBS23])222Note that for $n=1$, we have $\rho_{n}(p)\sim\frac{2}{3}$ as
$p\to\infty$, so the product $\prod_{p\leq A}\rho_{1}(p)$ goes to 0 in the
limit as $A\to\infty$., by truncating, i.e.

$\rho_{n}\approx\prod_{p\leq A}\rho_{n}(p).$

To determine how large to make $A$ and precisely estimate the error, we relate
$\rho_{n}$ to certain values of the Riemann zeta function and invoke classical
techniques for approximating $\zeta(s)$.

From the explicit description of $\rho_{n}(p)$ given in (6.3) – (6.16), we
extract the asymptotics

$1-\rho_{n}(p)=\frac{g_{n}(p)}{h_{n}(p)}\sim\frac{1}{\gamma_{n}p^{\delta_{n}}}$

for $2\leq n\leq 8$ with $\gamma_{n},\delta_{n}$ given below.

Table 7.1. Asymptotics for $\frac{g_{n}(p)}{h_{n}(p)}$ $n$ | $\gamma_{n}$ | $\delta_{n}$
---|---|---
2 | 3 | 3
3 | 3 | 10
4 | 9 | 22
5 | 9 | 43
6 | 9 | 78
7 | 27 | 129
8 | 27 | 201

An explicit check reveals that as a function on primes $p$,
$\frac{g_{n}}{h_{n}}$ approaches this asymptotic from below.

######  Lemma 7.1.

For all $2\leq n\leq 8$ and all primes $p$, we have

$\frac{g_{n}(p)}{h_{n}(p)}\leq\frac{1}{\gamma_{n}p^{\delta_{n}}}.$

Recall the Riemann zeta function and its usual Euler product,
$\zeta(s)=\sum_{m\geq
1}\frac{1}{m^{s}}=\prod_{p}\left(1-\frac{1}{p^{s}}\right)^{-1}$. For $A$ a
positive real number we denote by $\zeta_{>A}(s)$ the tail of the Euler
product,

$\zeta_{>A}(s)=\prod_{p>A}\left(1-\frac{1}{p^{s}}\right)^{-1}.$

It turns out that $\prod_{p}\rho_{n}(p)$ is close to its truncation when
$\zeta_{>A}(\delta_{n})$ is sufficiently close to 1.

######  Lemma 7.2.

Fix $2\leq n\leq 8$. Let $B\geq 1$ be a real number. If
$\zeta_{>A}(\delta_{n})\leq B$, then we have

$\left|\rho_{n}^{\mathrm{ELS}}-\prod_{p\leq A}\rho_{n}(p)\right|\leq
1-\frac{1}{B^{1/\gamma_{n}}}$

###### Proof.

We drop the subscripts $n$ for brevity. We have

$\displaystyle\left|\rho^{\mathrm{ELS}}-\prod_{p\leq A}\rho(p)\right|$
$\displaystyle=\left(\prod_{p\leq
A}\rho(p)\right)\left(1-\prod_{p>A}\rho(p)\right)$ $\displaystyle\leq
1-\prod_{p>A}\rho(p)$

since $\rho(p)$ is at most 1.

Suppose we would like to show that

(7.1) $1-\prod_{p>A}\rho(p)\leq\epsilon_{1}.$

This is equivalent to

(7.2)
$\log\left(\prod_{p>A}\rho(p)\right)\geq\epsilon_{2}=\log(1-\epsilon_{1}).$

Note that the quantities in (7.2) are negative. By the Taylor series for the
logarithm and absolute convergence of the involved sums, we may write

$\displaystyle\log\left(\prod_{p>A}\rho(p)\right)$
$\displaystyle=\sum_{p>A}\sum_{j\geq 1}\frac{-(g/h)^{j}}{j}$
$\displaystyle\geq\sum_{p>A}\sum_{j\geq 1}\frac{-1}{j\gamma^{j}p^{\delta j}}$
(by Lemma 7.1)
$\displaystyle\geq\frac{-1}{\gamma}\log\left(\zeta_{>A}(\delta)\right).$

Thus to establish (7.2), it suffices to show that

(7.3)
$\frac{1}{\gamma}\log\left(\zeta_{>A}(\delta)\right)\leq\epsilon_{3}=-\epsilon_{2}.$

Finally, we rearrange (7.3) to give the equivalent

(7.4) $\zeta_{>A}(\delta)\leq\epsilon_{4}=e^{\gamma\epsilon_{3}}.$

Suppose we know $\zeta_{>A}(\delta)<B$ for some choice of $A$. Note that such
a choice exists for $B$ arbitrarily close to 1, since $\zeta_{>A}(\delta)$ is
the tail of a convergent infinite product. Tracing through (7.1) – (7.4), we
establish (7.1) with $\epsilon_{1}=1-B^{-1/\gamma}$, from which the lemma
follows. ∎

An upper estimate for $\zeta_{>A}(s)$ follows from the classical
Euler–Maclaurin summation formula applied to $\zeta(s)$; see e.g. [Apo99,
Coh].

######  Lemma 7.3.

Fix a real number $A\geq 1$, an integer $M\geq 1$, an even integer $I\geq 2$,
and let $B_{2i}$ denote the $2i$-th Bernoulli number. For integral $s\geq 2$
we have

$\zeta_{>A}(s)\leq\scalebox{0.85}{$\displaystyle\left(\prod_{p\leq
A}\left(1-\frac{1}{p^{s}}\right)\right)\left(\sum_{m=1}^{M}\frac{1}{m^{s}}+\frac{1}{(s-1)M^{s-1}}-\frac{1}{2M^{s}}+\sum_{i=1}^{I}\frac{B_{2i}(s+2i-2)!}{(2i)!(s-1)!M^{s+2i-1}}+\left|\frac{B_{2I+2}(s+2I)!}{(2I+2)!(s-1)!M^{s+2I+1}}\right|\right)$}.$

######  Remark 7.4.

The upper bound for $\zeta_{>A}(s)$ has the advantage of being able to be
computed via exact arithmetic in Sage. Then when combining with Lemma 7.2 we
convert to inexact arithmetic only at the final step, choosing to round up.
See rho_numerics.ipynb in our GitHub repository [BK24].

Suppose we want our truncation $\prod_{p\leq A}\rho_{n}(p)$ to be accurate
within $10^{-D}$ of the true value of $\rho_{n}^{\mathrm{ELS}}$. Then we set
$B=\left(\frac{10^{D}}{10^{D}-1}\right)^{\gamma_{n}}$ and apply Lemmas 7.2 and
7.3 with various values of $D$ and $A$ with $M=1000,I=4$ to obtain the
accuracy data in Table 7.2.

Table 7.2. Accuracy of truncations $n$ | $A$ | $1-\prod_{p\leq A}\rho_{n}(p)\approx$ | $D$
---|---|---|---
2 | 61 | 0.0274 | 5
| 12919 |  | 10
3 | 11 | 0.00007328 | 10
| 503 |  | 26
4 | 5 | $5.022\cdot 10^{-9}$ | 16
| 179 |  | 50
5 | 3 | $1.343\cdot 10^{-15}$ | 21
| 17 |  | 53
6 | 3 | $3.502\cdot 10^{-26}$ | 38
| 19 |  | 100
7 | 3 | $5.152\cdot 10^{-42}$ | 62
| 7 |  | 110
8 | 3 | $6.222\cdot 10^{-64}$ | 97
| 5 |  | 141

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