# Hilbert properties under base change in small extensions

Lior Bary-Soroker  School of Mathematical Sciences, Tel Aviv University, Tel
Aviv 69978, Israel<EMAIL_ADDRESS>,  Arno Fehm  Institut für
Algebra, Technische Universität Dresden, 01062 Dresden, Germany  arno.fehm@tu-
dresden.de and  Sebastian Petersen  Institut für Mathematik, Universität
Kassel, 34121 Kassel, Germany<EMAIL_ADDRESS>
###### Abstract.

We study the preservation of the Hilbert property and of the weak Hilbert
property under base change in field extensions. In particular we show that
these properties are preserved if the extension is finitely generated or
Galois with finitely generated Galois group, and we also obtain some negative
results.

##  1\. Introduction

The Hilbert property HP of Colliot-Thélène–Sansuc and Serre [Ser08, CTS87] and
the weak Hilbert property WHP introduced later by Corvaja–Zannier [CZ17] are
derived from Hilbert’s irreducibility theorem and have been studied
intensively in recent years, see for example the list of references in the
introduction of [BFP24].

It is known that if a variety $X$ over a field $K$ of characteristic zero has
HP respectively WHP, then so does the base change $X_{L}$, for any finite
field extension $L/K$. The work [BFP24] proves that in the case of $X=A$ an
abelian variety over a number field, the same holds true (for WHP) also in
very specific infinite Galois extensions $L/K$.

The aim of this note is to provide a few more such base change results, which
are less arithmetic in nature but more general:

######  Theorem 1.1 (cf. Cor. 3.3).

Let $K$ be a field of characteristic zero, $X$ a normal $K$-variety, and $L/K$
a finitely generated field extension. If $X$ has HP (respectively WHP), then
so does $X_{L}$.

As an application we then give a new proof of the fact that the class of
varieties with HP is closed under products (Corollary 3.5).

Our second base change result is for field extensions $L/K$ that are small in
the concrete sense that $L/K$ is algebraic and has only finitely many
subextensions of degree $n$, for every $n\in\mathbb{N}$.

######  Theorem 1.2 (cf. Cor. 4.3).

Let $K$ be a field of characteristic zero, $X$ a normal $K$-variety and $L/K$
a small extension. If $X$ has HP (respectively WHP), then so does $X_{L}$.

In fact most of our results are stated for thin and strongly thin sets rather
than HP and WHP, adding flexibility. As opposed to many results in the
literature we also try to avoid assuming our varieties are proper and smooth.

In the last section we discuss further questions regarding base change of HP
and WHP and obtain a few negative results. In particular, HP and WHP do not go
down finite extensions (in a strong sense, see Proposition 5.6), and neither
“not HP” nor “not WHP” is preserved in unions of chains, even if the extension
is small (Proposition 5.8).

##  2\. Finite extensions

We begin by fixing notation and definitions for this note. Let $K$ always be a
field of characteristic zero. A $K$-variety is a separated integral scheme of
finite type over $K$. The degree $\deg(f)$ of a finite morphism $f\colon X\to
Y$ of $K$-varieties is the degree of the associated function field extension.
A cover is a finite surjective morphism of normal $K$-varieties. For a normal
$K$-variety $X$, a subset $T$ of $X(K)$ is thin (resp. strongly thin) in $X$
if $T\subseteq C(K)\cup\bigcup_{i=1}^{r}f_{i}(Z_{i}(K))$ for some proper
closed subscheme $C$ of $X$ and a finite collection $(f_{i}\colon
Z_{i}\rightarrow X)_{i=1,\dots,r}$ of covers with ${\rm deg}(f_{i})>1$ (resp.
with $f_{i}$ ramified) for each $i$. See [BFP24, §2] for basics on covers. A
normal111It makes sense to define HP also for non-normal varieties, but in
order for us to treat HP and WHP simultaneously we restrict to normal ones.
$K$-variety $X$ has the Hilbert property HP (resp. the weak Hilbert property
WHP) if $X(K)$ is not thin (resp. not strongly thin) in $X$. We can and will
always assume that the $Z_{i}$ are geometrically irreducible, since otherwise
$Z_{i}(K)=\emptyset$. Also note that every normal $K$-variety with WHP is
automatically geometrically irreducible.

The following lemma, which like many of our results in this work treats thin
and strongly thin sets simultaneously, should be considered known to experts,
and we include the proof only for completeness: For thin sets, this is
precisely [Ser08, Prop. 3.2.1], and for strongly thin sets, [CDJLZ22, Prop.
3.15] gives a proof for the case $X$ smooth proper and $T=X(L)$, which we
follow closely.

######  Lemma 2.1.

Let $X$ be a normal $K$-variety, let $L/K$ be a finite extension and let
$T\subseteq X(L)$. If $T$ is thin (resp. strongly thin) in $X_{L}$, then
$T\cap X(K)$ is thin (resp. strongly thin) in $X$.

###### Proof.

It suffices to consider the cases $T=C(L)$ and $T=f(Z(L))$ for $C$ a closed
proper subscheme of $X_{L}$ and $f\colon Z\to X_{L}$ a cover with $\deg(f)>1$
(resp. a ramified cover) and $Z$ geometrically irreducible. As the projection
$p\colon X_{L}\to X$ is closed, $C^{\prime}=p(C)$ is a proper closed subset of
$X$, and so $C(L)\cap X(K)\subseteq C^{\prime}(K)$ is strongly thin in $X$.
For the case $T=f(Z(L))$, we may assume that $X$ is affine. Then the Weil
restriction
$\mathrm{Res}_{L/K}(f)\colon\mathrm{Res}_{L/K}(Z)\to\mathrm{Res}_{L/K}(X_{L})$
exists and is a cover of $K$-varieties (see [Sch94, 4.10.2] for all properties
except for integrality, which is spelled out in [Die01, Lemma 1.10], and
normality, which can be deduced from [Wei82, Thm. 1.3.1]). With $\Delta\colon
X\to\mathrm{Res}_{L/K}(X_{L})$ the diagonal morphism (cf. [Sch94, 4.2.5]), the
cartesian square

$\textstyle{\mathrm{Res}_{L/K}(Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Res}_{L/K}(f)}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{\mathrm{Res}_{L/K}(X_{L})}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta}$

gives rise to the following commutative diagram in the category of sets:

$\textstyle{Z(L)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\mathrm{Res}_{L/K}(Z)(K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\mathrm{Res}_{L/K}(f)}$$\textstyle{V(K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{X(L)}$$\textstyle{\mathrm{Res}_{L/K}(X_{L})(K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{X(K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

The right hand square is cartesian and the composition of the lower horizontal
maps is just the inclusion, hence $f(Z(L))\cap X(K)\subseteq g(V(K))$, and so
it suffices to prove that $g(V(K))$ is thin (resp. strongly thin) in $X$, i.e.
that $g(W(K))$ is thin (resp. strongly thin) in $X$ for every irreducible
component $W$ of $V$. As $g$ is finite and therefore closed, we can assume
without loss of generality that $g|_{W}\colon W\to X$ is surjective. The
normalization $\pi\colon\overline{W}\to W$ is an isomorphism above a nonempty
open subset $W^{\prime}$ of $W$. Then $h:=g\circ\pi\colon\overline{W}\to X$ is
a cover and $B:=g(W\setminus W^{\prime})$ is a proper closed subset of $X$
with $g(W(K))\subseteq h(\overline{W}(K))\cup B(K)$. Thus it is enough to show
that $h(\overline{W}(K))$ is thin (resp. strongly thin) in $X$. For this we
have to check that $\deg(h)>1$ (resp. $h$ is ramified). Again we can assume
that the $K$-variety $\overline{W}$ is geometrically irreducible. The
following diagram commutes, where $p^{\prime}$ and $p^{\prime\prime}$ denote
the projections:

$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\mathrm{Res}_{L/K}(Z)_{L}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{\prime}}$$\scriptstyle{\mathrm{Res}_{L/K}(f)}$$\textstyle{\overline{W}_{L}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\scriptstyle{h_{L}}$$\textstyle{X_{L}}$$\textstyle{\mathrm{Res}_{L/K}(X_{L})_{L}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{\prime\prime}}$$\textstyle{X_{L}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta_{L}}$

As $p^{\prime\prime}\circ\Delta_{L}={\rm id}_{X_{L}}$ (see again [Sch94,
4.2.5]), the map $\alpha:=p^{\prime}\circ\delta\colon\overline{W}_{L}\to Z$ is
a cover with $h_{L}=f\circ\alpha$. This implies
$\deg(h)=\deg(h_{L})\geq\deg(f)>1$. If $f$ is ramified, then so is $h_{L}$
(cf. [BFP24, Lemma 2.15]) and thus also $h$. ∎

##  3\. Finitely generated extensions

In this section we discuss HP and WHP – and more precisely thin and strongly
thin sets – under base change in finitely generated extensions.

######  Lemma 3.1.

Let $S$ be a $K$-variety with $S(K)$ dense in $S$, and write $L=K(S)$. Let $X$
be a normal $K$-variety, and let $T\subseteq X(L)$. If $T$ is thin (resp.
strongly thin) in $X_{L}$, then $T\cap X(K)$ is thin (resp. strongly thin) in
$X$.

###### Proof.

It suffices to consider the case $T=C(L)\cup f(Z(L))$ with $C$ a closed proper
subscheme of $X_{L}$ and $f\colon Z\to X_{L}$ a cover with $\deg(f)>1$ (resp.
a ramified cover) and $Z$ geometrically irreducible. There is a nonempty open
subscheme $U$ of $X_{L}\setminus C$ such that $V:=f^{-1}(U)\to U$ is étale.

After replacing $S$ by a suitable nonempty open subscheme we can assume that
$S$ is normal, and that $Z$ and $U$ extend to $S$-schemes $\mathscr{Z}$ and
$\mathscr{U}$ of finite type (cf. [EGA4.3, IV.8.8.2]) in such a way that the
morphisms $\mathscr{Z}\to S$ and $\mathscr{U}\to S$ are flat, separated and
surjective (cf. [EGA4.3, IV.8.9.4, IV.8.10.5]). Moreover, as $Z$ and $U$ are
geometrically irreducible normal varieties, we can assume that all fibers of
$\mathscr{Z}\to S$ and of $\mathscr{U}\to S$ are geometrically irreducible
normal varieties (cf. [EGA4.3, IV.9.9.5, IV.9.7.7], [EGA3.1, III.9.1.13]).

After replacing $S$ by one of its nonempty open subschemes and replacing the
rest accordingly, $f\colon Z\to X_{L}$ extends to a finite surjective
$S$-morphism $F\colon\mathscr{Z}\to X_{S}$, and the inclusion $U\to X_{L}$
extends to an open $S$-immersion $\mathscr{U}\to X_{S}$ (cf. [EGA4.3,
IV.8.8.2, IV.8.10.5]). Then $V$ is the generic fiber of
$\mathscr{V}:=F^{-1}(\mathscr{U})$. For every $s\in S$ we have the following
diagrams with cartesian squares.

$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{X_{L}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Spec}(L)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f|_{V}}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Spec}(L)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\textstyle{X_{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S}$$\textstyle{\mathscr{V}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F|_{\mathscr{V}}}$$\textstyle{\mathscr{U}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S}$$\textstyle{\mathscr{Z}_{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{s}}$$\textstyle{X_{K(s)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Spec}(K(s))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{V}_{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{s}|_{\mathscr{V}_{s}}}$$\textstyle{\mathscr{U}_{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Spec}(K(s))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Since $f|_{V}$ is étale, after shrinking $S$ once more we can assume that for
every $s\in S$ the morphism
$F_{s}|_{\mathscr{V}_{s}}\colon\mathscr{V}_{s}\to\mathscr{U}_{s}$ is étale and
$\deg(F_{s}|_{\mathscr{V}_{s}})={\rm deg}(f)$ (cf. [EGA4.4, IV.17.7.11]). As
$\mathscr{Z}$ is flat over $S$, it follows that the morphism
$F|_{\mathscr{V}}\colon\mathscr{V}\to\mathscr{U}$ is étale (cf. [EGA4.4,
IV.17.8.2]). Finally, if $f$ is ramified, we can assume that $F_{s}$ is
ramified for every $s\in S$ [EGA4.4, IV.17.7.11].

As $S(K)$ is dense in $S$ there exists $s\in S(K)$, and

$T_{s}:=(X(K)\setminus\mathscr{U}_{s}(K))\cup F_{s}(\mathscr{Z}_{s}(K))$

is thin (resp. strongly thin) in $X$, so it suffices to prove that $T\cap
X(K)\subseteq T_{s}$. Let $x\in T\cap X(K)$. If $x\notin\mathscr{U}_{s}(K)$ we
are done, so assume $x\in\mathscr{U}_{s}(K)$. We prove that $x\in
F_{s}(\mathscr{Z}_{s}(K))$. Let

$\hat{x}\colon S\to X_{S}=X\times_{K}S,\ t\mapsto(x,t)$

be the constant $S$-morphism corresponding to $x$. The set
$\hat{x}^{-1}(\mathscr{U})$ is open in $S$ and contains $s$. We replace $S$ by
$\hat{x}^{-1}(\mathscr{U})$ and can thus view $\hat{x}$ as a section
$\hat{x}\colon S\to\mathscr{U}$ of $\mathscr{U}\to S$. Consider the cartesian
square

$\textstyle{F^{-1}(\hat{x})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{x}}$$\textstyle{\mathscr{V}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{U}}$

and note that $F^{-1}(\hat{x})$ is a finite étale $S$-scheme whose generic
fiber is $f^{-1}(x)$ and whose fiber over $s$ is $F_{s}^{-1}(x)$. In
particular, every connected component of $F^{-1}(\hat{x})$ is an étale cover
of $S$. So since $f^{-1}(x)\to\mathrm{Spec}(L)$ has a section by the
assumption $x\in T$, also $F^{-1}(\hat{x})\to S$ has a section, and this
implies that $F_{s}^{-1}(x)\to\mathrm{Spec}(K)$ has a section. It follows that
$x\in F_{s}(\mathscr{Z}_{s}(K))$ as desired. ∎

######  Proposition 3.2.

Let $X$ be a normal $K$-variety, $L/K$ a finitely generated field extension
and $T\subseteq X(L)$. If $T$ is thin (resp. strongly thin) in $X_{L}$, then
$T\cap X(K)$ is thin (resp. strongly thin) in $X$.

###### Proof.

Fix an intermediate field $K\subseteq F\subseteq L$ such that $F/K$ is purely
transcendental, i.e. $F=K(\mathbb{A}^{n})$ for some $n$, and $L/F$ is
algebraic, hence finite. The claim then follows from Lemma 2.1 applied to
$L/F$ and Lemma 3.1 applied to $S=\mathbb{A}^{n}$. ∎

######  Corollary 3.3.

Let $X$ be a normal $K$-variety and $L/K$ a finitely generated field
extension. If $X$ has HP (resp. WHP), then $X_{L}$ has HP (resp. WHP).

In [Ser08], Serre had asked whether the product of two varieties with HP has
HP again. This was positively answered in [BFP14] by proving and applying a
fibration theorem for HP. We now give another proof of this product theorem by
first proving a variant of the fibration theorem using the generic fiber and
then combining this with Corollary 3.3.

######  Proposition 3.4.

Let $p\colon\mathscr{X}\to S$ be a dominant morphism of normal $K$-varieties.
Assume that $S$ has HP and that the generic fiber $X$ of $p$ is normal and has
HP. Then $\mathscr{X}$ has HP.

###### Proof.

Let $\mathscr{U}$ be a nonempty open subset of $\mathscr{X}$, and let
$(F_{i}\colon\mathscr{Z}_{i}\to\mathscr{X})_{i=1,\dots,r}$ be a family of
covers with $\deg(F_{i})>1$ and $\mathscr{Z}_{i}$ geometrically irreducible
for all $i$. We have to show that
$\mathscr{U}(K)\setminus\bigcup_{i=1}^{r}F_{i}(\mathscr{Z}_{i}(K))$ is
nonempty. After replacing $\mathscr{U}$ by a smaller nonempty open set we can
assume that $f_{i}^{-1}(\mathscr{U})\to\mathscr{U}$ is étale for all $i$.
Thus, replacing $\mathscr{X}$ by $\mathscr{U}$, we can assume the $F_{i}$
étale right from the outset and have to prove that
$\mathscr{X}(K)\setminus\bigcup_{i=1}^{r}F_{i}(\mathscr{Z}_{i}(K))$ is not
empty. For each $i$, the generic fiber $f_{i}\colon Z_{i}\to X$ of $F_{i}$ is
an étale cover of $X$ with $\deg(f_{i})=\deg(F_{i})>1$. As $X$ has HP, there
exists a point

$x\in X(L)\setminus\bigcup_{i=1}^{r}f_{i}(Z_{i}(L)),$

where $L=K(S)$. After replacing $S$ by one of its nonempty open subsets (and
replacing the rest accordingly) $x$ extends to a section $\sigma\colon
S\to\mathscr{X}$ of $p$. Form the cartesian square

$\textstyle{\mathscr{Z}_{i,\sigma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{i,\sigma}}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{\mathscr{Z}_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{i}}$$\textstyle{\mathscr{X}}$

and note that the generic fiber of the finite étale morphism $F_{i,\sigma}$ is
$f_{i}^{-1}(x)$ and that $f_{i}^{-1}(x)\to\mathrm{Spec}(L)$ does not admit a
section by our choice of $x$. Thus $F_{i,\sigma}$ does not admit a section. It
follows that for every $i$ the connected components of
$\mathscr{Z}_{i,\sigma}$ are all of degree $>1$ over $S$. As $S$ has HP, there
exists $s\in
S(K)\setminus\bigcup_{i=1}^{r}F_{i,\sigma}(\mathscr{Z}_{i,\sigma}(K))$. Thus
$\sigma(s)\in\mathscr{X}(K)\setminus\bigcup_{i=1}^{r}F_{i}(\mathscr{Z}_{i}(K))$,
as desired. ∎

######  Corollary 3.5 ([BFP14, Cor. 3.4]).

Let $X$ and $Y$ be normal $K$-varieties. If both $X$ and $Y$ have HP, so does
$X\times_{K}Y$.

###### Proof.

Since $X$ has HP, by Corollary 3.3 also the generic fiber $X_{K(Y)}$ of the
projection $X\times_{K}Y\to Y$ has HP. Therefore $X\times_{K}Y$ has HP by
Proposition 3.4. ∎

By [CDJLZ22, Thm. 1.9], Corollary 3.5 holds for WHP (and $X,Y$ smooth proper
$K$-varieties) instead of HP when $K$ is finitely generated, but the case of
general $K$ is open. Such a product theorem for WHP would follow from
Corollary 3.3 if one could prove Proposition 3.4 for WHP instead of HP. Even
the following weaker version seems open:

######  Question 3.6.

In Proposition 3.4, if we assume that $X$ has only WHP instead of HP, does it
still follow that $\mathscr{X}$ has WHP?

##  4\. Small extensions

In this section we discuss HP and WHP – and again more precisely thin and
strongly thin sets – in small extensions as defined in the introduction. A
Galois extension is small if and only if its Galois group is small in the
sense that it has only finitely many open subgroups of index $n$ for every
$n\in\mathbb{N}$, and every (topologically) finitely generated profinite group
is small. See [FJ08, Chapter 16.10] for a discussion of small profinite groups
and their relation to finitely generated profinite groups. An arithmetically
important example of a small Galois extension is the maximal Galois extension
of a number field unramified outside a given finite set of primes, see [FJ08,
Example 16.10.9]. For a combinatorial sufficient condition for a Galois
extension to have small Galois group see [BF13, Proposition 2.5]. Small
absolute Galois groups play an important role in many model theoretic results,
see [FJ16].

######  Lemma 4.1.

Let $X$ be a normal $K$-variety and $f\colon Z\to X$ a cover of degree $n$.
Let $L/K$ be an algebraic extension and let $L_{0}$ be the compositum of all
subextensions of $L/K$ of degree at most $n$. Then $f(Z(L))\cap
X(K)=f(Z(L_{0}))\cap X(K)$.

###### Proof.

Since $X$ is normal, the finite surjective morphism $f$ is universally open
[EGA4.3, 14.4.4]. As ${\rm char}(K)=0$, [EGA4.3, 15.5.2] therefore gives for
$x\in X(K)$ that

$\sum_{z\in f^{-1}(x)}[K(z):K]\leq n,$

in particular $f^{-1}(x)(L)=f^{-1}(x)(L_{0})$. Thus $x\in f(Z(L))$ if and only
if $x\in f(Z(L_{0}))$. ∎

######  Proposition 4.2.

Let $X$ be a normal $K$-variety, $L/K$ a small extension and $T\subseteq
X(L)$. If $T$ is thin (resp. strongly thin) in $X_{L}$, then $T\cap X(K)$ is
thin (resp. strongly thin) in $X$.

###### Proof.

It suffices to consider the case $T=C(L)\cup f(Z(L))$ for a proper closed
subset $C$ of $X_{L}$ and $f\colon Z\to X_{L}$ a cover of degree $n>1$ (resp.
a ramified cover). Since $L/K$ is algebraic, there exists a finite extension
$K^{\prime}/K$ in $L$, a proper closed subset $C^{\prime}$ of $X_{K^{\prime}}$
and a cover $f^{\prime}\colon Z^{\prime}\to X_{K^{\prime}}$ of degree $n$
(resp. a ramified cover $f^{\prime}\colon Z^{\prime}\to X_{K^{\prime}}$) such
that $C^{\prime}_{L}=C$, $Z^{\prime}_{L}=Z$ and $f^{\prime}_{L}=f$. We will
now show that

$T^{\prime}:=T\cap
X(K^{\prime})=C^{\prime}(K^{\prime})\cup(f^{\prime}(Z^{\prime}(L))\cap
X(K^{\prime}))$

is thin (resp. strongly thin) in $X_{K^{\prime}}$, which by Lemma 2.1 will
prove that $T\cap X(K)=T^{\prime}\cap X(K)$ is thin (resp. strongly thin) in
$X$. Clearly $C^{\prime}(K^{\prime})$ is strongly thin in $X_{K^{\prime}}$,
and Lemma 4.1 applied to $f^{\prime}$ gives that

$f^{\prime}(Z^{\prime}(L))\cap X(K^{\prime})=f^{\prime}(Z^{\prime}(L_{0}))\cap
X(K^{\prime}),$

where $L_{0}$ is the compositum of all subextensions of $L/K^{\prime}$ of
degree at most $n$. As $Z^{\prime}_{L_{0}}$ is irreducible (because even
$Z^{\prime}_{L}$ is irreducible), and in particular $f^{\prime}_{L_{0}}$ is a
cover of degree $n$ (resp. a ramified cover, cf. [EGA4.2, IV.2.7.1]), we have
that $f^{\prime}(Z^{\prime}(L_{0}))$ is thin (resp. strongly thin) in
$X_{L_{0}}$. As the extension $L_{0}/K^{\prime}$ is finite because $L/K$ is
small, this implies that $f^{\prime}(Z^{\prime}(L_{0}))\cap X(K^{\prime})$ is
thin (resp. strongly thin) in $X_{K^{\prime}}$ by Lemma 2.1. Hence
$T^{\prime}$ is thin (resp. strongly thin) in $X_{K^{\prime}}$. ∎

######  Corollary 4.3.

Let $X$ be a normal $K$-variety and $L/K$ a small extension. If $X$ has HP
(resp. WHP), then $X_{L}$ has HP (resp. WHP).

The case $X=\mathbb{P}^{1}$ and $L/K$ Galois of this corollary appears already
as [FJ08, Prop. 16.11.1]. The proof, however, uses irreducible specializations
and therefore does not carry over from HP to WHP, which is why the
introduction of [BFP24] stated (in the special case of Galois extensions with
finitely generated Galois group) that at that point we did not know how to
prove it for WHP.

######  Example 4.4.

For example, if $L$ denotes the cyclotomic $\mathbb{Z}_{p}$-extension of
$\mathbb{Q}$, and $E$ is an elliptic curve over $\mathbb{Q}$ of positive rank,
then $E$ has WHP (as follows from Falting’s theorem, see [CZ17]), and thus by
Corollary 4.3 so has $E_{L}$. By [Kat04] (see also [MR03, Example 1.2]),
$E(L)$ is finitely generated. This is in stark contrast with the results from
[BFP24], where, for abelian extensions $M/\mathbb{Q}$, in order to prove that
$E_{M}$ has WHP we needed to assume that $E(M)$ has infinite rank, in fact
even that $E(M)/E(K)$ has positive rank for every number field $K\subseteq M$.
In particular, the results of [BFP24] do not give that $E_{L}$ has WHP.

##  5\. Counterexamples

It is obvious that neither HP nor WHP is preserved in unions of chains of
fields. In this final section we discuss two other potential preservation
theorems for these two properties and their negation. Both constructions will
make use of the following result for fields $K$ that are Hilbertian (i.e.
$\mathbb{P}^{1}_{K}$ has HP) and pseudo-algebraically closed (PAC, see e.g.
[FJ08, Chapter 11]):

######  Proposition 5.1.

Let $K$ be a Hilbertian PAC field. Then every normal $K$-variety $X$ has HP.

###### Proof.

By a theorem of Fried–Völklein [FV92], the absolute Galois group of a
countable Hilbertian PAC field is a free profinite group on countably many
generators, in particular $K$ is what is called $\omega$-free, and [FJ08,
Prop. 27.3.4] gives that over an $\omega$-free PAC field every variety has HP.
∎

Also the following Proposition 5.5, which is a strengthening of [Ser08, p. 20
Exercise 1], will be applied to both our examples:

######  Lemma 5.2.

Let $B,C\subseteq\mathbb{P}^{1}(\overline{K})$ be finite sets. There exists
$\tau\in{\rm Aut}(\mathbb{P}^{1}_{\mathbb{Q}})$ with $\tau(B)\cap
C=\emptyset$.

###### Proof.

Without loss of generality, $B,C\subseteq\mathbb{A}^{1}(\overline{K})$. Choose
$a\in\mathbb{Q}\setminus(B-C)$ and let $\tau$ be the translation by $a$. ∎

######  Lemma 5.3.

Let $g\colon X\to Y$ be a smooth surjective morphism of normal $K$-varieties,
and let be $T\subseteq Y(K)$ strongly thin in $Y$. If

  1. (a)

the generic fiber of $g$ is geometrically irreducible, or

  2. (b)

$g$ is proper,

then $\bar{T}:=\\{x\in X(K):g(x)\in T\\}$ is strongly thin in $X$.

###### Proof.

If $T=C(K)$ for some proper closed subset $C$ of $Y$, then
$\bar{C}:=g^{-1}(C)$ is a proper closed subset of $X$ containing $\bar{T}$,
thus $\bar{T}$ is strongly thin. It hence suffices to consider the case where
$T=f(Z(K))$ for a ramified cover $f\colon Z\to Y$ with $Z$ geometrically
irreducible. In the cartesian square

$\textstyle{Z\times_{Y}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{\prime}}$$\scriptstyle{f^{\prime}}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{Y}$

the fiber product $Z\times_{Y}X$ is normal because $g^{\prime}$ is smooth and
$Z$ is normal. As $\bar{T}$ is contained in $f^{\prime}((Z\times_{Y}X)(K))$,
it suffices to prove that the latter set is strongly thin in $X$ in both cases
(a) and (b).

First assume that the generic fiber $g^{-1}(\eta)$ of $g$ is geometrically
irreducible. Then if $\zeta$ denotes the generic point of $Z$,
$(g^{\prime})^{-1}(\zeta)=g^{-1}(\eta)\times_{k(\eta)}k(\zeta)$ is
irreducible. Thus as $(g^{\prime})^{-1}(\zeta)\to Z\times_{Y}X$ is dominant,
because it is the pullback of the dominant map $\mathrm{Spec}(k(\zeta))\to Z$
by the surjective morphism $g^{\prime}$, we get that $Z\times_{Y}X$ is
irreducible and thus $f^{\prime}$ is a cover. Since $f$ is ramified and $g$ is
surjective, it follows that $f^{\prime}$ is ramified [EGA4.4, 17.7.3(i)]. Thus
$f^{\prime}((Z\times_{Y}X)(K))$ is strongly thin in $X$.

Now assume instead that $g$ is proper. Let $Z_{1},\dots,Z_{r}$ be the
irreducible components of $Z\times_{Y}X$, and denote by $f_{j}^{\prime}\colon
Z_{j}\to X$ (resp. $g_{j}^{\prime}\colon Z_{j}\to Z$) the restrictions of
$f^{\prime}$ (resp. $g^{\prime}$) to $Z_{j}$. Each $Z_{j}$ is open in
$Z\times_{Y}X$ because $Z\times_{Y}X$ is normal. We have to show that
$\bar{T}_{j}:=f_{j}^{\prime}(Z_{j}(K))$ is strongly thin in $X$ for every $j$.
As $f_{j}^{\prime}$ is finite, hence closed, we can assume that
$f_{j}^{\prime}$ is surjective, and therefore a cover. It is enough to show
that $f_{j}^{\prime}$ is ramified. Since $g^{\prime}$ is smooth and proper,
hence open and closed, and $Z$ is connected, $g_{j}^{\prime}\colon Z_{j}\to Z$
is surjective and smooth. In particular $g_{j}^{\prime}$ is flat (cf. [EGA4.4,
17.5.1(i)]). As $f$ is not smooth it follows that $f\circ g_{j}^{\prime}$ is
not smooth (cf. [EGA4.4, 17.7.7(i)]). Hence $g\circ f_{j}^{\prime}$ is not
smooth and $f_{j}^{\prime}$ cannot be étale, as desired. ∎

######  Remark 5.4.

In the special case where $X$ and $Y$ are smooth and proper over $K$, and
$T=Y(K)$, the previous lemma is implied by [CDJLZ22, Thm. 3.7]; we need it in
the non-proper case however. By the proof of [CTS87, Prop. 7.13], Lemma 5.3(a)
holds for “thin” instead of “strongly thin”. Lemma 5.3(b) though does not hold
for “thin” instead of “strongly thin”: The set $T=\mathbb{Q}^{\times 2}$ is
thin in $X:=\mathbb{G}_{m,\mathbb{Q}}$, the morphism $g\colon X\to X,x\mapsto
x^{2}$ is smooth, proper and surjective, but $\\{x\in X(\mathbb{Q}):g(x)\in
T\\}=\mathbb{Q}^{\times}$ is not thin in $X$.

######  Proposition 5.5.

If some normal $K$-variety $X$ has WHP, then $K$ is Hilbertian.

###### Proof.

Let $(f_{j}\colon Z_{j}\to\mathbb{P}^{1}_{K})_{j=1,\dots,r}$ be a family of
covers with $\deg(f_{j})>1$ and $Z_{j}$ geometrically irreducible for every
$j$. Let $T=\bigcup_{j=1}^{r}f_{j}(Z_{j}(K))$, and let $U$ be a nonempty open
subset of $\mathbb{P}^{1}_{K}$. We have to show that $U(K)\setminus T$ is not
empty. By the Hurwitz formula, $f_{j}$ is ramified for every $j$. In
particular, $T$ is strongly thin in $\mathbb{P}^{1}_{K}$.

As ${\rm char}(K)=0$, there exists a nonempty open subset $X^{\prime}$ of $X$
and an étale cover $p\colon X^{\prime}\to V$ where $V$ is some nonempty open
subscheme of $\mathbb{A}^{n}_{K}$, $n=\dim(X)$. Let $q\colon
V\to\mathbb{A}_{K}^{1}\subseteq\mathbb{P}^{1}_{K}$ be the projection on the
first coordinate followed by the inclusion. Then $(q\circ p)(X^{\prime})$ is
open in $\mathbb{P}^{1}$. By Lemma 5.2 there exists an automorphism $\tau$ of
$\mathbb{P}^{1}_{K}$ such that $\tau(q(p(X^{\prime})))$ contains the finitely
many branch points of each $f_{j}$. Let $g=\tau\circ q\circ p$. Note that
$\tau\circ q$ is smooth and has geometrically connected fibers, and recall
that $p$ is an étale cover, in particular smooth and proper. By Lemma 5.3 we
conclude that $\bar{T}:=\\{x\in X^{\prime}(K):g(x)\in T\\}$ is strongly thin
in $X^{\prime}$. It follows that $\bar{T}$ is strongly thin in $X$. Since $X$
has WHP, $g^{-1}(U)(K)\setminus\bar{T}$ contains a point $x_{0}$, and
$g(x_{0})\in U(K)\setminus T$ as desired. ∎

Our first example shows that HP and WHP do not go down finite extensions. More
precisely, “not HP” and “not WHP” are not preserved under base change in
finite extensions $L/K$, in a strong sense:

######  Proposition 5.6.

Let $K=\mathbb{Q}^{\rm tr}$ be the maximal totally real Galois extension of
$\mathbb{Q}$ and let $L=\mathbb{Q}^{\rm tr}(i)$. Then every normal $L$-variety
$X$ has HP, but no normal $K$-variety has WHP.

###### Proof.

The first claim follows from Proposition 5.1 since $L$ is Hilbertian by a
theorem of Weissauer, and PAC by results of Moret-Bailly and
Green–Pop–Roquette, see [Jar11, Example 5.10.7]. The second claim follows from
Proposition 5.5 since $K$ is not Hilbertian, see e.g. [BF14, Thm. 1]. ∎

Our second example is motivated by attempts to prove a general product theorem
for WHP (see discussion before Question 3.6) at least over small extensions
$L$ of $\mathbb{Q}$: If $X,Y$ are $L$-varieties with WHP, one could try to
descend $X$ and $Y$ to varieties with WHP over a number field $K\subseteq L$,
apply the product theorem of [CDJLZ22] there, and then use Corollary 4.3 for
the small extension $L/K$ to conclude that $X\times_{L}Y$ has WHP. The
following Proposition 5.8 shows that this strategy fails.

######  Lemma 5.7.

Let $K$ be a countable Hilbertian field. There exists a PAC field that is a
small extension of $K$.

###### Proof.

Let $(C_{i})_{i\in\mathbb{N}}$ be an enumeration of the geometrically integral
$K$-curves. For each $i$ we apply the stabilizing basis theorem [FJ08, Theorem
18.9.3] to obtain an integer $n_{i}$ and a non-constant rational map
$f_{i}\colon C_{i}\to\mathbb{P}^{1}$ of degree $n_{i}$ with the following
properties:

  1. (1)

$5\leq n_{1}<n_{2}<n_{3}<\ldots$, and

  2. (2)

The monodromy group of $f_{i}$ (that is, the Galois group of the Galois
closure of $K(C_{i})/K(\mathbb{P}^{1})$ viewed as a permutation group acting
on the generic fiber) and the geometric monodromy group of $f_{i}$ (i.e., the
monodromy group over $\overline{K}$) are both $S_{n_{i}}$, for each $i$.

As in the proof of [FJ08, Lemma 16.2.6], since $K$ is Hilbertian, we may
inductively construct $a_{i}\in\mathbb{P}^{1}(K)$ such that, for each $i$,

  1. (3)

The fiber $f_{i}^{-1}(a_{i})$ of $f_{i}$ above $a_{i}$ is integral. In
particular, $f_{i}^{-1}(a_{i})=\mathrm{Spec}(E_{i})$ for some field extension
$E_{i}/K$ of degree $n_{i}$. (In the notation of [FJ08, Lemma 16.2.6],
$E_{i}=K(b_{i})$.)

  2. (4)

If $L_{i}$ denotes the Galois closure of $E_{i}/K$, we have
$\mathrm{Gal}(L_{i}/K)=S_{n_{i}}$ and $L_{i}$ is linearly disjoint over $K$
from the compositum $L_{1}\cdots L_{i-1}$.

Let $E=\prod_{i=1}^{\infty}E_{i}$ and $L=\prod_{i=1}^{\infty}L_{i}$ be the
field composita. By construction, $C_{i}(E)\neq\emptyset$ for every $i$, hence
$E$ is PAC [FJ08, Theorem 11.2.3]. Also, by construction, $L/K$ is Galois,
$\mathrm{Gal}(L/K)=\Gamma:=\prod_{i=1}^{\infty}S_{n_{i}}$, and
$\mathrm{Gal}(L/E)=\Gamma_{1}:=\prod_{i=1}^{\infty}S_{n_{i}-1}$, where
$S_{n-1}\leq S_{n}$ is the stabilizer of a point. Let
$\Delta:=\prod_{i=1}^{\infty}A_{n_{i}}\leq\Gamma$. Since the $A_{n_{i}}$ are
distinct, simple non-abelian, and generated by two elements, $\Delta$ is
topologically generated by two elements, so in particular the number
$N_{r}(\Delta)$ of open subgroups of $\Delta$ of index $r$ is finite, for
every $r\in\mathbb{N}$. On the other hand, since $n_{i}\geq 5\geq 3$,
$S_{n_{i}-1}\cdot A_{n_{i}}=S_{n_{i}}$ (where we denote $H_{1}\cdot
H_{2}=\\{h_{1}h_{2}:h_{1}\in H_{1},\ h_{2}\in H_{2}\\}$). So
$\Gamma_{1}\cdot\Delta=\Gamma$.

Let $\mathcal{G}$ be the set of open subgroups of $\Gamma$ containing
$\Gamma_{1}$. For $G\in\mathcal{G}$, the inclusion $\Delta\rightarrow\Gamma$
induces a bijection $\Delta/(G\cap\Delta)\rightarrow\Gamma/G$, hence
$(\Gamma:G)=(\Delta:G\cap\Delta)$. In particular, the map $\mathcal{G}\ni
G\mapsto G\cap\Delta$ is injective, which implies that there are at most
$N_{r}(\Delta)$ many open subgroups of $\Gamma$ of index $r$ containing
$\Gamma_{1}$. So by the Galois correspondence, there at most $N_{r}(\Delta)$
subextensions of $E/K$ of degree $r$, for every $r$, hence $E/K$ is small. ∎

######  Proposition 5.8.

There exists a small extension $L/\mathbb{Q}$ and a smooth proper
$\mathbb{Q}$-variety $X$ such that $X_{L}$ has HP but $X_{K}$ does not have
WHP for any number field $K$.

###### Proof.

Let $X$ be any smooth proper $\mathbb{Q}$-curve of genus at least $2$, e.g.
the Fermat curve $x^{4}+y^{4}=z^{4}$. By Lemma 5.7 there exists a small
extension $L$ of $\mathbb{Q}$ which is PAC. As $L/\mathbb{Q}$ is small and
$\mathbb{Q}$ is Hilbertian, so is $L$ (cf. Proposition 4.3). So by Proposition
5.1, $X_{L}$ has HP. By Falting’s theorem, $X(K)$ is not Zariski-dense in $X$,
so $X_{K}$ does not have WHP, for any number field $K$. ∎

In particular, “not HP” and “not WHP” do not go up in chains of number fields
even when the union of the chain is a small extension of $\mathbb{Q}$. We
remark that the extension $L/\mathbb{Q}$ constructed here has Galois closure
which is not small over $\mathbb{Q}$.

######  Question 5.9.

Does there exist a small Galois extension $L/\mathbb{Q}$ with $L$ PAC?

## Acknowledgements

A.F. would like to thank Philip Dittmann for helpful discussions around
Proposition 5.5. L.B.-S. was supported by the Israel Science Foundation (grant
no. 702/19). L.B.-S. and S.P. thank TU Dresden for hospitality during a
research visit in December 2023 where part of this work was done.

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#  Span of Restriction of Hilbert Theta Functions

Gabriele Bogo, Yingkun Li  Fachbereich Mathematik, Technische Universität
Darmstadt, Schlossgartenstrasse 7, D–64289 Darmstadt, Germany
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In this paper, we study the diagonal restrictions of certain Hilbert theta
series for a totally real field $F$, and prove that they span the
corresponding space of elliptic modular forms when the $F$ is quadratic or
cubic. Furthermore, we give evidence of this phenomenon when $F$ is quartic,
quintic and sextic.

##  1\. Introduction

Theta functions are classical examples of holomorphic modular forms. Given a
positive definite, unimodular $\mathbb{Z}$-lattice $L$ of rank $8m$ with
$m\in\mathbb{N}$, the associated theta function

(1.1) $\theta_{L}(\tau):=\sum_{\lambda\in
L}q^{Q(\lambda)},~{}q:={\mathbf{e}}(\tau):=e^{2\pi i\tau},$

is in $M_{4m}$, the space of elliptic modular forms of weight $4m$ on
${\mathrm{SL}}_{2}(\mathbb{Z})$. For example, the theta functions associated
to the $E_{8}$ lattice and Leech lattice $\Lambda_{24}$ are explicitly given
as

(1.2)
$\theta_{E_{8}}(\tau)=E_{4}(\tau),~{}\theta_{\Lambda_{24}}(\tau)=E_{4}(\tau)^{3}-720\Delta(\tau),$

where $E_{2k}(\tau)$ is the Eisenstein series of weight $2k$ and
$\Delta(\tau)$ is the Ramanujan $\Delta$-function.

For $N\in\mathbb{N}$, we denote

(1.3) ${\mathcal{M}}^{(N)}_{\mathbb{Q}}:=\bigoplus_{k\in\mathbb{N}}M_{Nk}$

the finitely generated graded algebra of elliptic modular forms with weights
divisible by $N$, and would like to consider the subalgebra
${\mathcal{M}}^{\theta}_{\mathbb{Q}}\subset{\mathcal{M}}^{(4)}_{\mathbb{Q}}$
generated by theta functions of unimodular lattices. Using the relation

(1.4) $\theta_{L_{1}\oplus
L_{2}}(\tau)=\theta_{L_{1}}(\tau)\theta_{L_{2}}(\tau).$

for any two unimodular lattices $L_{1},L_{2}$, we see that
${\mathcal{M}}^{\theta}_{\mathbb{Q}}$ is simply the span of such theta
functions. Equation (1.2) and the fact
${\mathcal{M}}^{(4)}_{\mathbb{Q}}=\mathbb{C}[E_{4},\Delta]$ imply that

(1.5) ${\mathcal{M}}^{\theta}_{\mathbb{Q}}={\mathcal{M}}^{(4)}_{\mathbb{Q}}.$

The construction of theta functions also extends to the case of Hilbert
modular forms. Let $F$ be a totally real field of degree $d$ with ring of
integers ${\mathcal{O}_{F}}$, and denote $\alpha_{j}\in\mathbb{R}$ the real
embeddings of $\alpha\in F$ for $1\leq j\leq d$. For $N\in\mathbb{N}$, denote
${\mathcal{M}}_{F}^{(N)}$ the algebra of holomorphic Hilbert modular forms of
parallel weight $Nk$ for $k\in\mathbb{N}$. Given a totally positive definite,
$\mathbb{Z}$-unimodular ${\mathcal{O}_{F}}$-lattice $L$ of rank $8m$ (see
Definition 1), the associated theta function

(1.6) $\theta_{L}(\tau):=\sum_{\lambda\in
L}\prod_{j=1}^{d}q_{j}^{Q(\lambda)_{j}},~{}\tau=(\tau_{1},\dots,\tau_{d})\in\mathbb{H}^{d},~{}q_{j}:={\mathbf{e}}(\tau_{j}),$

is a Hilbert modular form of parallel weight $4m$ on
${\mathrm{SL}}_{2}({\mathcal{O}_{F}})$. It is well-known that such lattice
exists precisely when

(1.7) $m\in\frac{1}{d_{2}}\mathbb{N},~{}d_{2}:=\gcd(2,d)$

(see Prop. 2.1). However, their explicit constructions and classification have
only been carried out when $d$ is small (see e.g. [Sch94, Wan14]). As a
result, the relationship between ${\mathcal{M}}_{F}^{(4/d_{2})}$ and the
subalgebra ${\mathcal{M}}^{\theta}_{F}$ generated by such $\theta_{L}$ is not
clear.

On the other hand, we have the following diagonal restriction map

$\displaystyle{\mathcal{M}}^{(N)}_{F}$
$\displaystyle\to{\mathcal{M}}^{(Nd)}_{\mathbb{Q}}$ $\displaystyle f$
$\displaystyle\mapsto f^{\Delta}(\tau):=f(\tau^{\Delta}),$

where $\tau^{\Delta}=(\tau,\dots,\tau)\in\mathbb{H}^{d}$. In this note, we
will investigate the question about the image of ${\mathcal{M}}^{\theta}_{F}$
under this map, which is denoted by $({\mathcal{M}}^{\theta}_{F})^{\Delta}$
and contained in ${\mathcal{M}}^{(4d/d_{2})}_{\mathbb{Q}}$. The main result is
as follows.

######  Theorem 1.1.

For a totally real field $F$ of degree $d=2,3$, we have

(1.8)
$({\mathcal{M}}_{F}^{\theta})^{\Delta}={\mathcal{M}}^{(4d/d_{2})}_{\mathbb{Q}}.$

Based on this, it is then natural to make the following conjecture.

######  Conjecture 1.

Equation (1.8) holds for any totally real field $F$ of degree $d$.

To prove Theorem 1.1, we apply an instance of the Siegel-Weil formula to see
that the Hecke Eisenstein series $E_{F,k}$ defined in (2.5) is contained in
${\mathcal{M}}^{\theta}_{F}$ for all $k\in(4/d_{2})\mathbb{N}$. Then we
calculate the Petersson inner product between the diagonal restriction of
$E_{F,k}$ and an elliptic cusp form. For $d=2$, this inner product is related
to Fourier coefficients of half-integral weight modular forms by a result of
Kohnen-Zagier [KZ84]. For $d\geq 3$, we give an expression for this inner
product in terms of a sum over the double coset
$\Gamma_{F,\infty}\backslash\Gamma_{F}/\Gamma_{\mathbb{Q}}$ (see Prop. 3.1).
When $d=3$, we related this double coset to orders in a cubic field $F$ (see
section 4). Using these results, we show that when $d=2,3$,
${\mathcal{M}}_{\mathbb{Q}}^{(4d/d_{2})}$ can be generated by
$E_{F,k}^{\Delta}$ and $\theta_{L}^{\Delta}$ for a $\mathbb{Z}$-unimodular
${\mathcal{O}_{F}}$-lattice $L$.

The same approach can be used to check conjecture 1 numerically when
$d\in\\{4,5,6,8,10\\}$. We list some results for $d=4,5,6$ and $F$ has small
discriminants in the last section (see Theorem 6.1).

Acknowledgement: We thank Jan Bruinier and Tonghai Yang for helpful discussion
about Prop. 2.3. The authors are supported by the LOEWE research unit USAG,
and by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative
Research Centre TRR 326 “Geometry and Arithmetic of Uniformized Structures”,
project number 444845124.

##  2\. Preliminary

Let $F$ be a totally real field of degree $d$ with ring of integers
${\mathcal{O}_{F}}$ and different $\mathfrak{d}_{F}$. Denote
${\mathrm{Cl}}(F)$ the (wide) class group of $F$. Let $(V,Q)$ be an
$F$-quadratic space of dimension $n$. We say that $V$ is totally positive if
$V\otimes_{\iota(F)}\mathbb{R}$ is totally positive for every real embedding
$\iota:F\hookrightarrow\mathbb{R}$. In that case,
${\mathrm{SO}}_{V}(\mathbb{R})$ is compact and the double quotient
${\mathrm{SO}}_{V}(F)\backslash{\mathrm{SO}}_{V}(\hat{F})/K$ is a finite set
for any open compact subgroup $K\subset{\mathrm{SO}}_{V}(\hat{F})$. Here
$\mathbb{A}_{F}$ and $\hat{F}$ are the adele and finite adele of $F$.

A finitely generated ${\mathcal{O}_{F}}$-module $L\subset V$ is called a
(${\mathcal{O}_{F}}$-)lattice if $L\otimes_{\mathcal{O}_{F}}F=V$. We denote
$\hat{L}:=L\otimes\hat{\mathbb{Z}}\subset\hat{V}=V\otimes\hat{\mathbb{Q}}$. If
$Q(L)\subset\mathfrak{d}_{F}^{-1}$, we say that $L$ is $\mathbb{Z}$-even
integral and call the lattice

(2.1) $L^{\prime}:=\\{y\in V:(y,L)\subset\mathfrak{d}_{F}^{-1}\\}$

its $\mathbb{Z}$-dual. Viewed as a $\mathbb{Z}$-lattice with respect to
$Q_{\mathbb{Q}}(x):=\mathrm{tr}_{F/\mathbb{Q}}Q(x)$, such $L$ is even integral
with dual $L^{\prime}$.

######  Definition 1.

An ${\mathcal{O}_{F}}$-lattice $L$ is said to be $\mathbb{Z}$-unimodular if
$L^{\prime}=L$.

As a convention, the trivial lattice in the trivial $F$-vector space is
totally positive and $\mathbb{Z}$-unimodular. Consider the monoid

(2.2) ${\mathcal{U}}^{+}_{F}:=\\{(L,Q):L\text{ is an even
}\mathbb{Z}\text{-unimodular }{\mathcal{O}_{F}}\text{-lattice and totally
positive definite}\\}$

with respect to $\oplus$, and denote
${\mathcal{U}}^{+,n}_{F}\subset{\mathcal{U}}^{+}_{F}$ the subset of lattices
of rank $n$. We first have the following result.

######  Proposition 2.1.

The set ${\mathcal{U}}^{+,n}_{F}$ is non-empty precisely when $(8/d_{2})\mid
n$.

###### Proof.

Satz 1 in [Cha70] implies that there exists definite, unimodular
${\mathcal{O}_{F}}$-lattices in the sense loc. cit. if and only if
$(8/d_{2})\mid n$. Furthermore since $n$ is even, all of the $2^{d}$ possible
definite signatures will appear in the set of definite, unimodular
${\mathcal{O}_{F}}$-lattices of rank $n$. One can then use the fact that the
class $\mathfrak{d}_{F}$ in the class group is a square to translate this
result to the existence $\mathbb{Z}$-unimodular lattices. (see the proof of
Prop. 2.5 in [Li21] for details). ∎

######  Remark 2.2.

For $L\in{\mathcal{U}}^{+,n}_{F}$ and
$h\in{\mathrm{SO}}_{V}(\hat{\mathbb{Q}})$ with
$V=L\otimes_{\mathcal{O}_{F}}F$, the lattice

(2.3) $h\cdot L:=(h\cdot\hat{L})\cap V\subset V$

is also in ${\mathcal{U}}^{+,n}_{F}$.

For each $L\in{\mathcal{U}}^{+,n}_{F}$, let $\theta_{L}(\tau)$ be the
associated theta function defined in (1.6). It is a Hilbert modular form of
parallel weight $n/2$ for ${\mathrm{SL}}_{2}({\mathcal{O}_{F}})$. Now, the
Siegel-Weil formula [Sie66, Wei65] gives us the following result.

######  Proposition 2.3.

Let $F$ be a totally real field of degree $d$. Then

(2.4)
$\int_{{\mathrm{SO}}_{V}(F)\backslash{\mathrm{SO}}_{V}(\mathbb{A}_{F})/{\mathrm{SO}}_{V}(\mathbb{R})}\theta_{h\cdot
L}(\tau)dh=\kappa E_{F,n/2}(\tau),$

for some positive constant $\kappa$, where $E_{F,k}$ is the Hecke Eisenstein
series of parallel weight $k$ defined by

(2.5)
$\begin{split}E_{F,k}(\tau)&:=1+\zeta_{F}(k)^{-1}\sum_{{\mathcal{A}}=[\mathfrak{a}]\in{\mathrm{Cl}}(F)}{\mathrm{Nm}}(\mathfrak{a})^{k}\sum_{(c,d)\in\mathfrak{a}^{2}/\mathcal{O}_{F}^{\times},~{}c\neq
0}\prod_{j=1}^{d}(c_{j}\tau_{j}+d_{j})^{-k}\end{split}$

In particular, $E_{F,k}\in{\mathcal{M}}^{\theta}_{F}$ for all
$k\in(4/d_{2})\mathbb{N}$.

######  Remark 2.4.

The Hecke Eisenstein series have the following well-known Fourier expansion
(see [Sie69, Zag76])

(2.6)
$E_{F,k}(\tau)=1+\frac{2^{d}}{\zeta_{F}(1-k)}\sum_{t\in\mathfrak{d}_{F}^{-1},~{}t\gg
0}\sigma_{k-1}(t\mathfrak{d}_{F})\prod_{j=1}^{d}q_{j}^{t_{j}\tau_{j}}$

with
$\sigma_{r}(\mathfrak{a}):=\sum_{{\mathfrak{b}|\mathfrak{a},~{}\mathfrak{b}\subset\mathcal{O}_{F}}}{{\mathrm{Nm}}(\mathfrak{b})^{r}}$
for any integral ideal $\mathfrak{a}$ and $r\in\mathbb{N}$.

###### Proof.

By the Siegel-Weil formula, the left hand side of (2.4) equals to the
Eisenstein series

$E_{L}(\tau)=v^{-n/4}\sum_{\gamma\in
B(F)\backslash{\mathrm{SL}}_{2}(F)}\Phi_{L}(\gamma g_{\tau},n/2-1),$

where $B\subset{\mathrm{SL}}_{2}$ is the standard Borel subgroup, and
$\Phi_{L}$ is the Siegel-Weil section associated to the lattice $L$ (see e.g.
[Kud08, section I.3]). For $t\in F^{\times}$, the $t$-th Fourier coefficient
of $E_{L}$ is given by

$\prod_{\mathfrak{p}<\infty}W_{t,\mathfrak{p}}(1,n/2-1,\Phi_{L,\mathfrak{p}})$

up to constant independent of $t$. Here $W_{t,\mathfrak{p}}(g,s,\phi)$ is the
local Whittaker function (see e.g. [Yan05]). Since $L$ is
$\mathbb{Z}$-unimodular, the local lattice
$L\otimes\mathcal{O}_{F,\mathfrak{p}}$ in $V\otimes F_{\mathfrak{p}}$ is self-
dual for every finite place $\mathfrak{p}$. Standard calculations (see e.g.
[KY10]) then gives us

$W_{t,\mathfrak{p}}(1,s,\Phi_{L,\mathfrak{p}})=\sum_{m=0}^{{\mathrm{ord}}_{\mathfrak{p}}(t\mathfrak{d}_{F_{\mathfrak{p}}})}{\mathrm{Nm}}(\mathfrak{p})^{s}$

when $t\in\mathfrak{d}_{F_{\mathfrak{p}}}^{-1}$, and zero otherwise. So up to
a constant, the Eisenstein series $E_{L}$ and $E_{F,n/2}$ have the same non-
constant term Fourier coefficients, hence agree. Now the left hand side of
(2.4) is just a sum of $\theta_{L_{j}}$ over certain
$L_{j}\in{\mathcal{U}}^{+,n}_{F}$ by Remark 2.2. Combining this with Prop. 2.1
finishes the proof. ∎

We can rewrite the Hecke-Eisenstein series $E_{F,k}$ as

$E_{F,k}(\tau):=1+\sum_{\begin{subarray}{c}{\mathcal{A}}=[\mathfrak{a}]\in{\mathrm{Cl}}(F)\\\
(c,d)\in\mathfrak{a}^{2}/\mathcal{O}_{F}^{\times}\\\ c\neq 0\\\
{\mathcal{O}_{F}}c+{\mathcal{O}_{F}}d=\mathfrak{a}\end{subarray}}\left(\frac{{\mathrm{Nm}}(\mathfrak{a})}{{\mathrm{Nm}}(c)}\right)^{k}\prod_{j=1}^{d}(\tau_{j}+d_{j}/c_{j})^{-k}$

For any $\beta\in F$, there is unique ${\mathcal{A}}=[\mathfrak{a}]$ and
$(c,d)\in\mathfrak{a}^{2}/\mathcal{O}_{F}^{\times}$ with $c\neq 0$ such that
$\mathfrak{a}={\mathcal{O}_{F}}c+{\mathcal{O}_{F}}d$ and $\beta=d/c$.
Therefore, we denote

(2.7)
$A_{\beta}:=\frac{{\mathrm{Nm}}(c)}{{\mathrm{Nm}}(\mathfrak{a})}\in\mathbb{Z}-\\{0\\}.$

It is easy to check this definition does not depend on the choice of the
representative $\mathfrak{a}$, and

(2.8) $A_{\beta+a,k}=A_{\beta,k}$

for all $a\in\mathbb{Z}$. Then we have

(2.9) $E_{F,k}(\tau)=1+\sum_{\beta\in
F}A_{\beta}^{-k}\prod_{j=1}^{d}(\tau_{j}+\beta_{j})^{-k}.$

##  3\. Petersson Inner Product Calculations

In this section, let $F/\mathbb{Q}$ be totally real with degree $d\geq 3$. We
will give an expression for the Petersson inner product between the diagonal
restriction of the Hecke Eisenstein series $E_{F,k}$ and an elliptic cusp form
$f$ of weight $dk$.

For $\alpha\in M_{m,n}(F)$ and $1\leq j\leq d$, we write $\alpha_{j}\in
M_{m,n}(\mathbb{R})$ with $1\leq j\leq d$ for the real embeddings of $\alpha$.
We identify $\mathbb{P}^{1}(F)\cong B(F)\backslash{\mathrm{SL}}_{2}(F)$ via

(3.1) $\beta\mapsto\begin{cases}\left(\begin{smallmatrix}*&*\\\
1&\beta\end{smallmatrix}\right)&\beta\in F,\\\ \left(\begin{smallmatrix}*&*\\\
0&1\end{smallmatrix}\right)&\beta=\infty.\end{cases}$

Let $S_{0}\cup\\{\infty\\}\subset\mathbb{P}^{1}(F)$ be a set of
representatives of the double coset
$B(F)\backslash{\mathrm{SL}}_{2}(F)/{\mathrm{SL}}_{2}(\mathbb{Z})$. Then
$S_{0}\subset F-\mathbb{Q}$ and we can use (2.9) to express the diagonal
restriction of $E_{F,k}$ as

(3.2) $E_{F,k}^{\Delta}(\tau)=E_{dk}+\sum_{\beta\in
S_{0}}E_{F,k,\beta}(\tau),~{}E_{F,k,\beta}(\tau):=\sum_{\gamma\in{\mathrm{SL}}_{2}(\mathbb{Z})}A_{-\gamma^{-1}\cdot(-\beta_{j})}^{-k}\prod_{j=1}^{d}(\tau-\gamma^{-1}\cdot(-\beta_{j}))^{-k}$

with $\tau\in\mathbb{H}$. Note that $E_{dk}$ is just the elliptic Eisenstein
series of weight $dk$.

Let $f(\tau)=\sum_{n\geq 1}c_{n}q^{n}\in S_{dk}$ be a cusp form. We are
interested in estimating its inner product with $E^{\Delta}_{F,k}$. By the
usual unfolding process, we obtain

$\displaystyle\langle E^{\Delta}_{F,k},f\rangle$ $\displaystyle=\sum_{\beta\in
S_{0}}\int_{\Gamma_{\infty}\backslash\mathbb{H}}E^{\infty}_{F,k,\beta}(\tau)\overline{f(\tau)}v^{dk}\frac{dudv}{v^{2}}$
$\displaystyle=\sum_{\beta\in S_{0}}\int_{0}^{\infty}\sum_{n\geq
1}\overline{c_{n}}a_{F,k,\beta}(n,v)e^{-2\pi nv}v^{dk-1}\frac{dv}{v},$

where $\Gamma_{\infty}:=B(\mathbb{Q})\cap{\mathrm{SL}}_{2}(\mathbb{Z})$ and

(3.3)
$\begin{split}E^{\infty}_{F,k,\beta}(\tau)&:=\sum_{\gamma\in\Gamma_{\infty}}A_{-\gamma^{-1}\cdot(-\beta)}^{-k}\prod_{j=1}^{d}(\tau-\gamma^{-1}\cdot(-\beta_{j}))^{-k}\\\
&=2A_{\beta}^{-k}\prod_{j=1}^{d}(\tau+\beta_{j}+b)^{-k}=\sum_{n\in\mathbb{Z}}a_{F,k,\beta}(n,v){\mathbf{e}}(nu).\end{split}$

for $\beta=d/c\in S_{0}$. Here we have $r_{-\gamma\cdot(-\beta)}=r_{\beta}$
for all $\gamma\in\Gamma_{\infty}$ by (2.8). It is easy to see that

(3.4)
$\begin{split}a_{F,k,\beta}(n,v)&=2A_{\beta}^{-k}\int_{\mathbb{R}}\prod_{j=1}^{d}(u+iv+\beta_{j})^{-k}{\mathbf{e}}(-nu)du\\\
&=4\pi i(-A_{\beta})^{-k}\sum_{z\in
Z(\beta)}\mathrm{Res}_{x=z}\left({\mathbf{e}}(nx)\prod_{j=1}^{d}(x-(\beta_{j}+iv))^{-k}\right),\end{split}$

where $Z(\beta):=\\{\beta_{j}+iv:1\leq j\leq d\\}\subset\mathbb{H}$ since

(3.5) $\sum_{z\in
Z(\beta)}\mathrm{Res}_{x=z}\left({\mathbf{e}}(nx)\prod_{j=1}^{d}(x-z_{j})^{-k}\right)=\frac{1}{2\pi
i}\int_{\mathbb{R}}{\mathbf{e}}(nx)\prod_{j=1}^{d}(x-z_{j})^{-k}dx.$

Suppose $\beta_{j}$’s are all distinct. Then

$\displaystyle\sum_{z\in Z(\beta)}\mathrm{Res}_{x=z}$
$\displaystyle\left({\mathbf{e}}(nx)\prod_{j=1}^{d}(x-(\beta_{j}+iv))^{-k}\right)=\frac{1}{\Gamma(k)}\sum_{j=1}^{d}\left(\frac{d}{dx}\right)^{k-1}\left(\frac{{\mathbf{e}}(nx)}{\prod_{j^{\prime}=1,~{}j^{\prime}\neq
j}^{d}(x-(\beta_{j^{\prime}}+iv))^{k}}\right)\mid_{x=\beta_{j^{\prime}}+iv}$
$\displaystyle=\frac{{\mathbf{e}}(niv)}{\Gamma(k)}\sum_{j=1}^{d}\sum_{\ell=0}^{k-1}(2\pi
in)^{k-1-\ell}{{\mathbf{e}}(n\beta_{j})e^{-2\pi
nv}}\binom{k-1}{\ell}\left(\frac{P_{d-1,k,\ell}}{Q_{d-1,k+\ell}}\right)(\beta_{j}-\beta_{1},\dots,\beta_{j}-\beta_{d}),$

where $P_{m,k,\ell},Q_{m,r}\in\mathbb{Q}[x_{1},\dots,x_{m}]$ are symmetric
polynomials of degrees $(m-1)\ell$ and $mr$ defined by

(3.6) $\begin{split}P_{m,k,\ell}(x_{1},\dots,x_{m})&:=(x_{1}\dots
x_{m})^{k+\ell}(\partial_{x_{1}}+\dots+\partial_{x_{m}})^{\ell}(x_{1}\dots
x_{m})^{-k},\\\ Q_{m,r}(x_{1},\dots,x_{m})&:=(x_{1}\dots
x_{m})^{r}.\end{split}$

Note that

(3.7)
$\frac{P_{m,k,\ell}}{Q_{m,k+\ell}}(x_{1},\dots,x_{m})=(-1)^{\ell}{\ell!}\sum_{r=(r_{j})\in\mathbb{N}^{m},~{}\sum_{j}r_{j}=\ell}\left(\\!\\!\binom{k}{r}\\!\\!\right)\prod_{j=1}^{m}x_{j}^{-k-r_{j}},$

where $\left(\\!\\!\binom{k}{r}\\!\\!\right):=\frac{k^{(r_{1})}\dots
k^{(r_{m})}}{r_{1}!\dots r_{m}!}$ for $r=(r_{1},\dots,r_{m})\in\mathbb{N}^{m}$
with $k^{(n)}:=k(k+1)\dots(k+n-1)$. Substituting this into the unfolding gives
us the following result.

######  Proposition 3.1.

Suppose $F$ is a totally real field of degree $d\geq 3$ and there is no
intermediate field between $F$ and $\mathbb{Q}$. For any $k\in 2\mathbb{N}$
and $f(\tau)=\sum_{n\geq 1}c(n)q^{n}\in S_{dk}$, we have

(3.8) $\begin{split}&\langle
E^{\Delta}_{F,k},f\rangle=\frac{i\Gamma(dk-1)}{(4\pi)^{dk-2}\Gamma(k)}\sum_{\ell=0}^{k-1}(2\pi
i)^{k-1-\ell}\sum_{\beta\in S_{0}}A_{\beta}^{-k}\\\
&\times\sum_{j=1}^{d}\left(\frac{P_{d-1,k,\ell}}{Q_{d-1,k+\ell}}\right)(\beta_{j}-\beta_{1},\dots,\beta_{j}-\beta_{j-1},\beta_{j}-\beta_{j+1},\dots,\beta_{j}-\beta_{d})\sum_{n\geq
1}\frac{{\mathbf{e}}(n\beta_{j})\overline{c_{n}}}{n^{(d-1)k+\ell}},\end{split}$

where the polynomials $P_{m,k,\ell}$ and $Q_{m,r}$ are defined in (3.6).

######  Remark 3.2.

The condition that there is no intermediate field between $F$ and $\mathbb{Q}$
implies that $\beta_{i}=\beta_{j}$ if and only if $i=j$ for all $\beta\in
F-\mathbb{Q}$. A similar but more complicated formula for the inner product
can be derived without this condition.

######  Example 3.3.

Let $d=3$ and $k=2$. Then

$\frac{P_{d-1,k,\ell}}{Q_{d-1,k+\ell}}(x,y)=\begin{cases}1/(xy)^{2},&\ell=0,\\\
-2(x+y)/(xy)^{3},&\ell=1.\end{cases}$

Set
$\gamma_{1}:=\beta_{2}-\beta_{3},\gamma_{2}:=\beta_{3}-\beta_{1},\gamma_{3}:=\beta_{1}-\beta_{2}$,
we have

$\displaystyle\sum_{\ell=0}^{k-1}(2\pi
in)^{k-1-\ell}\sum_{j=1}^{d}\frac{P_{d-1,k,\ell}}{Q_{d-1,k+\ell}}(\beta_{j}-\beta_{1},\dots,\beta_{j}-\beta_{d}){\mathbf{e}}(n\beta_{j})$
$\displaystyle=\left(\frac{2\pi
in}{(\gamma_{2}\gamma_{3})^{2}}+\frac{2(\gamma_{3}-\gamma_{2})}{(\gamma_{2}\gamma_{3})^{3}}\right){\mathbf{e}}(n\beta_{1})+\left(\frac{2\pi
in}{(\gamma_{1}\gamma_{3})^{2}}+\frac{2(\gamma_{1}-\gamma_{3})}{(\gamma_{1}\gamma_{3})^{3}}\right){\mathbf{e}}(n\beta_{2})+\left(\frac{2\pi
in}{(\gamma_{1}\gamma_{2})^{2}}+\frac{2(\gamma_{2}-\gamma_{1})}{(\gamma_{1}\gamma_{2})^{3}}\right){\mathbf{e}}(n\beta_{3}).$

For $d=3$ and $k-1\geq\ell\geq 0$, we can write explicitly

$\displaystyle\sum_{j=1}^{d}\frac{P_{d-1,k,\ell}}{Q_{d-1,k+\ell}}(\beta_{j}-\beta_{1},\dots,\beta_{j}-\beta_{d}){\mathbf{e}}(n\beta_{j})$
$\displaystyle=\frac{P_{2,k,\ell}}{Q_{2,k+\ell}}(\gamma_{3},-\gamma_{2}){\mathbf{e}}(n\beta_{1})+\frac{P_{2,k,\ell}}{Q_{2,k+\ell}}(-\gamma_{3},\gamma_{1}){\mathbf{e}}(n\beta_{2})+\frac{P_{2,k,\ell}}{Q_{2,k+\ell}}(-\gamma_{2},-\gamma_{1}){\mathbf{e}}(n\beta_{3}).$

Using the inequalities $k^{(a)}k^{(b)}\leq k^{(a+b)}$,
$(x_{1}+x_{2}+x_{3})^{2}\leq 3(x_{1}^{2}+x_{2}^{2}+x_{3}^{2})$,

(3.9) $\sum_{\sigma\in
S_{3}}x_{\sigma(1)}^{a}x_{\sigma(2)}^{b}x_{\sigma(3)}^{c}\leq\frac{a!b!c!}{(a+b+c)!}(x_{1}+x_{2}+x_{3})^{a+b+c},~{}x_{i},a,b,c\geq
0$

and Equation (3.7), we obtain the bound

$\displaystyle\left|\sum_{j=1}^{d}\frac{P_{d-1,k,\ell}}{Q_{d-1,k+\ell}}(\beta_{j}-\beta_{1},\dots,\beta_{j}-\beta_{d}){\mathbf{e}}(n\beta_{j})\right|$
$\displaystyle\leq\left|\frac{P_{2,k,\ell}}{Q_{2,k+\ell}}(\gamma_{3},-\gamma_{2})\right|+\left|\frac{P_{2,k,\ell}}{Q_{2,k+\ell}}(-\gamma_{3},\gamma_{1})\right|+\left|\frac{P_{2,k,\ell}}{Q_{2,k+\ell}}(-\gamma_{2},-\gamma_{1})\right|$
$\displaystyle\leq\frac{\ell!}{|\gamma_{1}\gamma_{2}\gamma_{3}|^{k+\ell}}\sum_{a+b=\ell}\frac{k^{(a)}k^{(b)}}{a!b!}\left(|\gamma_{1}^{b}\gamma_{2}^{a}\gamma_{3}^{k+\ell}|+|\gamma_{2}^{b}\gamma_{3}^{a}\gamma_{1}^{k+\ell}|+|\gamma_{3}^{b}\gamma_{1}^{a}\gamma_{2}^{k+\ell}|\right)$
$\displaystyle\leq\ell!\frac{(|\gamma_{1}|+|\gamma_{2}|+|\gamma_{3}|)^{k+2\ell}}{|\gamma_{1}\gamma_{2}\gamma_{3}|^{k+\ell}}\frac{(k+\ell)!}{(k+2\ell)!}\frac{\ell+1}{2}k^{(\ell)}\leq\frac{\binom{k-1+\ell}{\ell}}{\binom{k+2\ell}{\ell}}(\ell+1)!\frac{3^{k/2+\ell}}{2}\frac{(\gamma_{1}^{2}+\gamma_{2}^{2}+\gamma_{3}^{2})^{k/2+\ell}}{|\gamma_{1}\gamma_{2}\gamma_{3}|^{k+\ell}}.$

##  4\. Double Coset and Binary Cubic Forms

When $d=3$, we can identify the double coset
$B(F)\backslash{\mathrm{SL}}_{2}(F)/{\mathrm{SL}}_{2}(\mathbb{Z})-\\{\infty\\}$
with orders in ${\mathcal{O}_{F}}$ in the following way. Let

$\mathcal{Q}_{F}:=\\{f(X,Y)=AX^{3}+BX^{2}Y+CXY^{2}+DY^{3}\in\mathbb{Z}[X,Y]:f(\beta,1)=0\text{
for some }\beta\in F\backslash\mathbb{Q}\\}$

be the set of integral binary cubic forms with a root in $F-\mathbb{Q}$. A
form is primitive if its coefficients have no common factor. There is a
natural action of ${\mathrm{SL}}_{2}(\mathbb{Z})$ on $\mathcal{Q}_{F}$ that
preserves the discriminant

(4.1)
$\begin{split}\Delta(f)&:=A^{6}((\beta_{1}-\beta_{2})(\beta_{1}-\beta_{3})(\beta_{2}-\beta_{3}))^{2}\\\
&=18ABCD+B^{2}C^{2}-4AC^{3}-4B^{3}D-27A^{2}D^{2},\end{split}$

and the subset of primitive forms. The quantity

(4.2) $P(f):=B^{2}-3AC>0$

is the leading coefficient of the Hessian of $f$, which is a positive definite
quadratic form and a coinvariant of $f$. For every $f\in\mathcal{Q}_{F}$,
Prop. 2 in [Cre99] gives us $f^{\prime}\sim_{{\mathrm{SL}}_{2}(\mathbb{Z})}f$
satisfying

(4.3) $P(f^{\prime})\leq\sqrt{\Delta(f^{\prime})}=\sqrt{\Delta(f)}.$

Given $\beta\in F-\mathbb{Q}$, we can associate to it a primitive element
$f_{\beta}\in\mathcal{Q}_{F}$ defined by

(4.4)
$f_{\beta}(X,Y):=A_{\beta}\prod_{j=1}^{3}(X-\beta_{j}Y)=A_{\beta}X^{3}+B_{\beta}X^{2}Y+C_{\beta}XY^{2}+D_{\beta}Y^{3}\in\mathcal{Q}_{F}.$

Note that $f_{\beta}(\beta,1)=0$ and the right action of
${\mathrm{SL}}_{2}(\mathbb{Z})$ on $B(F)\backslash{\mathrm{SL}}_{2}(F)$
corresponds to its natural action on $\mathcal{Q}_{F}$.

To any binary cubic form $f$ with non-zero discriminant and $f(\beta,1)=0$ we
can associate the free $\mathbb{Z}$-module of rank 3

(4.5)
$\mathcal{O}_{f}:=\mathbb{Z}+\mathbb{Z}A\beta+\mathbb{Z}(A\beta^{2}+B\beta+C)\subset\mathbb{Q}(\beta),$

which is also a commutative ring. A classical result of Delone and Faddeev
tells us that this gives a bijection between
${\mathrm{GL}}_{2}(\mathbb{Z})$-classes of binary cubic forms with non-zero
discriminants and isomorphism classes of commutative rings that are free
$\mathbb{Z}$-modules of rank 3 [DF64]. If we restrict $\beta$ to be in a fixed
field $F$, then $\mathcal{O}_{f}$ is an order in ${\mathcal{O}_{F}}$, and
$\mathcal{O}_{f_{1}},\mathcal{O}_{f_{2}}\subset{\mathcal{O}_{F}}$ are the same
if and only if $f_{1},f_{2}\in\mathcal{Q}_{F}$ are
${\mathrm{GL}}_{2}(\mathbb{Z})$-equivalent (see e.g. [Nak98, Lemma 3.1]).
Furthermore, we have

(4.6)
$\Delta(f)=\Delta(\mathcal{O}_{f})=D_{F}[{\mathcal{O}_{F}}:\mathcal{O}_{f}]^{2}$

with $\Delta(\cdot)$ the discriminant. For
$s=[\beta]\in\mathbb{P}^{1}(F)/{\mathrm{SL}}_{2}(\mathbb{Z})-\\{\infty\\}$, we
then denote

(4.7)
$\mathcal{O}_{s}:=\mathcal{O}_{f_{\beta}},\Delta(s):=\Delta(\mathcal{O}_{s}).$

The discussions above lead to the following result.

######  Proposition 4.1.

The map

$\displaystyle\mathbb{P}^{1}(F)/{\mathrm{SL}}_{2}(\mathbb{Z})-\\{\infty\\}$
$\displaystyle\to\\{\mathcal{O}:\mathcal{O}\subset{\mathcal{O}_{F}}\text{ is
an order}\\}/\cong$ $\displaystyle s$ $\displaystyle\mapsto\mathcal{O}_{s}$

is well-defined and $(2|\mathrm{Aut}({\mathcal{O}_{F}})|)$-to-1.

######  Remark 4.2.

The quantity $|\mathrm{Aut}({\mathcal{O}_{F}})|$ is either 3 or 1 depending on
$F/\mathbb{Q}$ is Galois or not.

Finally, the following Dirichlet series

(4.8)
$\begin{split}\eta_{F}(s)&:=\sum_{\mathcal{O}\subset{\mathcal{O}_{F}}\text{
order}}[{\mathcal{O}_{F}}:\mathcal{O}]^{-s}=\sum_{\mathcal{O}\subset{\mathcal{O}_{F}}\text{
order}}\frac{D_{F}^{s/2}}{\Delta(\mathcal{O})^{s/2}}.\end{split}$

can be factorized in the following way by a result of Datskovsky and Wright
[DW86] (see [Nak98, Lemma 3.2])

(4.9) $\eta_{F}(s)=\frac{\zeta_{F}(s)}{\zeta_{F}(2s)}\zeta(2s)\zeta(3s-1).$

##  5\. Proof of Theorem 1.1

We are now ready to prove Theorem 1.1. The cases of $d=2,3$ are proved
separately.

###### Proof of Theorem 1.1 for $d=2$.

For $k=2,4$, the space $M_{2k}$ is 1-dimensional and spanned by the Eisenstein
series $E_{2k}$. Since $\theta_{L}^{\Delta}$ is non-trivial for any
$L\in{\mathcal{U}}^{+}_{F}$, the claim follows in these two base cases as
$M^{\theta}_{F,k}$ is non-trivial by Prop. 2.1 (see also [Sch94] for an
explicit construction). More generally, we know that
${\mathcal{M}}_{\mathbb{Q}}^{(4)}=\mathbb{Q}[E_{4},\Delta]$. Therefore, it
suffices to show that $\Delta\in S_{12}$ is in $(M^{\theta}_{F,6})^{\Delta}$.
As $M_{12}$ is 2-dimensional and

(5.1)
$E_{4}^{3}=E_{12}+\frac{432000}{691}\Delta\in(M^{\theta}_{F,6})^{\Delta},$

we just need to produce a form $f\in(M^{\theta}_{F,6})^{\Delta}$ linearly
independent from $E_{4}^{3}$. For this purpose, we apply Prop. 2.3 with $k=6$
to get

$f(\tau):=(E_{F,6}^{\Delta})(\tau)=1+\frac{4}{\zeta_{F}(-5)}\sum_{m\geq
1}q^{m}\sum_{\nu\in\mathfrak{d}_{F}^{-1},~{}\nu\gg
0,~{}\mathrm{tr}(\nu)=m}\sigma_{5}((\nu)\mathfrak{d}_{F}).$

By Theorem 6 in [KZ84], we know that

(5.2) $f=E_{12}-\frac{12}{691}\frac{c(D)}{\zeta_{F}(-5)}\Delta,$

where $c(D)$ is the $D$-th Fourier coefficient of the half-integral weight
form

$g(\tau)=\sum_{D\in\mathbb{N}}c(D)q^{D}:=\frac{1}{8\pi
i}(2E_{4}(4\tau)\theta^{\prime}(\tau)-E_{4}^{\prime}(4\tau)\theta(\tau))$

spanning the Kohnen plus space $S_{13/2}^{+}$. Now using the easy estimate
$L(k,\chi_{D})>2-\zeta(k)$ for $k\geq 2$ (see e.g. Equation (3) in [CK13]) we
know that
$\zeta_{F}(1-k)=D^{k-1/2}\frac{4\Gamma(k)^{2}}{(-4\pi)^{k}}\zeta_{F}(k)$
satisfies

$|\zeta_{F}(-5)|>0.01\cdot D^{11/2}.$

On the other hand, the Hecke bound for $c(D)$ yields

$|c(D)|\leq c\cdot
D^{13/4},~{}c:=e^{2\pi}\max_{\tau\in\mathbb{H}}|g(\tau)|v^{13/4}<10$

Comparing with (5.1), it is clear that $f$ and $E_{4}^{3}$ are linearly
independent for all fundamental discriminant $D>0$. This finishes the proof of
Theorem 1.1 for $d=2$. ∎

Using the calculation in section 3 and the correspondence in section 4, we can
prove the following lemma.

######  Lemma 5.1.

For $d=3,k\geq 3$ and $f(\tau)=\sum_{n\geq 1}c_{f}(n)q^{n}\in S_{3k}$, let
$c_{f}>0$ be a constant such that

$|c_{f}(n)|\leq c_{f}\cdot n^{3k/2}$

for all $n\geq 1$. Then we have the bound

(5.3) $|\langle E^{\Delta}_{F,k},f\rangle|\leq C_{k}c_{f}D_{F}^{-k/4}$

for all cubic field $F$, with
$C_{k}:=6c_{k}\frac{\zeta(k/2)^{3}}{\zeta(k)^{2}}\zeta(3k/2-1)$ and the
constant $c_{k}$ given in (5.4).

###### Proof.

Let $a_{k}:=\tfrac{\Gamma(3k-1)}{\Gamma(k)}(4\pi)^{2-3k}$. For $\beta\in
S_{0}\subset F$, recall that $f_{\beta}$ is the binary cubic form associated
to it in (4.4), which has coefficients
$A_{\beta},B_{\beta},C_{\beta},D_{\beta}$. Using (3.8), the estimate in
Example 3.3 and (4.3), we obtain the bound

$\displaystyle|\langle E^{\Delta}_{F,k},f\rangle|$ $\displaystyle\leq
a_{k}\sum_{\ell=0}^{k-1}(2\pi)^{k-1-\ell}\sum_{n\geq
1}\frac{|c_{f}(n)|}{n^{2k+\ell}}\sum_{\beta\in
S_{0}}A_{\beta}^{-k}\left|\sum_{j^{\prime}=1}^{d}\frac{P_{d-1,k,\ell}}{Q_{d-1,k+\ell}}(\beta_{j^{\prime}}-\beta_{1},\dots,\beta_{j^{\prime}}-\beta_{d}){\mathbf{e}}(n\beta_{j^{\prime}})\right|$
$\displaystyle\leq c_{f}\cdot
a_{k}\sum_{\ell=0}^{k-1}(2\pi)^{k-1-\ell}\zeta(k/2+\ell)\frac{\binom{k-1+\ell}{\ell}}{\binom{k+2\ell}{\ell}}(\ell+1)!\frac{3^{k/2+\ell}}{2}$
$\displaystyle\times\sum_{\beta\in
S_{0}}A_{\beta}^{-k}\frac{((\beta_{1}-\beta_{2})^{2}+(\beta_{2}-\beta_{3})^{2}+(\beta_{3}-\beta_{1})^{2})^{k/2+\ell}}{((\beta_{1}-\beta_{2})^{2}(\beta_{2}-\beta_{3})^{2}(\beta_{3}-\beta_{1})^{2})^{(k+\ell)/2}}$
$\displaystyle\leq 2^{-1}c_{f}\cdot
a_{k}\sum_{\ell=0}^{k-1}(2\pi)^{k-1-\ell}\zeta(k/2+\ell)\frac{\binom{k-1+\ell}{\ell}}{\binom{k+2\ell}{\ell}}(\ell+1)!6^{k/2+\ell}\sum_{\beta\in
S_{0}}\frac{P(f_{\beta})^{k/2+\ell}}{\Delta(f_{\beta})^{(k+\ell)/2}}$
$\displaystyle\leq c_{f}\cdot c_{k}\sum_{\beta\in
S_{0}}\Delta(f_{\beta})^{-k/4}\leq c_{f}\cdot c_{k}\cdot
2|{\mathrm{Aut}}(\mathcal{O}_{F})|\cdot
D_{F}^{-k/4}\eta_{F}\left(\tfrac{k}{2}\right)$

Here the constant $c_{k}$ is defined by

(5.4)
$c_{k}:=\frac{\Gamma(3k-1)}{2\Gamma(k)}(4\pi)^{2-3k}\sum_{\ell=0}^{k-1}(2\pi)^{k-1-\ell}\zeta(k/2+\ell)\frac{\binom{k-1+\ell}{\ell}}{\binom{k+2\ell}{\ell}}(\ell+1)!6^{k/2+\ell}.$

For the last steps, we used Prop. 4.1. Combining this with (4.9) and applying
$\zeta_{F}(s)\leq\zeta(s)^{3}$ for $s>1$, we have

$\displaystyle|\langle E^{\Delta}_{F,k},f\rangle|$ $\displaystyle\leq
c_{f}c_{k}2|{\mathrm{Aut}}(\mathcal{O}_{F})|\frac{\zeta_{F}\bigl{(}\tfrac{k}{2}\bigr{)}}{\zeta_{F}(k)}\zeta(k)\zeta\bigl{(}\tfrac{3k}{2}-1\bigr{)}D_{F}^{-k/4}\leq
6c_{f}c_{k}\frac{\zeta\bigl{(}\tfrac{k}{2}\bigr{)}^{3}}{\zeta(k)^{2}}\zeta\bigl{(}\tfrac{3k}{2}-1\bigr{)}D_{F}^{-k/4}$

for $k\geq 3$. This finishes the proof. ∎

######  Remark 5.2.

For $k=4$, the bound above gives $C_{4}<5.79$. We can obtain a better bound by
estimating the second to the last line in Example 3.3 case by case for each
$\ell=0,1,2,3$, instead of using (3.9). The improved bound is

$|\langle E^{\Delta}_{F,4},f\rangle|\leq 0.067c_{f}D_{F}^{-1}$

for all totally real cubic field $F$.

Now we are ready to prove Theorem 1.1 in the cubic case.

###### Proof of Theorem 1.1 for $d=3$.

Since ${\mathcal{M}}_{\mathbb{Q}}^{(12)}=\mathbb{C}[E_{12},\Delta]$, we only
have to check that
$({\mathcal{M}}^{\theta}_{F})^{\Delta}\cap{\mathcal{M}}_{\mathbb{Q}}^{(12)}$
is 2 dimensional. For any $L\in{\mathcal{U}}^{+}_{F}$, the diagonal
restriction $\theta^{\Delta}_{L}$ is the theta function for a unimodular
lattice $P$ over $\mathbb{Z}$. So we know that
$\theta_{P}\in({\mathcal{M}}^{\theta}_{F})^{\Delta}$ for some Niemeier lattice
$P$. To see that it is linearly independent from
$E_{F,4}^{\Delta}=1+c(1)q+O(q^{2})$, it suffices to show that $c(1)$ is not
integral. We have checked this numerically for any cubic $F$ with
$D_{F}<70000$.

More generally, we have

$\theta_{P}=E_{12}+(N_{2}(P)-65520/691)\Delta,$

with $N_{2}(P)$ is the number of norm 2 vectors in $P$. From Table V in
[CS82], we obtain a list of $N_{2}(P)$ and

$|\langle\theta_{P},\Delta\rangle|=|N_{2}(P)-65520/691|\langle\Delta,\Delta\rangle>1.22\times
10^{-6}$

for any Niemeier lattice $P$. On the other hand by taking $c_{\Delta}=1$, the
upper bound found in Lemma 5.1 and improved in Remark 5.2 gives us

$|\langle E^{\Delta}_{F,4},\Delta\rangle|\;<\;\frac{0.067}{D_{F}}.$

So $E^{\Delta}_{F,4}$ and $\theta_{P}$ are linearly independent for $D_{F}\geq
60000$. This finishes the proof. ∎

##  6\. Numerical Evidence for Conjecture 1

In this section, we approach numerically Conjecture 1 in the case $F$ is a
totally real field of degree $d\in\\{4,5,6\\}$. For these choices of $d$ the
space $\mathcal{M}_{\mathbb{Q}}^{(4d/d_{2})}$ can be in principle generated by
the restriction of Eisenstein series and of (at most) one theta function
$\theta_{L}$ of rank $8/d_{2}$. Conjecture 1 reduces then to the verification
of the linear independence of $\theta_{L}^{\Delta}$ and
$E_{F,4/d_{2}}^{\Delta}$ for $d=5,6$, and of monomials in
$\theta_{L}^{\Delta},E_{F,4d/d_{2}}^{\Delta}$ and $E_{F,k}^{\Delta}$ in
general for suitable weights $k$. This approach gives data supporting
Conjecture 1 in the case $d=4,5$, and in the case $d=6$ except for two fields
$F$. Our result, for which evidence is given in this final section, is the
following.

######  Theorem 6.1.

Conjecture 1 holds for

  1. (1)

$d=4$ and $D_{F}\leq 10^{5}$;

  2. (2)

$d=5$ and $D_{F}\leq 2\times 10^{6}$;

  3. (3)

$d=6$ and $D_{F}\leq 5\times 10^{6}$ except for the fields of discriminant
$453789$ and $1397493$.

###  6.1. A note on the computations

For $l,k\in\mathbb{Z}_{\geq 0}$, let $\sigma_{k-1}$ be as in Remark 2.4 and
define

$s_{l}^{F}(k)\;:=\;\sum_{\begin{subarray}{c}\nu\in\mathfrak{d}_{F}^{-1}\\\
\nu\gg 0\\\
\mathrm{tr}(\nu)=l\end{subarray}}\sigma_{k-1}\bigl{(}(\nu)\mathfrak{d}_{F}\bigr{)}\,.$

Then the diagonal restriction of $E_{F,k}$ has the following $q$-expansion at
$\infty$ by (2.6)

(6.1)
$E_{F,k}^{\Delta}(\tau)\;=\;1\;+\;\frac{2^{d}}{\zeta_{K}(1-k)}\sum_{l=0}^{\infty}{s^{F}_{l}(k)}\,.$

We computed the first few coefficients of the above expansion with PARI/GP
[The21]. As (6.1) shows, this reduces to the determination of the functions
$s^{F}_{l}(k)$ for small values of $l$ (up to $l=5$ in the case $d=5$) and
different values of $k$. The main difficulty is to find the totally positive
$\nu\in\mathfrak{d}_{F}^{-1}$ of fixed trace $l$. Let
$(\nu_{1},\dots,\nu_{d})$ be an integral basis for $\mathfrak{d}_{F}^{-1}$.
Then any $\nu\in\mathfrak{d}_{F}^{-1}$ is of the form
$\nu=v_{1}\nu_{1}+\cdots+v_{d}\nu_{d}$ for
$(v_{1},\dots,v_{d})\in\mathbb{Z}^{d}$ and conversely every vector in
$\mathbb{Z}^{d}$ gives an element $\nu\in\mathfrak{d}_{F}^{-1}$. If
$Q(x_{1},\dots,x_{d})$ denotes the quadratic form $x_{1}^{2}+\dots x_{d}^{2}$,
we have, for a totally positive $\nu\in\mathfrak{d}_{F}^{-1}$, that
$Q(\sigma_{1}(\nu),\dots,\sigma_{d}(\nu))<\mathrm{tr}(\nu)^{2}$. This implies
that if $A=(\sigma_{i}(\nu_{j}))_{i,j}$ denotes the matrix of the real
embeddings of the basis of $\mathfrak{d}_{F}^{-1}$, we can search the totally
positive $\nu\in\mathfrak{d}_{F}^{-1}$ of fixed trace $l$ among of vectors
$v=(v_{1},\dots,v_{d})\in\mathbb{Z}^{d}$ satisfying

$v^{T}(A^{T}A)v\;=\;Q(\nu)\;<\;l^{2}\,.$

This gives a finite (but large as $l$ and $D_{F}$ grow) set of vectors on
which we can perform the final search. Once the suitable
$\nu\in\mathfrak{d}_{F}^{-1}$ have been determined, it is straightforward to
compute $\sigma_{k}\bigl{(}(\nu)\mathfrak{d}_{F}\bigr{)}$ for every value of
$k$ by using the basic PARI functions.

######  Remark 6.2.

It is possible to investigate also the cases $d=8,10$ with the method outlined
at the beginning of this section. For the case $d=8$ we need to compute five
coefficients of the $q$-expansion (6.1), while for $d=10$ we need to compute
six coefficients. This, together with the size of the discriminants of these
fields ($D_{F}\geq 282300416$ for $d=8$ and $D_{F}\geq 443952558373$ for
$d=10$), makes it hard to collect significant data in these cases.

###  6.2. Tables

#### d=4

Let $F$ be a totally real field with $[F:\mathbb{Q}]=4$. In this case, the
proof of Conjecture 1 reduces to the statement that
$\mathcal{M}_{\mathbb{Q}}^{(8)}$ is spanned by restrictions of Hilbert
Eisenstein series on $\Gamma_{F}$. It is easy to see that $\\{E_{4}^{2},\Delta
E_{4},\Delta^{2}\\}$is a generating set for $\mathcal{M}_{\mathbb{Q}}^{(8)}$.
By a dimension argument, $E_{F,2}^{\Delta}=E_{4}^{2}$. It follows that $\Delta
E_{4}$ and $\Delta^{2}$ can be obtained by restriction of Eisenstein series on
$\Gamma_{F}$ respectively if the sets
$\\{E_{F,4}^{\Delta},(E_{F,2}^{\Delta})^{2}\\}$, and
$\\{(E_{F,2}^{\Delta})^{3},E_{F,2}^{\Delta}E_{F,4}^{\Delta},E_{F,6}^{\Delta}\\}$
are both linearly independent.

In order to study this problem, we compute the restriction of $E_{F,k}$ for
$k=4,6$. As bases for $M_{16}$ and $M_{24}$, we choose
$\\{E_{4}^{4},E_{4}\Delta\\}$ and $\\{E_{4}^{6},E_{4}^{3}\Delta,\Delta^{2}\\}$
respectively. We have

(6.2) $\displaystyle E_{F,4}^{\Delta}$
$\displaystyle\;=\;E_{4}^{4}\;+\;bE_{4}\Delta\,,$ $\displaystyle
E_{F,6}^{\Delta}$
$\displaystyle\;=\;E_{4}^{6}\;-\;c_{1}E_{4}^{3}\Delta\;+\;c_{2}\Delta^{2}\,,$

for some coefficients $b,c_{1},c_{2}\in\mathbb{Q}$ that depend on $F$. To
prove Conjecture 1, it suffices to check that $b$ and $c_{2}$ are both non-
zero. We computed the coefficients $b,c_{1},c_{2}$ for the first $30$ totally
real quartic fields $F$. The results are reported in Table 1. For these fields
it is enough to specify the discriminant $D_{F}$ to uniquely identify the
field $F$ (check the number field database [LMF22]). This remark applies also
for the fields we consider in the cases $d=5,6$.

It turns out that the numerical values of $b,c_{1}$, and $c_{2}$ are very
close to $955,1439$, and $-129930$ respectively. These numbers are related to
the Eisenstein series of weight $16$ and $24$ since

$E_{16}\;=\;E_{4}^{4}+b(E_{16})E_{4}\Delta\,,\quad
E_{24}\;=\;E_{4}^{5}+c_{1}(E_{24})E_{4}^{2}\Delta+c_{2}(E_{24})\Delta^{2},$

with

$b(E_{16})=-\tfrac{3456000}{3617}\sim
955\,,\;c_{1}(E_{24})=\tfrac{340364160000}{236364091}\sim
1439\,,\;c_{2}(E_{24})=-\tfrac{30710845440000}{236364091}\sim 129930\,.$

In other words, it seems that the diagonal restriction of $E_{F,4}$ and
$E_{F,6}$ are close to $E_{16}$ and $E_{24}$ respectively. In analogy with the
proof of Theorem 1.1 in the case $d=3$, Conjecture 1 holds for $D_{F}\gg 0$ if
the Petersson products of $E_{F,4}^{\Delta}$ and $E_{F,6}^{\Delta}$ with all
cusp forms of weight $16$ and $24$ respectively can be bounded by small
quantities as $D_{F}\to\infty$. If $F$ ranges over the totally real quartic
fields with no non-trivial subfields, the decay of the Petersson products as
$D_{F}\to\infty$ can be observed from the data. We expect similar strategy for
the proof of Theorem 1.1 when $d=3$ to work in this case. When $F$ ranges
instead over extensions of the form $\mathbb{Q}\subset K\subset F$, where $K$
is a fixed real quadratic field, the data suggest that

$\langle E_{F,k}^{\Delta},f\rangle\rightarrow\langle
E_{K,2k}^{\Delta},f\rangle\quad\text{as }\mathrm{disc}(F)\to\infty\,.$

The proof of Conjecture 1 may be obtained then in two steps: first proving
that $E_{F,k}$ restrict to the Hilbert Eisenstein series $E_{K,2k}$ on
$\Gamma_{K}$ as $F\to\infty$, and then using Theorem 1.1 for the real
quadratic field $K$.

Table 1. $d=4$

$\begin{array}[]{lcccc}\hline\cr\hline\cr D_{F}&\lx@intercol\hfil
E_{F,4}^{\Delta}\hfil\lx@intercol&\lx@intercol\hfil
<EMAIL_ADDRESS>&-b&|b-b(E_{16})|&|c_{1}-c_{1}(E_{24})|&|c_{2}-c_{2}(E_{24})|\\\ \hline\cr
725&\frac{518400}{541}&2.7375349&0.00050313732&25.886498\\\\[4.30554pt]
1125&\frac{1209600}{1261}&3.7507260&0.00054525118&81.739221\\\\[4.30554pt]
1600&\frac{16588800}{17347}&0.80418080&0.00021600333&72.207992\\\\[4.30554pt]
1957&\frac{3379968}{3541}&0.96439255&0.00038594892&17.453573\\\\[4.30554pt]
2000&\frac{3628800}{3793}&1.2217550&0.00025214822&55.822134\\\\[4.30554pt]
2048&\frac{83358720}{87439}&2.1522766&0.00086000436&17.157301\\\\[4.30554pt]
2225&\frac{4406400}{4601}&2.2168733&0.00044417599&65.944997\\\\[4.30554pt]
2304&\frac{6996480}{7337}&1.8993132&0.00078107824&34.635539\\\\[4.30554pt]
2525&\frac{40953600}{42787}&1.6625629&0.00038679430&60.388956\\\\[4.30554pt]
2624&\frac{31242240}{32681}&0.48766988&0.00016760431&11.280096\\\\[4.30554pt]
2777&\frac{30326400}{31739}&0.0052682944&2.49791\times
10^{-5}&3.1916173\\\\[4.30554pt]
3600&\frac{3940800}{4117}&1.7138725&0.00032163274&63.391164\\\\[4.30554pt]
3981&\frac{22598400}{23651}&0.0065088042&1.13683\times
10^{-5}&16.924484\\\\[4.30554pt]
4205&\frac{81112320}{84937}&0.51758364&7.99169\times
10^{-5}&20.235531\\\\[4.30554pt]
4225&\frac{31168800}{32567}&1.5789962&0.00036468807&64.303710\\\\[4.30554pt]
4352&\frac{14613696}{15301}&0.40686765&0.00042977636&2.3880196\\\\[4.30554pt]
4400&\frac{287193600}{300017}&1.7697819&0.00034047188&63.576584\\\\[4.30554pt]
4525&\frac{315705600}{329717}&2.0167958&0.00041971303&62.764065\\\\[4.30554pt]
4752&\frac{94772160}{99107}&0.77303737&0.00019694052&7.4011277\\\\[4.30554pt]
4913&\frac{358572096}{375437}&0.40870317&4.15983\times
10^{-5}&2.8025931\\\\[4.30554pt]
5125&\frac{24364800}{25453}&1.7587165&0.000037508012&63.196773\\\\[4.30554pt]
5225&\frac{262310400}{273971}&1.9505876&0.00039428701&63.490510\\\\[4.30554pt]
5725&\frac{716947200}{748883}&1.8674479&0.00042362838&63.447454\\\\[4.30554pt]
5744&\frac{727626240}{761737}&0.26820601&7.42018\times
10^{-5}&5.6160966\\\\[4.30554pt]
6125&\frac{454636800}{474913}&1.8174699&0.00042240976&63.162128\\\\[4.30554pt]
6224&\frac{204809472}{214357}&0.028287048&2.32205\times
10^{-5}&5.7256495\\\\[4.30554pt]
6809&\frac{87570720}{91723}&0.75775312&0.00019944285&7.0978686\\\\[4.30554pt]
7053&\frac{1504154880}{1573751}&0.28894424&0.00013645348&2.0848135\\\\[4.30554pt]
7056&\frac{191034720}{200123}&0.90144417&0.00037862134&11.709300\\\\[4.30554pt]
7168&\frac{670104576}{701855}&0.72584168&0.00033000104&3.6107465\\\\[4.30554pt]
\hline\cr\hline\cr\end{array}$

#### d=5

Let $F$ be a totally real field of degree $5$. The space
$\mathcal{M}_{\mathbb{Q}}^{(20)}$ is generated by the set
$\\{E_{20},E_{8}\Delta,E_{4}\Delta^{3},\Delta^{5}\\}$. In order to get this
space by restriction of Hilbert theta series (Conjecture 1), we only need to
consider a Hilbert theta function $\theta_{L}$ for
$L\in{\mathcal{U}}^{+,8}_{F}$ and the Eisenstein series $E_{F,4},E_{F,8},$ and
$E_{F,12}$. Fixing basis for $M_{20},M_{40},$ and $M_{60}$, we find the
expressions

(6.3) $\displaystyle E_{F,4}^{\Delta}$
$\displaystyle\;=\;E_{4}^{5}\;+\;bE_{4}^{2}\Delta\,,$ $\displaystyle
E_{F,8}^{\Delta}$
$\displaystyle\;=\;E_{4}^{10}\;+\;c_{1}E_{4}^{7}\Delta\;+\;c_{2}E_{4}^{4}\Delta^{2}\;+\;c_{3}E_{4}\Delta^{3}\,,$
$\displaystyle E_{F,12}^{\Delta}$
$\displaystyle\;=\;E_{4}^{15}\;+\;d_{1}E_{4}^{12}\Delta\;+\;d_{2}E_{4}^{9}\Delta^{2}\;+\;d_{3}E_{4}^{6}\Delta^{3}\;+\;d_{4}E_{4}^{3}\Delta^{4}\;+\;d_{5}\Delta^{5}\,,$

for $b,c_{i},d_{i}\in\mathbb{Q}$ that depends on $F$. Since
$\theta_{L}^{\Delta}=1+\sum_{n\geq 1}{a_{n}q^{n}}$ with $a_{n}\in\mathbb{Z}$,
in order to prove linear independence of $\theta_{L}^{\Delta}$ and
$E_{F,4}^{\Delta}$, it suffices to show that $b\not\in\mathbb{Z}$. If this
holds true, we only need that $c_{3}\neq 0$ and $d_{5}\neq 0$ to prove
Conjecture 1. The results of the computation of $b,c_{3},$ and $d_{5}$, for
the first 30 totally real quintic fields $F$ (ordered by discriminant) can be
found in Table (2). Similarly to the case $d=4$, the numerical values of
$b,c_{i},d_{i}$ are close to the coefficients appearing in the expression of
the Eisenstein series $E_{20},E_{40}$, and $E_{60}$ with respect to the bases
specified above:

$E_{20}=E_{4}^{5}+b(E_{20})E_{4}^{2}\Delta\,,\;E_{40}=E_{4}^{10}+\sum_{i=1}^{3}{c_{i}(E_{40})E_{4}^{10-3i}\Delta^{i}}\,,\;E_{60}=E_{4}^{15}+\sum_{i=1}^{5}{d_{i}(E_{60})E_{4}^{15-3i}\Delta^{i}}\,,$

the relevant values being

$\displaystyle b(E_{20})$ $\displaystyle=\tfrac{209520000}{174611}\sim
1199\,,\quad
c_{3}(E_{40})=\tfrac{27014542428753690624000000000}{261082718496449122051}\sim
103471200$ $\displaystyle d_{5}(E_{60})$
$\displaystyle=\tfrac{1423152253904739393602157818174020937318400000000000000}{1215233140483755572040304994079820246041491}\sim
1171094011917\,.$

In Table 2 we do not write the numerical values of $c_{3},d_{5}$, but of their
difference with the coefficients $c_{3}(E_{40})$ and $d_{5}(E_{60})$
respectively. Analogously to the case $d=4$, it seems that the diagonal
restriction of $E_{F,4},E_{F,8}$, and $E_{F,12}$ are close to the Eisenstein
series $E_{20},E_{40}$, and $E_{60}$ respectively. In particular, since
$\langle E_{20},E_{4}^{2}\Delta\rangle=0$, this implies that the Petersson
product

$\langle
E_{F,4}^{\Delta},E_{4}^{2}\Delta\rangle\;=\bigl{|}b-b(E_{20})\bigr{|}\langle
E_{4}\Delta^{2},E_{4}^{2}\Delta\rangle\ $

is small for any field $F$ and may decay as $D_{F}\to\infty$. Similar
considerations apply to the cases $E_{F,8}^{\Delta}$ and $E_{F,12}^{\Delta}$.

Table 2. $d=5$

$\begin{array}[]{ccccc}\hline\cr\hline\cr D_{F}&\lx@intercol\hfil
E_{F,4}^{\Delta}\hfil\lx@intercol&E_{F,8}^{\Delta}&E_{F,12}^{\Delta}\\\
&-b&|b-b(E_{20})|&|c_{3}-c_{3}(E_{40})|&|d_{5}-d_{5}(E_{60})|\\\ \hline\cr
14641&\frac{1017360000}{847811}&0.060027104&20.049846&602.44929\\\\[4.30554pt]
24217&\frac{539084160}{449263}&0.0056153314&3.0959986&626.69793\\\\[4.30554pt]
36497&\frac{228998016}{190847}&0.020731861&3.7691249&625.79357\\\\[4.30554pt]
38569&\frac{1372671360}{1144027}&0.065169297&7.1399961&53.593811\\\\[4.30554pt]
65657&\frac{17909631360}{14926259}&0.050318395&1.4956754&21.447253\\\\[4.30554pt]
70601&\frac{22786945920}{18989939}&0.023943997&6.3509437&57.580627\\\\[4.30554pt]
81509&\frac{1255163040}{1046047}&0.013653680&0.14871660&33.681371\\\\[4.30554pt]
81589&\frac{157427145}{131198}&0.0040921029&3.7633773&5.5844793\\\\[4.30554pt]
89417&\frac{3299933520}{2750093}&0.010842438&0.88686411&7.1415794\\\\[4.30554pt]
101833&\frac{27422375040}{22853437}&0.00095029875&2.8922290&147.56474\\\\[4.30554pt]
106069&\frac{8416776960}{7014301}&0.020817098&1.2397693&6.2320866\\\\[4.30554pt]
117688&\frac{72647616960}{60544963}&0.029096490&0.024328539&11.486101\\\\[4.30554pt]
122821&\frac{2646596160}{2205599}&0.019992669&3.3965204&46.863235\\\\[4.30554pt]
124817&\frac{169474446720}{141236923}&0.0057786083&1.2672930&6.7827247\\\\[4.30554pt]
126032&\frac{186909793920}{155769041}&0.0082151643&0.36230593&11.688370\\\\[4.30554pt]
135076&\frac{39368816640}{32809823}&0.014954319&0.35027520&21.034030\\\\[4.30554pt]
138136&\frac{42439256640}{35368523}&0.0084035031&0.15151378&21.202474\\\\[4.30554pt]
138917&\frac{30923687520}{25771127}&0.010994082&3.3294478&176.27092\\\\[4.30554pt]
144209&\frac{35105335200}{29256611}&0.013203831&1.2246131&5.0019384\\\\[4.30554pt]
147109&\frac{79422612480}{66189911}&0.0041847312&1.4237043&8.3262747\\\\[4.30554pt]
149169&\frac{316249522560}{263551583}&0.028634117&1.8194260&15.033097\\\\[4.30554pt]
153424&\frac{24509153664}{20425187}&0.023174277&3.4016473&25.402792\\\\[4.30554pt]
157457&\frac{76544072064}{63790577}&0.0031668548&1.0412335&7.2747406\\\\[4.30554pt]
160801&\frac{411236196480}{342716341}&0.0072814568&0.29819082&10.519558\\\\[4.30554pt]
161121&\frac{6653973120}{5545309}&0.0038819428&0.12885785&18.108623\\\\[4.30554pt]
170701&\frac{125695281600}{104754347}&0.019242412&2.2046583&104.80927\\\\[4.30554pt]
173513&\frac{530059904640}{441734773}&0.026193754&0.10356861&42.247614\\\\[4.30554pt]
176281&\frac{187387136640}{156166489}&0.0054077065&1.9881737&26.798380\\\\[4.30554pt]
176684&\frac{60248727936}{50210921}&0.011580917&1.0387400&12.861239\\\\[4.30554pt]
179024&\frac{638510843520}{532132229}&0.014289468&0.34114599&1.2188617\\\\[4.30554pt]
\hline\cr\hline\cr\end{array}$

#### d=6

We have that $\mathcal{M}_{\mathbb{Q}}^{12}=\mathbb{C}[E_{4}^{3},\Delta]$. We
have only to check that

$E_{F,2}^{\Delta}\;=\;E_{4}^{3}\;+\;b\cdot\Delta$

is not the restriction of a Hilbert theta function $\theta_{L}$. We know this
is the case if $b$ is not an integer, as explained in the proof of Theorem 1.1
in the case $d=3$. However, looking at the values of $b$ computed for the
first $30$ totally real sextic fields $F$ in Table (3), this is not always the
case. Since $\theta_{L}^{\Delta}=1+N_{2}(L)q+\cdots$, we have to compare, for
integral values of $b$, the number $720+b$ with the possible values of
$N_{2}(L)$ listed in table V of [CS82] to check whether they differ or not.
This happens in all cases but two: the field of discriminant $453789$ has
$720+b=0=N_{2}(\Lambda_{24})$, the field of discriminant $1397493$ has
$720+b=72=N_{2}(A^{12}_{2})$. For these fields our argument can not confirm
the validity of Conjecture 1. We checked fields up to $D_{F}=5\times 10^{6}$
(144 fields) and found no other such instances.

As in the cases $d=4,5$, in table (3) we also compare the value of $b$ with
$b(E_{12})=-\tfrac{432000}{691}$ (see (5.1)).

Table 3. $d=6$ $\begin{array}[]{ccc}\hline\cr\hline\cr D_{F}&\lx@intercol\hfil
E_{F,2}^{\Delta}\hfil\lx@intercol\\\ &-b&|b-b(E_{12})|\\\ \hline\cr
300125&\frac{21600}{37}&41.397113\\\\[4.30554pt]
371293&\frac{11808}{19}&3.7072130\\\\[4.30554pt]
434581&\frac{8352}{13}&17.280641\\\\[4.30554pt]
453789&720&94.819103\\\\[4.30554pt]
485125&\frac{7200}{11}&29.364557\\\\[4.30554pt]
592661&672&46.819103\\\\[4.30554pt]
703493&\frac{2048}{3}&57.485769\\\\[4.30554pt]
722000&\frac{4800}{7}&60.533388\\\\[4.30554pt]
810448&\frac{3456}{5}&66.019103\\\\[4.30554pt]
820125&\frac{43200}{73}&33.400075\\\\[4.30554pt]
905177&\frac{3348}{5}&44.419103\\\\[4.30554pt]
966125&675&49.819103\\\\[4.30554pt] 980125&675&49.819103\\\\[4.30554pt]
1075648&\frac{8352}{13}&17.280641\\\\[4.30554pt]
1081856&\frac{3072}{5}&10.780897\\\\[4.30554pt]
\hline\cr\hline\cr\end{array}\quad\begin{array}[]{ccc}\hline\cr\hline\cr
D_{F}&\lx@intercol\hfil E_{F,2}^{\Delta}\hfil\lx@intercol\\\
&-b&|b-b(E_{12})|\\\ \hline\cr 1134389&684&58.819103\\\\[4.30554pt]
1202933&608&17.180897\\\\[4.30554pt]
1229312&\frac{27264}{43}&8.8656144\\\\[4.30554pt]
1241125&\frac{28800}{47}&12.414940\\\\[4.30554pt]
1259712&\frac{17280}{31}&67.761542\\\\[4.30554pt]
1279733&\frac{11736}{17}&65.172044\\\\[4.30554pt]
1292517&\frac{16416}{29}&59.111932\\\\[4.30554pt]
1312625&\frac{9000}{13}&67.126795\\\\[4.30554pt]
1387029&696&70.819103\\\\[4.30554pt] 1397493&648&22.819103\\\\[4.30554pt]
1416125&\frac{12000}{19}&6.3980501\\\\[4.30554pt]
1528713&\frac{12096}{19}&11.450682\\\\[4.30554pt]
1541581&\frac{8352}{13}&17.280641\\\\[4.30554pt]
1683101&\frac{65088}{103}&6.7414328\\\\[4.30554pt]
1767625&\frac{25200}{41}&10.546751\\\\[4.30554pt]
\hline\cr\hline\cr\end{array}$

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