2 Preliminaries

Fix an algebraic closure $\bar {\mathbb {Q}}$ of $\mathbb {Q}$ and let E be an elliptic curve defined over $\mathbb {Q}$ . Let p be a prime number and set $\mathbb {F}_p=\mathbb {Z}/p\mathbb {Z}$ to denote the finite field with p elements. For $n\geq 1$ , denote by $E[p^n]$ the kernel of the multiplication map $\times p^n: E(\bar {\mathbb {Q}})\rightarrow E(\bar {\mathbb {Q}})$ and set $E[p^\infty ]:=\bigcup _{n} E[p^n]$ . The absolute Galois group $\operatorname {G}_{\mathbb {Q}}:=\operatorname {Gal}(\bar {\mathbb {Q}}/\mathbb {Q})$ acts on $E[p^n]$ . Choosing an isomorphism $E[p^n]\simeq (\mathbb {Z}/p^n \mathbb {Z})^2$ , let $\rho _n$ be the associated Galois representation on $E[p^n]$ ,

$$ \begin{align*}\rho_n:\operatorname{G}_{\mathbb{Q}}\rightarrow \operatorname{GL}_2(\mathbb{Z}/p^n\mathbb{Z}).\end{align*} $$

Set $\rho :=\varprojlim _n \rho _n$ to be the Galois representation on the p-adic Tate-module $\operatorname {T}_p(E):=\varprojlim _n E[p^n]$ . Set $\bar {\rho }$ to denote the mod-p reduction of the characteristic-zero representation $\rho :\operatorname {G}_{\mathbb {Q}}\rightarrow \operatorname {GL}_2(\mathbb {Z}_p)$ . We refer to $\bar {\rho }$ as the residual representation and identify $\rho _1:\operatorname {G}_{\mathbb {Q}}\rightarrow \operatorname {GL}_2(\mathbb {F}_p)$ with $\bar {\rho }$ . Let $\chi :\operatorname {G}_{\mathbb {Q}}\rightarrow \mathbb {Z}_p^\times $ be the p-adic cyclotomic character and $\bar {\chi }:\operatorname {G}_{\mathbb {Q}}\rightarrow \mathbb {F}_p^\times $ its mod-p reduction.

Given an integer r, set $\mathbb {Z}_p(r):=\mathbb {Z}_p(\chi ^r)$ and given a $\mathbb {Z}_p[\operatorname {G}_{\mathbb {Q}}]$ -module M, set $M(r):=M\otimes _{\mathbb {Z}_p} \mathbb {Z}_p(\chi ^r)$ to denote the r th Tate-twist of M. Note that if M is an $\mathbb {F}_p[\operatorname {G}_{\mathbb {Q}}]$ -module, then, $M(r)=M\otimes _{\mathbb {F}_p} \mathbb {F}_p(r)$ . Let $V_{\bar {\rho }}=E[p]$ be the underlying vector space for $\bar {\rho }$ . Given $i,j\in \mathbb {Z}$ lying in the range $1\leq i\leq (p-1)$ and $1\leq j\leq (p-2)$ , we set $V^{i,j}=V^{i,j}_{p,E}$ to denote $\operatorname {Sym}^i(V_{\bar {\rho }})(j)$ . Throughout, we assume that $\bar {\rho }$ is irreducible. However, for the sake of discussion, let us consider the special case when $\bar {\rho }$ is surjective. Then, upon choosing a basis of $V_{\bar {\rho }}=E[p]$ , we may identify the image of $\bar {\rho }$ with $\operatorname {GL}_2(\mathbb {F}_p)$ and thus the module $V^{i,j}$ is viewed as an irreducible representation of $\operatorname {GL}_2(\mathbb {F}_p)$ . Note that any irreducible representation of $\operatorname {GL}_2(\mathbb {F}_p)$ over $\mathbb {F}_p$ is of the form $V^{i,j}$ . A semisimple module M over $\mathbb {F}_p[\operatorname {G}_{\mathbb {Q}}]$ decomposes into a direct sum

$$ \begin{align*}M\simeq \bigoplus_{i,j} (V^{i,j})^{r_{i,j}(M)},\end{align*} $$

where $r_{i,j}(M)\geq 0$ .

Let $K=\mathbb {Q}(E[p])$ be the splitting field of $\bar {\rho }$ , taken to be the Galois extension of $\mathbb {Q}$ given by $K=\bar {\mathbb {Q}}^{\operatorname {ker}\bar {\rho }}$ . Set $G:=\operatorname {Gal}(K/\mathbb {Q})$ and identify G with the image of $\bar {\rho }$ . Note that when $\bar {\rho }$ is surjective, G is identified with $\operatorname {GL}_2(\mathbb {F}_p)$ . There is a natural action of G on the class group $\operatorname {Cl}_K$ , and thus on the mod-p class group $\operatorname {Cl}_K/p\operatorname {Cl}_K$ . The study of the Galois module structure of $\operatorname {Cl}_K/p\operatorname {Cl}_K$ is the primary focus of this paper. When $\bar {\rho }$ is surjective, its semisimplification decomposes into a direct sum of irreducible representations of $\operatorname {GL}_2(\mathbb {F}_p)$ ,

$$ \begin{align*}(\operatorname{Cl}_K/p \operatorname{Cl}_K)^{\operatorname{ss}}\simeq \bigoplus_{i,j} (V^{i,j})^{n_{i,j}}.\end{align*} $$

We are specifically interested in the space of homomorphisms $\operatorname {Hom}_G(\operatorname {Cl}_K{\kern-1.2pt}/{\kern-1.2pt}p\operatorname {Cl}_K, E[p])$ . Note that $\operatorname {Hom}_G(\operatorname {Cl}_K/p\operatorname {Cl}_K, E[p])\neq 0$ precisely when there is a G-stable quotient of $\operatorname {Cl}_K/p \operatorname {Cl}_K$ that is isomorphic to $E[p]$ . By class field theory, this corresponds to the existence of an unramified $(\mathbb {Z}/p\mathbb {Z})^2$ -extension L of K that is Galois over $\mathbb {Q}$ , such that $\operatorname {Gal}(L/K)\simeq E[p]$ as a module over $G=\operatorname {Gal}(K/\mathbb {Q})$ . We have that

$$ \begin{align*}n_{1,0}\geq \operatorname{dim}_{\mathbb{F}_p} \operatorname{Hom}_G(\operatorname{Cl}_K/p\operatorname{Cl}_K, E[p]),\end{align*} $$

with equality in the special case when the representation of G on $\operatorname {Cl}_K/p\operatorname {Cl}_K$ is semisimple.

Given a prime number $\ell $ , let $\mathcal {E}$ be the Néron model of E over $\mathbb {Z}_\ell $ . Let $E^0(\mathbb {Q}_\ell )$ be the subset of points of $E(\mathbb {Q}_\ell )=\mathcal {E}(\mathbb {Z}_\ell )$ that reduce modulo $\ell $ to the identity component of $E_{/\mathbb {F}_\ell }$ . Fix an absolute value $|\cdot |_p:\mathbb {Q}_p^\times \rightarrow \mathbb {Q}^\times $ , normalized by $|p|_p^{-1}=p$ . The Tamagawa number at $\ell $ is set to be $c_\ell (E):=[E(\mathbb {Q}_\ell ):E^0(\mathbb {Q}_\ell )]$ , and set $c_\ell ^{(p)}(E):=|c_\ell (E)|_p^{-1}$ . Therefore, $c_\ell (E)$ is a unit in $\mathbb {Z}_p$ if and only if $c_\ell ^{(p)}(E)=1$ . We denote by $\operatorname {Sel}_p(E/\mathbb {Q})$ the p-Selmer group of E over $\mathbb {Q}$ defined by

$$ \begin{align*}\operatorname{Sel}_p(E/\mathbb{Q}):=\operatorname{ker}\{H^1(\bar{\mathbb{Q}}/\mathbb{Q}, E[p])\rightarrow \prod_{\ell} H^1(\bar{\mathbb{Q}}_\ell/\mathbb{Q}_\ell, E(\bar{\mathbb{Q}}_\ell))[p]\},\end{align*} $$

where the restriction map for the prime $\ell $ is the composite

$$ \begin{align*}H^1(\bar{\mathbb{Q}}/\mathbb{Q}, E[p])\rightarrow H^1(\bar{\mathbb{Q}}_\ell/\mathbb{Q}_\ell, E[p])\rightarrow H^1(\bar{\mathbb{Q}}_\ell/\mathbb{Q}_\ell, E(\bar{\mathbb{Q}}_\ell))[p].\end{align*} $$

The Tate–Shafarevich group

fits into an exact sequence

see [Reference Coates and SujathaCoa00, page 8, (1)].

Note that $\operatorname {dim}_{\mathbb {F}_p} E(\mathbb {Q})/p E(\mathbb {Q})=\operatorname {rank} E(\mathbb {Q})+\operatorname {dim}_{\mathbb {F}_p} E(\mathbb {Q})[p]$ , and since $\bar {\rho }$ is irreducible, $E(\mathbb {Q})[p]=0$ , and thus, $\operatorname {dim}_{\mathbb {F}_p} E(\mathbb {Q})/p E(\mathbb {Q})=\operatorname {rank} E(\mathbb {Q})$ . Hence, we find that

Due to the Cassels–Tate pairing, the $\mathbb {F}_p$ -dimension of is even. The following result shows that there is an explicit relationship between $\operatorname {Cl}_K/p\operatorname {Cl}_K$ and the p-torsion in the Tate–Shafarevich group , and is key to constructing quotients of $\operatorname {Cl}_K/p\operatorname {Cl}_K$ that are isomorphic to $E[p]$ .

Theorem 2.1 (Prasad and Shekhar).

Let E be an elliptic curve over $\mathbb {Q}$ and p be an odd prime number at which E has good reduction. With respect to the notation above, suppose that the following assumptions hold:

1. (1) $c_\ell ^{(p)}(E)=1$ for all primes $\ell \neq p$ ;

2. (2) $E(\mathbb {Q}_p)[p]=0$ .

Then,

(2-1)

In particular, under the above assumptions, if , or $\operatorname {rank} E(\mathbb {Q})\geq 2$ , then, $\operatorname {Hom}(\operatorname {Cl}_K/p\operatorname {Cl}_K, E[p])$ is nonzero.

Theorem 2.3 (Prasad and Shekhar).

Let $E_{/\mathbb {Q}}$ be an elliptic curve and p an odd prime at which the conditions of Theorem 2.1 are satisfied. Let $\mathcal {I}$ be the finite set from Definition 2.2. Then, the following bound is satisfied

Theorem 2.1 is crucially used in the proof of Theorem 4.4, which is our main result. However, both Theorems 2.1 and 2.3 are applied to prove results in the case when E is a fixed elliptic curve and p varies over all primes at which E has good reduction; see Section 5.

3 Density results for Weierstrass equations with local conditions

In this section, we recall results due to Cremona and Sadek [Reference Cremona and SadekCS23] for the density of elliptic curves $E_{/\mathbb {Q}}$ satisfying local conditions at a prescribed set of primes. This set of primes may in fact be infinite. The results in this section are used to study the proportion of elliptic curves $E_{/\mathbb {Q}}$ , ordered according to height, that satisfy the conditions of Theorem 2.1. As always, we fix an odd prime p. Given a tuple of integers $a=(a_1,a_2,a_3,a_4,a_6)$ , we have an associated elliptic curve $E_a$ defined by the long Weierstrass equation

$$ \begin{align*}Y^2+a_1XY+a_3Y=X^3+a_2X^2+a_4X+a_6.\end{align*} $$

The height of $E_a$ is defined as follows:

$$ \begin{align*}\operatorname{ht}(a)=\operatorname{ht}(E_a):=\operatorname\max_i\{|a_i|^{{1}/{i}}\}.\end{align*} $$

We let

$$ \begin{align*} b_2& =a_1^2+4 a_2, b_4=a_1 a_3+2 a_4, b_6=a_3^2+4a_6, \end{align*} $$

$$ \begin{align*} b_8& =a_1^2 a_6+4a_2a_6-a_1a_3a_4+a_2 a_3^2-a_4^2, \end{align*} $$

$$ \begin{align*} \Delta(a)& =-b_2^2 b_8-8 b_4^3-27 b_6^2+9b_2 b_4 b_6. \end{align*} $$

Given a ring R, we let

$$ \begin{align*}\mathcal{W}(R) = R^5 = \{a=(a_1,a_2,a_3,a_4,a_6)\,|\,a_i\in R\}, \end{align*} $$

and let $\Delta (a)=\Delta (E_a)$ be the associated discriminant. Given a prime $\ell $ , let $\mathcal {W}_{M}(\mathbb {Z}_\ell )$ be the subset of $\mathcal {W}(\mathbb {Z}_\ell )$ consisting of equations that are minimal in the sense of [Reference SilvermanSil09, Section VII.1]. Recall that if E is an elliptic curve over $\mathbb {Q}_\ell $ , the Kodaira type is one of the following choices:

• • $\operatorname {I}_0$ if E has good reduction;

• • $\operatorname {I}_{\ge 1}$ if E has bad multiplicative reduction, of type $\operatorname {I}_m$ for some $m\ge 1$ ;

• • finally, if E has bad additive reduction, we have the following choices: $\mathrm {II}$ , $\mathrm {III}$ , $\mathrm {IV}$ , $\mathrm {II}^*$ , $\mathrm {III}^*$ , $\mathrm {IV}^*$ , $\mathrm {I}_0^*$ , $\mathrm {I}_{\ge 1}^*$ , the latter meaning type $\mathrm {I}_m^*$ for some $m\geq 1$ .

We refer to [Reference SilvermanSil94, IV, Section 9] for a comprehensive study of Kodaira types and how they may be detected via Tate’s algorithm. For $a\in \mathcal {W}(\mathbb {Z}_\ell )$ , we say that a has Kodaira type T if the associated elliptic curve $E_a$ has Kodaira type T. Note that this definition applies to nonminimal Weierstrass equations as well. Given a Kodaira type T, let $\mathcal {W}^T(\mathbb {Z}_\ell )$ be the subset of $\mathcal {W}(\mathbb {Z}_\ell )$ of tuples a of Kodaira type T and set $\mathcal {W}_{M}^T(\mathbb {Z}_\ell )=\mathcal {W}^T(\mathbb {Z}_\ell )\cap \mathcal {W}_{M}(\mathbb {Z}_\ell )$ . Let $\mu $ be the Haar measure on $\mathcal {W}(\mathbb {Z}_\ell )\simeq \mathbb {Z}_\ell ^5$ and set $\rho ^M(\ell ):=\mu (\mathcal {W}_{M}(\mathbb {Z}_\ell ))$ . Given a Kodaira type T at the prime $\ell $ , set

$$ \begin{align*} \rho^M_T(\ell)& :=\mu(\mathcal{W}_M^T(\mathbb{Z}_\ell)),\\ \rho_T(\ell)& :=\mu(\mathcal{W}^T(\mathbb{Z}_\ell)). \end{align*} $$

According to [Reference Cremona and SadekCS23, Proposition 2.6], we have that $\rho _T(\ell )=(1-\ell ^{-10})^{-1}\rho ^M_T(\ell )$ . These local densities are calculated in loc. cit., and these calculations are summarized here.

Theorem 3.1 (Cremona and Sadek).

Let $\ell $ be a prime, then $\rho ^M(\ell )=1-\ell ^{-10}$ . For each Kodaira type T, the local density $\rho _T^M(\ell )$ is given by the values in Table 1.

Table 1 Local densities: Kodaira types.

Let $\mathcal {S}$ be a subset of $\mathcal {W}(\mathbb {Z})\simeq \mathbb {Z}^5$ , the density of $\mathcal {S}$ is taken to be the following limit

(3-1) $$ \begin{align}\mathfrak{d}(\mathcal{S})&:=\lim\limits_{x\rightarrow \infty} \dfrac{\#\{a\in \mathcal{S}\mid \operatorname{ht}(a)<x\}}{\#\{a\in \mathbb{Z}^5\mid \operatorname{ht}(a)<x\}},\nonumber \\&=\lim\limits_{x\rightarrow \infty} 2^{-5}x^{-16}\#\{a\in \mathcal{S}\mid |a_i|<x^i \text{ for }i=1,2,3,4,6\}. \end{align} $$

Note that the above limit may not exist. We let $\overline {\mathfrak {d}}(\mathcal {S})$ and $\underline {\mathfrak {d}}(\mathcal {S})$ be the upper and lower limits, respectively. Note that $\overline {\mathfrak {d}}(\mathcal {S})$ and $\underline {\mathfrak {d}}(\mathcal {S})$ are defined by replacing the limit by $\limsup $ and $\liminf $ , respectively, and that these limits do exist unconditionally. If $\mathcal {S}$ is a set of elliptic curves $E_{/\mathbb {Q}}$ , we by abuse of notation denote by $\mathcal {S}$ the subset of $\mathcal {W}(\mathbb {Z})$ consisting of all tuples $a=(a_1,a_2,a_3,a_4,a_6)$ such that $E_a\in \mathcal {S}$ . Note that since we work with long Weierstrass equations, the choice of a for a given isomorphism class of elliptic curves is not unique. According to [Reference Cremona and SadekCS23, Theorem 1.1], the proportion of integral Weierstrass equations that are globally minimal is $1/\zeta (10)=93\,555/\pi ^{10}\approx 99.99\%$ . Let $\Phi $ be a possibly infinite set of prime numbers and for each prime $\ell \in \Phi $ , let $U_\ell $ be a subset of $\mathcal {W}(\mathbb {Z}_\ell )$ defined by a set of congruence classes. In other words, $U_\ell $ is the inverse image of a subset of $\mathcal {W}(\mathbb {Z}/\ell ^M \mathbb {Z})$ for some integer $M>0$ . Let U be the family of conditions $\{U_\ell \mid \ell \in \Phi \}$ . Given $N>0$ , let

$$ \begin{align*}Z_N(U):=\{a\in \mathcal{W}(\mathbb{Z})\mid a\in U_\ell \text{ for some prime }\ell>N\}.\end{align*} $$

Definition 3.2. The family $U=\{U_\ell \mid \ell \in \Phi \}$ is said to be admissible if

$$ \begin{align*}\lim_{N\rightarrow \infty} \mathfrak{d}(Z_N(U))=0.\end{align*} $$

Let $\mathcal {W}_U$ be the set of integral Weierstrass equations $a\in \mathcal {W}(\mathbb {Z})$ not satisfying $U_\ell $ at any prime $\ell \in \Phi $ .

Proposition 3.3. Let U be an admissible family and $s_\ell =\mu (U_\ell )$ . Then, $\sum _{\ell \in \Phi } s_\ell $ converges and

$$ \begin{align*}\mathfrak{d}(\mathcal{W}_U)=\prod_{\ell\in \Phi} (1-s_\ell).\end{align*} $$

Lemma 3.5. Let T be a Kodaira type such that $U(T)$ is admissible, then, the density of $\mathcal {W}_{U(T)}$ exists and equals

$$ \begin{align*}\mathfrak{d}(\mathcal{W}_{U(T)})=\prod_{\ell\neq p}(1-\rho_T(\ell))\geq 1-\sum_{\ell\neq p} \rho_T(\ell).\end{align*} $$

We come to the main result of this section, which is subsequently used in the proof of our main result in the next section.

Let $\Delta \in \mathbb {Z}_{<0}$ with $\Delta \equiv 0,1\, \pmod {\,4}$ , and set

$$ \begin{align*}B(\Delta):=\{ax^2+bxy+cy^2\in \mathbb{Z}[x,y]\mid a>0, b^2-4ac=\Delta\}.\end{align*} $$

The group $\operatorname {SL}_2(\mathbb {Z})$ acts on $\mathbb {Z}[x,y]$ , where a matrix $\sigma = ( {\begin {smallmatrix} p & q \\ r & s \\ \end {smallmatrix} } )$ sends $x\mapsto (px+qy)$ and $y\mapsto (rx+sy)$ . Thus, if $f=ax^2+bxy+cy^2$ , the matrix $\sigma $ acts by

$$ \begin{align*}f\circ \sigma= a(px+qy)^2+b(px+qy)(rx+sy)+c(rx+sy)^2.\end{align*} $$

We note that $B(\Delta )$ is stable under the action of $\operatorname {SL}_2(\mathbb {Z})$ and $B(\Delta )/\operatorname {SL}_2(\mathbb {Z})$ is finite; see [Reference SchoofSch87] for additional details.

Let $d(p)$ be the number of tuples $a\in \mathcal {W}(\mathbb {F}_p)$ such that $E_a(\mathbb {F}_p)[p]\neq 0$ . Note that

$$ \begin{align*}\frac{d(p)}{\#\mathcal{W}(\mathbb{F}_p)}=\frac{d(p)}{p^5}.\end{align*} $$

Proposition 3.12. With respect to the notation above,

$$ \begin{align*}\overline{\mathfrak{d}}(\mathcal{S}_p')\leq \begin{cases}\bigg( \dfrac{p-1}{2p^2}\bigg)H(1-4p) & \text{if }p\geq 7,\\ \bigg(\dfrac{p-1}{2p^2}\bigg)(H(1-4p)+H(p^2+1-6p)) & \text{if }p\leq 5. \end{cases} \end{align*} $$

Proof. The number of isomorphism classes of elliptic curves E over $\mathbb {F}_p$ such that $E(\mathbb {F}_p)[p]\neq 0$ is

$$ \begin{align*} \begin{cases}H(1-4p) & \text{if }p\geq 7,\\ H(1-4p)+H(p^2+1-6p) & \text{if }p\leq 5; \end{cases} \end{align*} $$

see [Reference Ray and SujathaRS23, Corollary 3.11] for additional details. Given $a\in \mathcal {W}(\mathbb {F}_p)$ and associated integral Weierstrass model $E_a$ , then, after a transformation $(X,Y){\kern-1.2pt}\mapsto{\kern-1.2pt} (X+r, Y+sX+t)$ , we obtain an elliptic curve with a short Weierstrass equation. Note that two elliptic curves

$$ \begin{align*} E_{c,d}& : Y^2=X^3+cX+d\\ E_{c',d'}& : Y^2=X^3+c'X+d' \end{align*} $$

are isomorphic over $\mathbb {F}_p$ if there exists $s\in \mathbb {F}_p^\times $ such that

$$ \begin{align*}c'=s^4 c\quad\text{and}\quad d'=s^6 d. \end{align*} $$

It is thus easy to see that the number of Weierstrass equations in a given isomorphism class is at most $p^3(({p-1})/{2})$ . Thus, we find that

$$ \begin{align*} d(p)\leq \begin{cases}p^3\bigg(\dfrac{p-1}{2}\bigg)H(1-4p) & \text{if }p\geq 7,\\ p^3\bigg(\dfrac{p-1}{2}\bigg)(H(1-4p)+H(p^2+1-6p) ) & \text{if }p\leq 5. \end{cases}\\[-41pt] \end{align*} $$

4 Results for a fixed prime and varying elliptic curve

In this section, we prove our main results. Throughout, we fix an odd prime number p. Given a set of rational elliptic curves $\mathcal {S}$ , we obtain a subset of $\mathcal {W}(\mathbb {Z})$ consisting of all a such that $E_a\in \mathcal {S}$ . By abuse of notation, we refer to this set as $\mathcal {S}$ as well. The density $\mathfrak {d}(\mathcal {S})$ is defined as in (3-1). Recall that the upper and lower densities, denoted $\overline {\mathfrak {d}}(\mathcal {S})$ and $\underline {\mathfrak {d}}(\mathcal {S})$ , are always defined even if $\mathfrak {d}(\mathcal {S})$ need not be. Let $\mathcal {E}_p$ be the set of elliptic curves $E_{/\mathbb {Q}}$ such that $\operatorname {Hom}_G(\operatorname {Cl}_K, E[p])\neq 0$ . Note that for $E\in \mathcal {E}_p$ , we have in particular that $p\,|\,\#\operatorname {Cl}_K$ .

Recall from Definition 3.7 that $\mathcal {S}_{p}$ is the set of elliptic curves $E_{/\mathbb {Q}}$ such that $p|c_\ell (E)$ for some prime $\ell \neq p$ . Recall from Definition 3.9 that $\mathcal {S}_p'$ is the set of elliptic curves E for which $p\nmid \Delta (E)$ and $E(\mathbb {Q}_p)[p]\neq 0$ . Denote by $\mathcal {S}_p"$ the set of elliptic curves $E_{/\mathbb {Q}}$ with bad reduction at p. Let $\mathfrak {T}_p$ be the elliptic curves E such that is finite and .

Proposition 4.1. Let p be an odd prime. Then,

$$ \begin{align*}\underline{\mathfrak{d}}(\mathcal{E}_p)\geq \underline{\mathfrak{d}}(\mathfrak{T}_p\backslash (\mathcal{S}_p\cup \mathcal{S}_p'\cup \mathcal{S}_p")).\end{align*} $$

Throughout, we assume that for any elliptic curve $E_{/\mathbb {Q}}$ , the p-primary part of the Tate–Shafarevich group is finite. The density of elliptic curves $E_{/\mathbb {Q}}$ such that was studied by Delaunay [Reference Delaunay, Conrey, Farmer, Mezzadri and SnaithDel07].

Conjecture 4.2 (Delaunay).

With respect to the notation above,

$$ \begin{align*}\underline{\mathfrak{d}}(\mathfrak{T}_p)\geq 1-\prod_{i\geq 1}(1-p^{-(2i-1)})> p^{-1}+p^{-3}-p^{-4}.\end{align*} $$

Moreover, given a set $\mathcal {E}$ of elliptic curves defined by local congruence conditions,

$$ \begin{align*}\underline{\mathfrak{d}}(\mathfrak{T}_p\cap \mathcal{E})=\underline{\mathfrak{d}}(\mathfrak{T}_p)\mathfrak{d}(\mathcal{E}).\end{align*} $$

Remark 4.3. For the purposes of this article, we use only the second lower bound predicted by the above conjecture: that is, we assume that

$$ \begin{align*}\underline{\mathfrak{d}}(\mathfrak{T}_p\cap \mathcal{E})>( p^{-1}+p^{-3}-p^{-4} )\mathfrak{d}(\mathcal{E}),\end{align*} $$

where $\mathcal {E}$ is taken to be the complement of $\mathcal {S}_p\cup \mathcal {S}_p'\cup \mathcal {S}_p"$ .

Theorem 4.4. Let p be an odd prime and assume the above conjecture. Then,

$$ \begin{align*}\underline{\mathfrak{d}}(\mathcal{E}_p)>(p^{-1}+p^{-3}-p^{-4})(1-p^{-1}-\mathfrak{d}_p-\mathfrak{d}_p'),\end{align*} $$

where

$$ \begin{align*}\mathfrak{d}_p=\begin{cases} \zeta(p)-1 & \text{if }p\geq 5,\\ (\zeta(3)-1)+(\zeta(4)-1)+(\zeta(7)-1)& \text{if }p=3,\\ \end{cases}\end{align*} $$

$$ \begin{align*}\mathfrak{d}_p'=\begin{cases}\bigg( \dfrac{p-1}{2p^2}\bigg)H(1-4p) & \text{if }p\geq 7,\\ \bigg(\dfrac{p-1}{2p^2}\bigg)(H(1-4p)+H(p^2+1-6p)) & \text{if }p\leq 5.\\ \end{cases}\end{align*} $$

Proof. Let $\mathcal {E}$ be the set of elliptic curves in the complement of $\mathcal {S}_p\cup \mathcal {S}_p'\cup \mathcal {S}_p"$ . We have from Theorem 3.8 (respectively Proposition 3.12) that $\mathfrak {d}(\mathcal {S}_p)=\mathfrak {d}_p$ (respectively $\mathfrak {d}(\mathcal {S}_p')=\mathfrak {d}_p'$ ). Furthermore, we have from Theorem 3.1 and Proposition 3.3 that $\mathfrak {d}(\mathcal {S}_p")={1}/{p}$ . From Proposition 4.1, we have that $\underline {\mathfrak {d}}(\mathcal {E}_p)\geq \underline {\mathfrak {d}}(\mathfrak {T}_p\backslash (\mathcal {S}_p\cup \mathcal {S}_p'\cup \mathcal {S}_p"))$ . According to our assumption,

$$ \begin{align*} &\underline{\mathfrak{d}}(\mathfrak{T}_p\backslash (\mathcal{S}_p\cup \mathcal{S}_p'\cup \mathcal{S}_p"))=\underline{\mathfrak{d}}(\mathfrak{T}_p\cap \mathcal{E})\\ &\quad\geq \bigg(1-\prod_{i\geq 1}(1-p^{-(2i-1)})\bigg)\mathfrak{d}(\mathcal{E})\\ &\quad> ( p^{-1}+p^{-3}-p^{-4} )\mathfrak{d}(\mathcal{E})\\ &\quad= ( p^{-1}+p^{-3}-p^{-4} ) (1-p^{-1}-\mathfrak{d}_p-\mathfrak{d}_p'),\end{align*} $$

and the result follows.

Corollary 4.5. Assume Conjecture 4.2. Then,

$$ \begin{align*}\underline{\mathfrak{d}}(\mathcal{E}_p)\geq p^{-1}+O(p^{-3/2+\epsilon}).\end{align*} $$

In other words, for any choice of $\epsilon>0$ , there is a constant $C>0$ , depending on $\epsilon $ and independent of p, such that

$$ \begin{align*}\underline{\mathfrak{d}}(\mathcal{E}_p)\geq p^{-1}-Cp^{-3/2+\epsilon}.\end{align*} $$

Proof. According to Theorem 4.4,

$$ \begin{align*}\begin{split} \underline{\mathfrak{d}}(\mathcal{E}_p)&> (p^{-1}+p^{-3}-p^{-4})(1-p^{-1}-\mathfrak{d}_p-\mathfrak{d}_p')\\ &> p^{-1}-p^{-1}(p^{-1}+\mathfrak{d}_p+\mathfrak{d}_p'). \end{split}\end{align*} $$

We have that

$$ \begin{align*}\zeta(p)-1=2^{-p}+\sum_{n\geq 3} n^{-p}< 2^{-p}+\int_{2}^\infty x^{-p}dx=2^{-p}\bigg(\dfrac{p+1}{p-1}\bigg),\end{align*} $$

and hence,

$$ \begin{align*}\mathfrak{d}_p=O(2^{-p})=O(p^{-1/2+\epsilon}).\end{align*} $$

However, it follows from known results that

$$ \begin{align*}\mathfrak{d}_p'=O(p^{-1/2}\operatorname\log p(\operatorname\log \operatorname\log p)^2)=O(p^{-1/2+\epsilon}),\end{align*} $$

(see [Reference LenstraLJ87, Proposition 1.9]) and the result follows.

5 Results for a fixed elliptic curve and varying prime

In this section, we prove results for a fixed elliptic curve $E_{/\mathbb {Q}}$ and varying prime p. We assume throughout that E does not have complex multiplication, and that the Tate–Shafarevich group is finite. Given a prime p, we let $K_p=\mathbb {Q}(E[p])$ . We make a number of observations with regards to the variation of $\operatorname {Cl}_{K_p}/p\operatorname {Cl}_{K_p}$ , where p varies over all primes p.

Proof. Note that for all but finitely many primes p:

1. (1) E has good reduction at p;

2. (2) $E(\mathbb {Q}_\ell )[p]=0$ for all primes $\ell $ at which E has bad reduction;

3. (3) ;

4. (4) the representation on $E[p]$ is irreducible;

5. (5) $c_\ell ^{(p)}(E)=1$ for all primes $\ell $ at which E has bad reduction.

Thus, it follows from Theorem 2.3 that we have that $\operatorname {Hom}_G(\operatorname {Cl}_{K_p}/p\operatorname {Cl}_{K_p}, E[p])=0$ provided $E(\mathbb {Q}_p)[p]=0$ . Let $\widetilde {E}$ be the reduction of E over $\mathbb {F}_p$ . The formal group $\widehat {E}(\mathbb {Z}_p)$ of E at p has no nontrivial p-torsion. Thus, if p is a prime of good reduction, the natural map $E(\mathbb {Q}_p)\rightarrow \widetilde {E}(\mathbb {F}_p)$ induces an injection

$$ \begin{align*}E(\mathbb{Q}_p)[p]\rightarrow \widetilde{E}(\mathbb{F}_p)[p].\end{align*} $$

A prime p at which E has good reduction is called an anomalous prime if $\widetilde {E}(\mathbb {F}_p)[p]\neq 0$ . It is well known that for a non-CM elliptic curve, $100\%$ of primes are nonanomalous (see for instance [Reference Murty and MotohashiMur97]). Thus, $E(\mathbb {Q}_p)[p]=0$ for $100\%$ of primes p, and thus the result follows.

Theorem 5.2. Let $E_{/\mathbb {Q}}$ be an elliptic curve and assume that the following conditions are satisfied:

1. (1) is finite;

2. (2) $\operatorname {rank} E(\mathbb {Q})\geq 2$ ;

3. (3) E does not have complex multiplication.

Then, for $100\%$ of primes p,

$$ \begin{align*}\operatorname{dim}\operatorname{Hom}_G(\operatorname{Cl}_{K_p}/p\operatorname{Cl}_{K_p}, E[p])\geq \operatorname{rank} E(\mathbb{Q})-1.\end{align*} $$

Proof. It follows from the proof of Theorem 5.1 that for $100\%$ of primes p, all of the following conditions are satisfied:

1. (1) E has good reduction at p;

2. (2) the representation on $E[p]$ is irreducible;

3. (3) $c_\ell ^{(p)}(E)=1$ for all primes $\ell $ at which E has bad reduction;

4. (4) $E(\mathbb {Q}_p)[p]=0$ .

It follows from Theorem 2.1 that

$$ \begin{align*}\operatorname{dim}\operatorname{Hom}_G(\operatorname{Cl}_{K_p}/p\operatorname{Cl}_{K_p}, E[p])\geq \operatorname{rank} E(\mathbb{Q})-1.\\[-37pt] \end{align*} $$

6 Computational results

In this section, we present some computations of $\operatorname {Cl}_{K}/3\operatorname {Cl}_K$ , where $K=\mathbb {Q}(E[3])$ , for certain families of elliptic curves E. Due to the difficulty of computing statistics for larger extensions, we primarily restrict ourselves to the case where $\operatorname {Gal}(K/\mathbb {Q})\subset \operatorname {GL}_2(\mathbb {F}_3)$ is the normalizer of a split Cartan subgroup. This is the smallest irreducible subgroup of $\operatorname {GL}_2(\mathbb {F}_3)$ . We include also a few additional calculations where the Galois group is the normalizer of a nonsplit Cartan subgroup.

We note that the hypotheses of [Reference Prasad and ShekharPS21] are often violated in these cases. Specifically, a majority of such E has bad reduction at $3$ . In addition, all ramification at $3$ is tame as the Galois group has order prime to $3$ : thus, any such E for which $a_3(E) \equiv 1\, \pmod {\,3}$ has $3$ as a local torsion prime. We computed our examples without regard to these considerations. We have verified that every computation violating the lower bound of [Reference Prasad and ShekharPS21] does not satisfy their hypotheses.

6.1 Normalizer of a split Cartan: varying j-invariants

We use the formula of Zywina [Reference ZywinaZyw15, Theorem 1.2],

$$ \begin{align*}j_2(t) = 27\frac{(t+1)^3(t-3)^3}{t^3}\end{align*} $$

such that any elliptic curve with such a j-invariant has $\operatorname {Gal}(\mathbb {Q}(E[3])/\mathbb {Q})$ contained in the normalizer of a split Cartan subgroup. For any t, let $E_t$ denote the elliptic curve of smallest conductor with j-invariant $j_2(t)$ and let $K_t$ denote the field $\mathbb {Q}(E_t[3])$ . We computed $\operatorname {Sel}_3(E_t/\mathbb {Q})$ as well as the Galois module structure of $\operatorname {Cl}_{K_t}/3$ for the 599 elliptic curves in the set

$$ \begin{align*}S_0 = \{E_{a/b} \mid |a| \leq 50, 1 \leq b \leq 10, [K_t:\mathbb{Q}]=8 \}.\end{align*} $$

Let $L_t$ denote the unique biquadratic subfield of $K_t$ ; note that $L_t$ necessarily contains $\mathbb {Q}(\!\sqrt {-3})$ . In each case, there was $d_t \geq 0$ such that there was an isomorphism

$$ \begin{align*}\operatorname{Cl}_{K_t}\!/3 \cong \operatorname{Cl}_{L_t}\!/3 \oplus E_t[3]^{d_t}\end{align*} $$

as $\operatorname {Gal}(K_t/\mathbb {Q})$ -modules. Note that $\operatorname {Gal}(K_t/\mathbb {Q})$ acts on $\operatorname {Cl}_{L_t}\!/3$ via its abelianization.

Our interest here is primarily in comparing $\operatorname {Sel}_3(E_t/\mathbb {Q})$ and $d_t$ . We briefly report the data for $\operatorname {Cl}_{L_t}\!/3$ in Table 2. The only curve in our sample with $\dim \operatorname {Cl}_{L_t}\!/3 = 3$ was $E_{-46/3}$ with $L_t = \mathbb {Q}(\!\sqrt {-3}, \sqrt {2917})$ .

Table 2 Mod 3 class group data.

Please see Table 3 for the other quantities.

Table 3 3 Selmer group data (i).

We remark that the restriction to E with specified $3$ -torsion fields obviously biases the distribution of $3$ -Selmer groups. Although it is unwise to extrapolate too much from this limited data, we note that for t with $\dim \operatorname {Sel}_3(E_t/\mathbb {Q}) \leq 1$ , the dimension of $\operatorname {Cl}_{K_t}\!/3$ appears to be roughly the same (being nonzero around $19\%$ of the time) whether the Selmer group has dimension zero or one. By contrast, for Selmer groups of dimension two, the proportion having $d_t \geq 1$ is much larger at $44\%$ . For Selmer groups of dimension three, it is much larger than that in this limited sample.

Granting the limits of this data set, it certainly appears that $\operatorname {Sel}_3(E_t/\mathbb {Q})$ has a significant effect on $d_t$ , but there are additional influences as well that are not currently clear.

6.2 Normalizer of a split Cartan: fixed j-invariants

We also investigated two quadratic twist families. Let $E_1$ denote the elliptic curve

$$ \begin{align*}y^2+xy+y = x^3-141x+624\end{align*} $$

with $j(E_1) = 857\,375/8$ and conductor $10\,082=2 \cdot 71^2$ . For any squarefree t, let $E_{1,t}$ denote the quadratic twist of $E_1$ by t and let $K_{1,t}$ denote the field $\mathbb {Q}(E_{1,t}[3])$ . Then, $\operatorname {Gal}(K_{1,t}/\mathbb {Q})$ is the normalizer of a split Cartan subgroup of $\operatorname {GL}_2(\mathbb {F}_3)$ . We note that each $K_{1,t}$ contains the biquadratic field $L_1=\mathbb {Q}(\!\sqrt {-3},\sqrt {-71})$ , which has trivial $3$ -class group.

Consider the set of 608 elliptic curves

$$ \begin{align*}S_1 = \{ E_{1,t} \mid 1 \leq t \leq 1000, t \text{~squarefree} \}.\end{align*} $$

In each case, we computed a $\operatorname {Gal}(K_{1,t}/\mathbb {Q})$ -isomorphism

$$ \begin{align*}\operatorname{Cl}_{K_{1,t}}\!/3 \cong E_{1,t}[3]^{d_{1,t}}\end{align*} $$

for some $d_{1,t}$ . We computed the data as seen in Table 4.

Table 4 3 Selmer group data (ii).

We note here that the dependence of $d_{1,t}$ on $\mathrm {Sel}_3(E_{1,t}/\mathbb {Q})$ is more clear than in the case of varying j-invariants: the lower bound of [Reference Prasad and ShekharPS21] (although, as noted, the hypotheses are often violated in this data set) appears to control most of the behaviour of $d_{1,t}$ .

We computed analogous data for the elliptic curve $E_2$ given by

$$ \begin{align*}y^2 = x^3-83\,667\,346\,875x - 10\,711\,930\,420\,406\,250\end{align*} $$

with $j(E_2)= -42\,875/8$ and conductor $6962 = 2 \cdot 59^2$ . With notation as above, we consider the set of 608 quadratic twists

$$ \begin{align*}S_2 = \{ E_{2,t} \mid 1 \leq t \leq 1000, t \text{~squarefree} \}.\end{align*} $$

Each $3$ -division field field $K_{2,t}$ has Galois group the normalizer of a split Cartan subgroup and contains the biquadratic field $L_2=\mathbb {Q}(\sqrt {-3},\sqrt {-59}).$ This field has class group $\mathbb {Z}/3$ , and for each t, we computed a $\operatorname {Gal}(K_{2,t}/\mathbb {Q})$ -isomorphism

$$ \begin{align*}\operatorname{Cl}_{K_{2,t}}\!/3 \cong \operatorname{Cl}_{L_2}\!/3 \oplus E_{2,t}[3]^{d_{2,t}}\end{align*} $$

for some $d_{2,t}$ (see Table 5).

Table 5 3 Selmer group data (iii).

This behaviour appears somewhat less straightforward than for $S_1$ . The distribution of $\mathrm {Sel}_3(E_{2,t}/\mathbb {Q})$ is obviously quite different from that of $\mathrm {Sel}_3(E_{1,t}/\mathbb {Q})$ and the effect on $d_{2,t}$ is less clear. We do note that the distributions of $d_{1,t}$ and $d_{2,t}$ ignoring Selmer groups are similar, which is somewhat curious.

6.3 Normalizer of a nonsplit Cartan: fixed j-invariant

We compiled very limited data in one case where the mod $3$ Galois image is the normalizer of a nonsplit Cartan subgroup. Consider the elliptic curve $E_3$ given by

$$ \begin{align*}y^2 = x^3+x^2-2x-8\end{align*} $$

with $j(E_3)=-64$ and conductor $1568=2^5 \cdot 7^2$ . We consider its set of 61 twists

$$ \begin{align*}S_3 = \{ E_{3,t} \mid 1 \leq t \leq 100, t \text{~squarefree}\}.\end{align*} $$

For each t, we have an isomorphism of $\operatorname {Gal}(K_{3,t}/\mathbb {Q})$ -modules

$$ \begin{align*}\operatorname{Cl}_{K_{3,t}}\!/3 \cong E_{3,t}[3]^{d_{3,t}}\end{align*} $$

for some $d_{3,t}$ (see Table 6).

Table 6 3 Selmer group data (iv).

There is too little data here to provide anything more than wild speculations; unfortunately the computations rapidly became very time consuming beyond this point.