2.1 Recollections from Iwasawa theory

For details, we refer the reader to standard texts in Iwasawa theory (for example, [Reference WashingtonWas97, Ch. 13]). Let p be a fixed prime. Consider the (unique) cyclotomic ${\mathbb Z}_p$ -extension of ${\mathbb Q}$ , denoted by ${\mathbb Q}_{\operatorname {\mathrm {cyc}}}$ . Set $\Gamma :=\operatorname {\mathrm {Gal}}({\mathbb Q}_{\operatorname {\mathrm {cyc}}}/{\mathbb Q})\simeq {\mathbb Z}_p$ . The Iwasawa algebra $\Lambda $ is the completed group algebra ${\mathbb Z}_p [\![ \Gamma ]\!] :=\varprojlim _n {\mathbb Z}_p[\Gamma /\Gamma ^{p^n}]$ . Fix a topological generator $\gamma $ of $\Gamma $ ; there is the following isomorphism of rings:

$$ \begin{align*} \Lambda&\xrightarrow{\sim}{\mathbb Z}_p [\![ T ]\!] \\ \gamma &\mapsto 1 +T. \end{align*} $$

Let M be a cofinitely generated cotorsion $\Lambda $ -module. The structure theorem of $\Lambda $ -modules asserts that the Pontryagin dual of M, denoted by $M^{\vee }$ , is pseudo-isomorphic to a finite direct sum of cyclic $\Lambda $ -modules. In other words, there is a map of $\Lambda $ -modules

$$ \begin{align*} M^{\vee}\longrightarrow \bigg(\bigoplus_{i=1}^s \Lambda/(p^{m_i})\bigg)\oplus \bigg(\bigoplus_{j=1}^t \Lambda/(h_j(T)) \bigg) \end{align*} $$

with finite kernel and cokernel. Here, $m_i>0$ and $h_j(T)$ is a distinguished polynomial (that is, a monic polynomial with nonleading coefficients divisible by p). The characteristic ideal of $M^{\vee }$ is (up to a unit) generated by the characteristic element,

$$ \begin{align*} f_{M}^{(p)}(T) := p^{\sum_{i} m_i} \prod_j h_j(T). \end{align*} $$

The $\mu $ -invariant of M is defined as the power of p in $f_{M}^{(p)}(T)$ . More precisely,

$$ \begin{align*} \mu(M) = \mu_p(M):=\begin{cases}0 & \textrm{if } s=0,\\ \displaystyle\sum_{i=1}^s m_i & \textrm{if } s>0. \end{cases} \end{align*} $$

The $\lambda $ -invariant of M is the degree of the characteristic element, that is,

$$ \begin{align*} \lambda(M) = \lambda_p(M) := \sum_{j=1}^t \deg h_j(T). \end{align*} $$

Let $E_{/F}$ be an elliptic curve with good reduction at p. We assume throughout that the prime p is odd. Let $N=N_E$ denote the conductor of E and denote by S the (finite) set of primes that divide $Np$ . Let $F_S$ be the maximal algebraic extension of F that is unramified at the primes $v\notin S$ . Set $E[p^{\infty }]$ to be the Galois module of all p-power torsion points in $E(\overline {F})$ . For a prime $v\in S$ and any finite extension $L/F$ contained in the unique cyclotomic ${\mathbb Z}_p$ -extension of F (denoted by $F_{\operatorname {\mathrm {cyc}}}$ ), write

$$ \begin{align*} J_v(E/L) = \bigoplus_{w|v} H^1( G_{L_w}, E)[p^{\infty}], \end{align*} $$

where the direct sum is over all primes w of L lying above v. Then, the p-primary Selmer group over L is defined as follows:

$$ \begin{align*} \operatorname{\mathrm{Sel}}_{p^{\infty}}(E/L):=\ker\bigg\{ H^1(\operatorname{\mathrm{Gal}}(F_S/L),E[p^{\infty}])\longrightarrow \bigoplus_{v\in S} J_v(E/L)\bigg\}. \end{align*} $$

It is easy to see that $\operatorname {\mathrm {Sel}}_{p^{\infty }}(E/L) = \varinjlim _{n}\operatorname {\mathrm {Sel}}_{p^n}(E/L)$ , see for example [Reference Coates and SujathaCS00, Section 1.7]. By taking direct limits of Equation (2-1), the p-primary Selmer group over F fits into a short exact sequence,

(2-2)

Next, define

$$ \begin{align*} J_v(E/F_{\operatorname{\mathrm{cyc}}}) = \varinjlim J_v(E/L), \end{align*} $$

where L ranges over finite extensions contained in $F_{\operatorname {\mathrm {cyc}}}$ and the inductive limit is taken with respect to the restriction maps. The p-primary Selmer group over $F_{\operatorname {\mathrm {cyc}}}$ is defined as follows:

$$ \begin{align*} \operatorname{\mathrm{Sel}}_{p^{\infty}}(E/F_{\operatorname{\mathrm{cyc}}}):=\ker\bigg\{ H^1(\operatorname{\mathrm{Gal}}(F_S/F_{\operatorname{\mathrm{cyc}}}),E[p^{\infty}])\longrightarrow \bigoplus_{v\in S} J_v(E/F_{\operatorname{\mathrm{cyc}}})\bigg\}. \end{align*} $$

When E is an elliptic curve defined over ${\mathbb Q}$ , p is an odd prime of good ordinary reduction, and $F/{\mathbb Q}$ is an abelian extension, K. Kato proved (see [Reference KatoKat04, Theorem 14.4]) that the p-primary Selmer group $\operatorname {\mathrm {Sel}}_{p^{\infty }}(E/F_{\operatorname {\mathrm {cyc}}})$ is a cofinitely generated cotorsion $\Lambda $ -module. Therefore, in view of the structure theorem of $\Lambda $ -modules, we can define the $\mu $ and $\lambda $ -invariants, which we denote as $\mu _p(E/F)$ and $\lambda _p(E/F)$ , respectively.

Given a number field F, we set $F_{\operatorname {\mathrm {cyc}}}$ to be the composite of F with ${\mathbb Q}_{\operatorname {\mathrm {cyc}}}$ . It is the unique ${\mathbb Z}_p$ -extension of F that is contained in the infinite cyclotomic field $F(\mu _{p^{\infty }})$ . Given $n\geq 0$ , let $F_n$ be the subfield of $F_{\operatorname {\mathrm {cyc}}}$ such that $[F_n:F]=p^n$ . The following lemma relating the $\lambda $ -invariant of the Selmer group to the rank of the elliptic curve is well known but we include it for the sake of completeness.

Proof. Denote by $\Gamma _n$ the Galois group $\operatorname {\mathrm {Gal}}(F_{\operatorname {\mathrm {cyc}}}/F_n)$ and let $r_n$ denote the ${\mathbb Z}_p$ -corank of $\operatorname {\mathrm {Sel}}_{p^{\infty }}(E/F_{\operatorname {\mathrm {cyc}}})^{\Gamma _n}$ . Since $\operatorname {\mathrm {Sel}}_{p^{\infty }}(E/F_{\operatorname {\mathrm {cyc}}})$ is cotorsion over the Iwasawa algebra, it has finite ${\mathbb Z}_p$ -corank, and it is an easy consequence of the structure theorem that

$$ \begin{align*} \lambda_p(E/F)=\operatorname{\mathrm{corank}}_{{\mathbb Z}_p}( \operatorname{\mathrm{Sel}}_{p^{\infty}}(E/F_{\operatorname{\mathrm{cyc}}}) ). \end{align*} $$

We deduce from Equation (2-2) that

(2-3) $$ \begin{align} \operatorname{\mathrm{corank}}_{{\mathbb Z}_p} \operatorname{\mathrm{Sel}}_{p^{\infty}}(E/F_n)\geq \operatorname{\mathrm{rank}}_{{\mathbb Z}} (E(F_n)), \end{align} $$

with equality if is finite. It follows from the structure theory of $\Lambda $ -modules that $\lambda _p(E/F_n)\geq r_n$ . It suffices to show that $r_n\geq \operatorname {\mathrm {rank}}_{{\mathbb Z}} (E(F_n))$ . This is indeed the case, since Mazur’s control theorem asserts that there is a natural map

$$ \begin{align*} \operatorname{\mathrm{Sel}}_{p^{\infty}}(E/F_n)\rightarrow \operatorname{\mathrm{Sel}}_{p^{\infty}}(E/F_{\operatorname{\mathrm{cyc}}})^{\Gamma_n} \end{align*} $$

with finite kernel. From Equation (2-3), we see that $r_n\geq \operatorname {\mathrm {rank}}_{{\mathbb Z}} (E(F_n))$ and the result follows.

Let $L/{\mathbb Q}$ be a degree-p Galois extension. The following theorem relates the $\lambda $ -invariants of $\operatorname {\mathrm {Sel}}_{p^{\infty }}(E/{\mathbb Q}_{\operatorname {\mathrm {cyc}}})$ and $\operatorname {\mathrm {Sel}}_{p^{\infty }}(E/L_{\operatorname {\mathrm {cyc}}})$ .

Theorem 2.3. Let $p\geq 5$ be a fixed prime. Let $L/{\mathbb Q}$ be a Galois extension of degree a power of p disjoint from the cyclotomic ${\mathbb Z}_p$ -extension of ${\mathbb Q}$ . Let $E_{/{\mathbb Q}}$ be a fixed elliptic curve with good ordinary reduction at p and suppose that $\operatorname {\mathrm {Sel}}_{p^{\infty }}(E/{\mathbb Q}_{\operatorname {\mathrm {cyc}}})$ is a cofinitely generated ${\mathbb Z}_p$ -module. Then, $\operatorname {\mathrm {Sel}}_{p^{\infty }}(E/L_{\operatorname {\mathrm {cyc}}})$ is also a cofinitely generated ${\mathbb Z}_p$ -module. Moreover, their respective $\lambda $ -invariants are related by the following formula:

$$ \begin{align*} \lambda_p(E/L) = p\lambda_p(E/{\mathbb Q}) + \sum_{w \in P_1} (e_{L_{\operatorname{\mathrm{cyc}}}/{\mathbb Q}_{\operatorname{\mathrm{cyc}}}}(w) -1 ) + 2\sum_{w\in P_2} (e_{L_{\operatorname{\mathrm{cyc}}}/{\mathbb Q}_{\operatorname{\mathrm{cyc}}}}(w) -1 ), \end{align*} $$

where $e_{L_{\operatorname {\mathrm {cyc}}}/{\mathbb Q}_{\operatorname {\mathrm {cyc}}}}(w)$ is the ramification index, and $P_1, P_2$ are sets of primes in $L_{\operatorname {\mathrm {cyc}}}$ such that

$$ \begin{align*} P_1 & = \{ w : \ w\nmid p, \ E \textrm{ has split multiplicative reduction at }w \},\\ P_2 & = \{ w : \ w\nmid p, \ E \textrm{ has good reduction at }w, \ E(L_{\operatorname{\mathrm{cyc}},w}) \textrm{ has a point of order }p \}. \end{align*} $$

We record the following result which was first proven by Greenberg.

2.2 Results, conjectures, and heuristics on ranks of elliptic curves

There are several important conjectures in the theory of elliptic curves. The first one of interest is the rank distribution conjecture which claims that over any number field, half of all elliptic curves (when ordered by height) have Mordell–Weil rank zero and the remaining half have Mordell–Weil rank one. Finally, higher Mordell–Weil ranks constitute zero percent of all elliptic curves, even though there may exist infinitely many such elliptic curves. Therefore, a suitably defined average rank would be $1/2$ . The best results in this direction are by Bhargava and Shankar (see [Reference Bhargava and ShankarBS15a, Reference Bhargava and ShankarBS15b]). They show that the average rank of elliptic curves over ${\mathbb Q}$ is strictly less than one, and that both rank zero and rank one cases comprise nonzero densities across all elliptic curves over ${\mathbb Q}$ (see [Reference Bhargava and ShankarBS13]).

Given an elliptic curve E defined over ${\mathbb Q}$ and base-changed to F, we have the associated Hasse–Weil L function, $L_E(s,F)$ . Let $F/{\mathbb Q}$ be an abelian extension with Galois group G and conductor f. Let $\hat {G}$ be the group of Dirichlet characters, $\chi : ({\mathbb Z}/f{\mathbb Z})^{\times } \rightarrow {\mathbb C}^{\times }$ . We further know that

$$ \begin{align*} L_E(s, F) = \prod_{\chi\in \hat{G}}L_E(s,\chi), \end{align*} $$

where the terms appearing on the right-hand side are the L-functions of $E_{/{\mathbb Q}}$ twisted by the character $\chi $ .

Conjecture 2.7 (Birch and Swinnerton-Dyer).

The Hasse–Weil L-function has analytic continuation to the whole complex plane, and

$$ \begin{align*} \operatorname{\mathrm{ord}}_{s=1} L_E(s,F) = \operatorname{\mathrm{rank}}_{{\mathbb Z}}(E(F)). \end{align*} $$

It follows from the Birch and Swinnerton-Dyer (BSD) conjecture that the vanishing of the twisted L-functions $L_E(s,\chi )$ at $s=1$ is equivalent to the existence of rational points of infinite order on $E(F)$ .

The following conjecture of David, Fearnley, and Kisilevsky predicts that given an elliptic curve over ${\mathbb Q}$ , the rank ‘rarely’ jumps in ${\mathbb Z}/p{\mathbb Z}$ -extensions with $p\neq 2$ . More precisely, we have the following conjecture.

Conjecture 2.8 [Reference David, Fearnley, Kisilevsky, Conrey, Farmer, Mezzadri and SnaithDFK07, Conjecture 1.2].

Let p be an odd prime and $E_{/{\mathbb Q}}$ be an elliptic curve. Define

$$ \begin{align*} N_{E,p}(X) := \# \{\chi \textrm{ of order }p \mid \operatorname{\mathrm{cond}}(\chi)\leq X \textrm{ and } L_{E}(1,\chi)=0 \}. \end{align*} $$

1. (1) If $p=3$ , then as $X\rightarrow \infty $ ,

   $$ \begin{align*} \log N_{E,p}(X) \sim \tfrac{1}{2}\log X. \end{align*} $$

2. (2) If $p=5$ , then as $X\rightarrow \infty $ , the set $N_{E,p}(X)$ is unbounded but $N_{E,p}(X)\ll X^{\epsilon }$ for any $\epsilon>0$ .

3. (3) If $p\geq 7$ , then $N_{E,p}(X)$ is bounded.

Under BSD, $\# N_{E,p}(X)$ can be rewritten as

$$ \begin{align*} (p-1)\#\{ F/{\mathbb Q} \textrm{ is cyclic of degree }p \mid \operatorname{\mathrm{cond}}(F)\leq X \textrm{ and } \operatorname{\mathrm{rank}}_{{\mathbb Z}}(E(F))>\operatorname{\mathrm{rank}}_{{\mathbb Z}}(E({\mathbb Q})) \}. \end{align*} $$

In [Reference DokchitserDok07, Theorem 1], T. Dokchitser showed that given an elliptic curve $E_{/{\mathbb Q}}$ , there are infinitely many ${\mathbb Z}/3{\mathbb Z}$ extensions where the rank jumps. More recently, B. Mazur and K. Rubin have shown (see [Reference Mazur, Rubin and LarsenMRL18, Theorem 1.2]) that given an elliptic curve E, there is a positive density set of primes (call it $\mathcal {S}$ ) such that for each $p\in \mathcal {S}$ , there are infinitely many cyclic degree-p extensions over ${\mathbb Q}$ with $\operatorname {\mathrm {rank}}_{{\mathbb Z}}(E(L)) = \operatorname {\mathrm {rank}}_{{\mathbb Z}}(E({\mathbb Q}))$ . However, the result is unable to provide a positive proportion. Mazur and Rubin revisited this conjecture in a recent preprint (see [Reference Mazur and RubinMR19]) and their heuristics, based on the distribution of modular symbols, predicts the same statement as the conjecture of David, Fearnley, and Kisilevsky.

3.1

Let $E_{/{\mathbb Q}}$ be a fixed rank 0 elliptic curve of conductor N with good ordinary reduction at a fixed prime $p\geq 5$ such that $\mu _p(E/{\mathbb Q}) = \lambda _p(E/{\mathbb Q})=0$ . Recall that the rank distribution conjecture predicts that half of the elliptic curves (ordered by height or conductor) have rank 0. Moreover, Proposition 2.5 asserts that there are density one good ordinary primes satisfying the condition of vanishing Iwasawa invariants. Lastly, when an elliptic curve does not have CM, density one of the primes are good ordinary; in the CM case, the good ordinary and the good supersingular primes each have density 1/2.

Throughout this section, $L/{\mathbb Q}$ denotes a ${\mathbb Z}/p{\mathbb Z}$ -extension disjoint from the cyclotomic ${\mathbb Z}_p$ -extension. Let q be a prime number distinct from p such that E has good reduction at q, that is, $\gcd (q,N)=1$ . Let $w|q$ be a prime in L, $L_w$ be the completion at w, and $\kappa $ be the residue field of characteristic q. We know that there is an exact sequence of abelian groups (see [Reference SilvermanSil09, Proposition VII.2.1]),

$$ \begin{align*} 0 \rightarrow E_1(L_w) \rightarrow E_0(L_w) \rightarrow \widetilde{E}_{\operatorname{\mathrm{ns}}}(\kappa) \rightarrow 0, \end{align*} $$

where $\widetilde {E}_{\operatorname {\mathrm {ns}}}(\kappa )$ is the set of nonsingular points of the reduced elliptic curve, $E_0(L_w)$ is the set of points with nonsingular reduction, and $E_1(L_w)$ is the kernel of the reduction map. Since $p\neq q$ , we know that $E_1(L_w)[p]$ is trivial (see [Reference SilvermanSil09, VII.3.1]). Because q is assumed to be a prime of good reduction, $E_0(L_w) = E(L_w)$ . Hence,

$$ \begin{align*} E(L_w)[p] \simeq \widetilde{E}(\kappa)[p]. \end{align*} $$

Since $L/{\mathbb Q}$ is a ${\mathbb Z}/p{\mathbb Z}$ -extension, the residue field is either $\mathbb {F}_{q^p}$ or $\mathbb {F}_{q}$ depending on whether q is inert in the extension or not.

As explained above, (under standard hypotheses) we know that $\lambda _p(E/L)\geq \operatorname {\mathrm {rank}}_{{\mathbb Z}}(E(L))$ . Since we have assumed that $\mu _p(E/{\mathbb Q})=0$ , it follows from Theorem 2.3 that $\mu _p(E/L)=0$ . To show that $\lambda _p(E/L)=0$ , it suffices to show that

$$ \begin{align*} \sum_{w\in P_1} ( e_{L_{\operatorname{\mathrm{cyc}}}/{\mathbb Q}_{\operatorname{\mathrm{cyc}}}}(w) -1) = \sum_{w\in P_2} ( e_{L_{\operatorname{\mathrm{cyc}}}/{\mathbb Q}_{\operatorname{\mathrm{cyc}}}}(w) -1)=0. \end{align*} $$

Recall that all primes in the cyclotomic ${\mathbb Z}_p$ -extension are finitely decomposed and the only primes that ramify are those above p. Moreover, $L\cap {\mathbb Q}_{\operatorname {\mathrm {cyc}}} ={\mathbb Q}$ and $p\not \in P_1 \cup P_2$ (recall the definition of these sets from Theorem 2.3). Therefore, $e_{L_{\operatorname {\mathrm {cyc}}}/{\mathbb Q}_{\operatorname {\mathrm {cyc}}}} = e_{L/{\mathbb Q}}$ . Since $p\geq 5$ , the reduction type does not change upon base-change. In particular, if $q(\neq p)$ is a prime of additive reduction for $E_{/{\mathbb Q}}$ , then it has additive reduction over $L_{\operatorname {\mathrm {cyc}}}$ (see [Reference Serre and TateST68, page 498] or [Reference Hachimori and MatsunoHM99, page 587]). Finally, since $L_{\operatorname {\mathrm {cyc}},w}/L_w$ is a pro-p group, we have that $E(L_{\operatorname {\mathrm {cyc}},w})[p^{\infty }]=0$ if and only if $E(L_{w})[p^{\infty }]=0$ . Thus, it suffices to show that

(3-1) $$ \begin{align} \sum_{w\in P_1} ( e_{L/{\mathbb Q}}(w) -1) = \sum_{w\in P_2} ( e_{L/{\mathbb Q}}(w) -1)=0, \end{align} $$

where $P_1, P_2$ are now sets of primes in L. More precisely,

$$ \begin{align*} P_1 & = \{ w\in L : \ w\nmid p, \ E \textrm{ has split multiplicative reduction at }w \},\\ P_2 & = \{ w\in L : \ w\nmid p, \ E \textrm{ has good reduction at }w, \ E(L_{w}) \textrm{ has a point of order }p \}. \end{align*} $$

It is often possible that $P_1 = \emptyset $ but the set $P_2$ is never empty. In fact, it is known that (see for example [Reference CojocaruCoj04, Section 2])

$$ \begin{align*} \lim_{X\rightarrow\infty} \frac{\# \{q \leq X \mid q\nmid pN, \ p\mid \#\widetilde{E}({\mathbb F}_q)\}}{\pi(X)} \approx \frac{1}{p}. \end{align*} $$

Here, $\pi (X)$ is the prime counting function.

The above discussion can be summarized in the result below.

The conditions imposed in Proposition 3.2 will be required throughout this section. Therefore, we make the following definition.

Let E be an elliptic curve and let $p,q$ be two distinct primes. Given a triple $(E, p, q)$ , we aim to understand when Equation (3-1) holds. We begin by recalling the following well-known result.

In fact, it follows from class field theory (see [Reference Mantilla-Soler and MonsurròMSM16, Lemma 2.5]) that the primes that ramify in a ${\mathbb Z}/p{\mathbb Z}$ -extension are either p or precisely those of the form $q\equiv 1\ \pmod {\,p}$ . By local class field theory, the discriminant of $L/{\mathbb Q}$ (denoted $d(L/{\mathbb Q})$ ) is given by (see [Reference Mantilla-Soler and MonsurròMSM16, Lemma 2.4])

$$ \begin{align*} d(L/{\mathbb Q}) = \begin{cases} \displaystyle\prod_{i=1}^r q_i^{p-1} & \textrm{if }q_i \textrm{ is ramified,}\\ \displaystyle\prod_{i=1}^r q_i^{p-1}p^{2(p-1)} & \textrm{if }q_i \textrm{ and } p \textrm{ are ramified.} \end{cases} \end{align*} $$

If $q\ (\neq p)$ ramifies in a ${\mathbb Z}/p$ -extension, then the ramification is tame.

For primes of the form $q\equiv 1\ \pmod {\,p}$ (that is, primes that can ramify in ${\mathbb Z}/p{\mathbb Z}$ -extensions), we introduce the notion of friendly and enemy primes.

Primes $w|q$ in L will also be called an enemy or a friendly prime depending on the behavior of q. The following lemma will play a crucial role in the subsequent discussion.

Proof. Recall that $P_1$ consists of primes $w\nmid p$ such that E has split multiplicative reduction at w. Observe that

$$ \begin{align*} \sum_{w\in P_1}( e_{L/{\mathbb Q}}(w) -1 ) \neq 0 \end{align*} $$

if and only if there exists a ramified prime in $L/{\mathbb Q}$ of split multiplicative type. The assertion is immediate from the definition of an enemy prime.

Since L is disjoint from ${\mathbb Q}_{\operatorname {\mathrm {cyc}}}$ , we know that if p is ramified in $L/{\mathbb Q}$ , there must be at least one other prime that is also ramified. Now,

$$ \begin{align*} \sum_{w\in P_2}( e_{L/{\mathbb Q}}(w) -1 )\neq 0 \end{align*} $$

precisely when there exists a $q\ (\neq p)$ satisfying all of the following conditions:

1. (i) q is a prime of good reduction for E;

2. (ii) q is ramified in the extension $L/{\mathbb Q}$ ; and

3. (iii) w is a prime above q with $E(L_w)[p]\neq 0$ .

The conclusion of the lemma is now straightforward.

3.1.1

For a given pair $(E,p)$ , we denote the set of enemy primes by ${\mathcal E}_{(E,p)}$ and the set of friendly primes by ${\mathcal F}_{(E,p)}$ . Write $\mathcal {N}_{(E,p)}$ for the set of primes of the form $q\equiv 1\ \pmod {\,p}$ , where E has bad reduction not of split multiplicative type. The three sets are disjoint. We further subdivide the first two of these sets into disjoint sets, namely

$$ \begin{align*} {\mathcal F}_{(E,p)} &= {\mathcal F}_{(E,p)}^{\operatorname{\mathrm{ord}}} \cup {\mathcal F}_{(E,p)}^{\operatorname{\mathrm{ss}}} \textrm{ and }\\ {\mathcal E}_{(E,p)} &= {\mathcal E}_{(E,p)}^{\operatorname{\mathrm{split}}} \cup {\mathcal E}_{(E,p)}^{\operatorname{\mathrm{ord}}} \cup {\mathcal E}_{(E,p)}^{\operatorname{\mathrm{ss}}}. \end{align*} $$

Here, ${\mathcal F}_{(E,p)}^{\operatorname {\mathrm {ord}}}$ (respectively ${\mathcal F}_{(E,p)}^{\operatorname {\mathrm {ss}}}$ ) is the set of primes of the form $q\equiv 1 \ \pmod {\,p}$ such that q is a prime of good ordinary (respectively supersingular) reduction and $p\nmid \#\widetilde {E}({\mathbb F}_q)$ . The set ${\mathcal E}_{(E,p)}^{\operatorname {\mathrm {split}}}$ consists of all the primes $q \equiv 1\,\mod {p}$ of split multiplicative reduction. Finally, ${\mathcal E}_{(E,p)}^{\operatorname {\mathrm {ord}}}$ (respectively ${\mathcal E}_{(E,p)}^{\operatorname {\mathrm {ss}}}$ ) is the set of primes of the form $q\equiv 1 \ \pmod {\,p}$ such that q is a prime of good ordinary (respectively supersingular) reduction and $p| \#\widetilde {E}({\mathbb F}_q)$ .

Lemma 3.8. Let $E_{/{\mathbb Q}}$ be an elliptic curve and $p\geq 5$ be a relevant prime. Then, ${\mathcal E}_{(E,p)}^{\operatorname {\mathrm {ss}}} = \emptyset $ . In particular, ${\mathcal E}_{(E,p)} = {\mathcal E}_{(E,p)}^{\operatorname {\mathrm {split}}} \cup {\mathcal E}_{(E,p)}^{\operatorname {\mathrm {ord}}}$ .

Proof. When $q \geq 7$ is a prime of supersingular reduction, then it follows from the Hasse bound that $a_q =0$ . Therefore,

$$ \begin{align*} \# \widetilde{E}({\mathbb F}_q) = q+ 1 - a_ q = q+1. \end{align*} $$

Note that we require that $q \equiv 1\ \pmod {\,p}$ . Thus, $p \nmid \# \widetilde {E}({\mathbb F}_q)$ .

For any subset $\mathcal {S}'$ of the set of primes, let $\mathfrak {d}(\mathcal {S}')$ denote the Dirichlet density of $\mathcal {S}'$ . With notation as above, we have that

(3-2) $$ \begin{align} \mathfrak{d}({\mathcal E}_{(E,p)}) + \mathfrak{d}({\mathcal F}_{(E,p)}) + \mathfrak{d}(\mathcal{N}_{(E,p)})= \frac{1}{\varphi(p)} = \frac{1}{p-1}. \end{align} $$

The first equality follows from Dirichlet’s theorem on primes in arithmetic progressions, which asserts that the proportion of primes that are congruent to $1$ modulo p is $1/\varphi (p)$ . The set $\mathcal {N}_{(E,p)}$ is finite, and hence $\mathfrak {d}(\mathcal {N}_{(E,p)}) = 0$ . We henceforth disregard the primes q that are of nonsplit multiplicative and additive reduction type. For the same reason, we may also disregard (for the purpose of proportion) the primes of split multiplicative reduction. As $X\rightarrow \infty $ , the contribution to $\mathcal {E}_{(E,p)}$ is primarily from primes q such that:

1. (a) q is a prime of good ordinary reduction;

2. (b) $q\equiv 1\ \pmod {\,p}$ ; and

3. (c) $p\mid \# \widetilde {E}({\mathbb F}_q)$ .

For non-CM elliptic curves, the set of primes of good ordinary reduction has density $1$ . In the case of non-CM elliptic curves with surjective residual Galois representation at p, we know how to calculate the proportion of primes satisfying conditions (a)–(c).

Lemma 3.9. Let $E_{/{\mathbb Q}}$ be a non-CM elliptic curve and $p\geq 5$ be a fixed prime of good ordinary reduction such that the residual Galois representation at p is surjective. Then,

$$ \begin{align*} \mathfrak{d}({\mathcal E}_{(E,p)})= \frac{p}{(p-1)^2(p+1)}. \end{align*} $$

Proof. The result is well known and follows from the proof of [Reference Garcia-Fritz and PastenGFP20, Proposition 4.6]. For the convenience of the reader, we briefly sketch the details here. A more detailed argument is provided in Section 4.2. Since $p\geq 5$ , it follows from the proof of Lemma 3.8 that if conditions (b) and (c) are satisfied, then condition (a) is automatically satisfied for any prime q of good reduction. Since there are only finitely many primes q at which E has bad reduction, we may as well assume that q is a prime of good reduction.

Let $\operatorname {\mathrm {Frob}}_q$ denote the Frobenius at q and set $\mathcal {S}$ to be the set of matrices $A\in \operatorname {\mathrm {GL}}_2({\mathbb F}_p)$ such that $\operatorname {\mathrm {trace}}(A)=2$ and $\det (A)=1$ . Let $\bar {\rho }:\operatorname {\mathrm {Gal}}(\bar {{\mathbb Q}}/{\mathbb Q})\rightarrow \operatorname {\mathrm {GL}}_2({\mathbb F}_p)$ denote the residual representation at p, that is, the representation on the group of p-torsion points $E[p]$ . By assumption, $\bar {\rho }$ is surjective.

Since q is a prime of good reduction, $\bar {\rho }$ is unramified at q and the characteristic polynomial of $\bar {\rho }(\operatorname {\mathrm {Frob}}_q)$ is given by

$$ \begin{align*}\det\,(\operatorname{\mathrm{Id}}-T \bar{\rho}(\operatorname{\mathrm{Frob}}_q))=T^2-(q+1-\#\tilde{E}({\mathbb F}_q)) T+q.\end{align*} $$

Thus, q satisfies both conditions (b) and (c) above if and only if $\bar {\rho }(\operatorname {\mathrm {Frob}}_q)$ is contained in $\mathcal {S}$ . According to Lemma 4.8, the cardinality of $\mathcal {S}$ is $p^2$ . The cardinality of $\operatorname {\mathrm {Image}}(\bar {\rho })=\operatorname {\mathrm {GL}}_2({\mathbb F}_p)$ is $(p^2-1)(p^2-p)$ . The result follows from the Chebotarev density theorem, according to which the density of $\mathcal {E}_{(E,p)}$ is

$$ \begin{align*}\frac{\#\mathcal{S}}{\# \operatorname{\mathrm{GL}}_2({\mathbb F}_q)}=\frac{p}{(p-1)^2(p+1)}.\\[-42pt] \end{align*} $$

We performed calculations using SageMath [Sag20] to get an estimate of the proportion of enemy primes in the CM case. More precisely, fix $5 \leq p\leq 50$ . Fix an elliptic curve $E_{/{\mathbb Q}}$ of rank 0 and conductor less than 100. Running through all primes q less than 200 million, we computed the proportion of enemy primes. The results are recorded at the end of the article, in Table 2. The data suggest that for elliptic curves with CM, the proportion of enemy primes is half of that in the non-CM case. We know from Deuring’s criterion that for a CM elliptic curve, the density of the set of good ordinary primes is $1/2$ . It therefore seems reasonable to assume that the enemy primes are equally likely to be primes of good ordinary or good supersingular reduction. More precisely, we make the following assumption.

Table 1 Values for Equation (3-4).

Table 2 Proportion of enemy primes.

Hypothesis 3.10. Let $E_{/{\mathbb Q}}$ be an elliptic curve with CM and p be a fixed odd prime of good ordinary reduction. Then, $\mathfrak {d}({\mathcal E}_{(E,p)})= {p}/{2(p-1)^2(p+1)}$ .

We can now prove the main result of this section.

Theorem 3.11. Let $(E,p)$ be a given pair of a rank 0 elliptic curve over ${\mathbb Q}$ and a relevant prime $p\geq 5$ . Then, the following assertions hold.

1. (1) Suppose that the residual representation at p is surjective. Then, there are infinitely many cyclic number fields of degree-p in which the $\lambda $ -invariant does not jump. In particular, there are infinitely many ${\mathbb Z}/p{\mathbb Z}$ -extensions $L/{\mathbb Q}$ in which the rank does not jump in $L_n$ for all $n\geq 0$ .

2. (2) Let $(E,p)$ be a pair of a rank 0 elliptic curve (defined over ${\mathbb Q}$ ) with complex multiplication and a relevant prime $p \geq 5$ . Suppose further that Hypothesis 3.10 holds. Then the same conclusions hold as in the non-CM case.

Proof. Since p is assumed to be a relevant prime, we know that $\mu _p(E/{\mathbb Q})=\lambda _p(E/{\mathbb Q})=0$ . It follows from Theorem 2.3 that $\mu _p(E/L)=0$ for every degree p extension $L/{\mathbb Q}$ . The result will follow from Proposition 3.2 if we can show that there are infinitely many cyclic degree p extensions such that Equation (3-1) holds.

Observe that

$$ \begin{align*} \mathfrak{d}({\mathcal F}_{(E,p)}) = \dfrac{1}{p-1} - \mathfrak{d}({\mathcal E}_{(E,p)}) = \begin{cases} \dfrac{p^2-p-1}{(p-1)^2(p+1)} & \text{if }E\text{ does not have CM,}\\[8pt] \dfrac{2p^2-p-2}{2(p-1)^2(p+1)} & \text{if }E\text{ does not have CM.} \end{cases} \end{align*} $$

It follows from Lemma 3.7 that Equation (3-1) holds when every ramified prime is a friendly prime. From the above discussion, we see that there are infinitely many friendly primes. We have also shown in Proposition 3.4 that corresponding to each friendly prime (say q), there is one ${\mathbb Z}/p{\mathbb Z}$ -extension where only q ramifies. This completes the proof.

Note that condition (1) in Theorem 3.11, requiring that the residual representation is surjective, implies that the elliptic curve does not have complex multiplication. The Galois representations associated to CM elliptic curves are studied, for instance, in [Reference Lozano-RobledoLR22]. The following application of our theorem was pointed out by J. Morrow. We begin by stating a result of González-Jiménez and Najman.

Theorem 3.12. Let $E_{/{\mathbb Q}}$ be an elliptic curve and $p>7$ be a prime number. Let $L/{\mathbb Q}$ be a Galois extension with Galois group $G\simeq {\mathbb Z}/p{\mathbb Z}$ . Then, $E(L)_{\operatorname {\mathrm {tors}}}=E({\mathbb Q})_{\operatorname {\mathrm {tors}}}$ .

Proof. See [Reference González-Jiménez and NajmanGJN20, Theorem 7.2].

Combining Theorems 3.11 and 3.12, the following corollary is immediate.

Corollary 3.13. Let $(E,p)$ be a given pair of a rank 0 elliptic curve over ${\mathbb Q}$ and a relevant prime $p> 7$ . If $E_{/{\mathbb Q}}$ is an elliptic curve without CM, suppose that the residual representation at p is surjective. If $E_{/{\mathbb Q}}$ is an elliptic curve with CM, suppose that Hypothesis 3.10 holds. Then, there are infinitely many ${\mathbb Z}/p{\mathbb Z}$ -extensions of ${\mathbb Q}$ where the Mordell–Weil group does not grow.

Remark 3.14. To show that there are ‘infinitely many’ ${\mathbb Z}/p{\mathbb Z}$ -extensions with no $\lambda $ -jump, we only counted those where exactly one friendly prime is ramified. In particular, we counted only those ${\mathbb Z}/p{\mathbb Z}$ -extensions that are contained in the cyclotomic field ${\mathbb Q}(\mu _q)$ where q is a prime of the form $1 \ \pmod {\,p}$ . Our count ignored the contribution from ${\mathbb Z}/p{\mathbb Z}$ -extensions where two or more primes are ramified, all of which are either friendly primes or the prime p. It was pointed out to us by R. Lemke Oliver that Theorem 3.11 can be made explicit using standard analytic number theory techniques (see for example [Reference SerreSer74, Théorème 2.4]); however, our approach is still likely to fall short of proving a positive proportion.

A recent and significant result in this direction is by Mazur and Rubin (see [Reference Mazur, Rubin and LarsenMRL18, Theorem 1.2]). They show that given an elliptic curve E, there is a positive density set of primes (call it $\mathcal {S}$ ) such that for each $p\in \mathcal {S}$ , there are infinitely many ${\mathbb Z}/p{\mathbb Z}$ -extensions over ${\mathbb Q}$ with $E(L) = E({\mathbb Q})$ . In the case of rank 0 elliptic curves (defined over ${\mathbb Q}$ ), our result is stronger, in the sense that for $p>7$ , Corollary 3.13 holds unconditionally for density 1 primes if E is an elliptic curve without CM. However, for CM elliptic curves, the result is conditional and applies to a set of primes of density $1/2$ .

3.1.2 Example: CM case

Now, we work out a particular example in the CM case. As before, let $p\geq 5$ be a fixed prime and $q\equiv 1 \ \pmod {\,p}$ be a different prime. Let $k\not \equiv 0 \ \pmod {\,q}$ and consider the family of curves

$$ \begin{align*} E_k: y^2 = x^3 - kx. \end{align*} $$

Then, either of the following two statements is true (see [Reference WashingtonWas08, Section 4.4]).

1. (i) If $q\equiv 3 \ \pmod {\,4}$ , then $\# \widetilde {E_k}(\mathbb {F}_q) = q +1$ .

2. (ii) If $q\equiv 1 \ \pmod {\,4}$ , write $q= s^2 + t^2$ with $s\in {\mathbb Z}$ , $t\in 2{\mathbb Z}$ and $s+t \equiv 1 \ \pmod {\,4}$ . Then,

   $$ \begin{align*} \# \widetilde{E_k}(\mathbb{F}_q) = \begin{cases} q + 1 - 2s & \textrm{if } k \textrm{ is a fourth power mod }q,\\ q + 1 + 2s & \textrm{if } k \textrm{ is a square mod }q \textrm{ but not a fourth power},\\ q + 1 \pm 2t & \textrm{if } k \textrm{ is not a square mod }q. \end{cases} \end{align*} $$

For this family of elliptic curves, the primes $q\equiv 3 \ \pmod {\,4}$ are supersingular. Recall from Lemma 3.8 that if q is a supersingular prime and $q\equiv 1 \ \pmod {\,p}$ , then the primes $w|q$ are friendly (except for possibly finitely many). For the sake of concreteness, suppose that $k=1$ (the argument goes through more generally). By the Chinese remainder theorem, we know that if

$$ \begin{align*} q&\equiv 3 \ \pmod{\,4} \textrm{ and }\\ q&\equiv 1 \ \pmod{\,p}, \end{align*} $$

then $q \equiv 2p+1 \ \pmod {\,4p}$ . By Dirichlet’s theorem for primes in arithmetic progressions, we know that there are infinitely many primes satisfying this congruence condition. Moreover, the proportion of such primes is

$$ \begin{align*} \frac{1}{\varphi(4p)} = \frac{1}{2(p-1)}. \end{align*} $$

By Proposition 3.4, we know that corresponding to each such q, there is exactly one cyclic degree-p extension $L/{\mathbb Q}$ in which q is the unique ramified prime. Therefore, we have produced infinitely many ${\mathbb Z}/p{\mathbb Z}$ -extensions where the ramified primes are friendly. By Proposition 3.2, if $\operatorname {\mathrm {rank}}_{{\mathbb Z}}(E_k/{\mathbb Q}) =0$ and $\mu _p(E/{\mathbb Q})=0$ (for example, when $k=1$ ), then there are infinitely many ${\mathbb Z}/p{\mathbb Z}$ -extensions over ${\mathbb Q}$ where the rank does not jump.

We record a specific case of the above discussion.

Theorem 3.15. Let $p\geq 5$ be a fixed prime, and consider the elliptic curve

$$ \begin{align*} E: y^2 = x^3 - x. \end{align*} $$

Then, there are infinitely many ${\mathbb Z}/p{\mathbb Z}$ -extensions $L/{\mathbb Q}$ such that $\operatorname {\mathrm {Sel}}_{p^{\infty }}(E/L_{\operatorname {\mathrm {cyc}}})=0$ . In particular, there are infinitely many ${\mathbb Z}/p{\mathbb Z}$ -extensions such that $\operatorname {\mathrm {rank}}_{{\mathbb Z}}(E(L))=0$ .

3.2

In the last section, we fixed a rank 0 elliptic curve over ${\mathbb Q}$ and analyzed in how many ${\mathbb Z}/p{\mathbb Z}$ -extensions of ${\mathbb Q}$ did the rank jump. Now, we ask the following question.

Let E be a rank 0 elliptic curve of conductor N and p be a relevant prime. Throughout this section, we assume that the Shafarevich–Tate group of rank 0 elliptic curves defined over ${\mathbb Q}$ is finite. Proposition 2.5 asserts that density one good ordinary primes are relevant. Further assume that N is divisible by at least one prime of the form $1\ \pmod {\,p}$ . The following lemma is a special case of Proposition 3.2.

It follows from the proof of the above result that to obtain a lower bound of the proportion of rank 0 elliptic curves for which there is at least one degree-p cyclic extension over ${\mathbb Q}$ such that $\operatorname {\mathrm {Sel}}_{p^{\infty }}(E/L)=0$ , it is enough to count elliptic curves with the following properties:

1. (i) E has good ordinary reduction at $p\geq 5$ ;

2. (ii) E has any reduction type at primes $q\not \equiv 1 \ \pmod {\,p}$ ;

3. (iii) E has at least one prime of bad reduction that is not of split multiplicative type at a prime $q\equiv 1\ \pmod {\,p}$ .

Before proceeding with such a count, we need to define the notion of height. For an elliptic curve defined over ${\mathbb Q}$ , consider the long Weierstrass equation:

$$ \begin{align*} y^2+a_1 xy+a_3 y=x^3+a_2 x^2+a_4 x+a_6. \end{align*} $$

Define the height of this Weierstrass equation with integer coefficients $\boldsymbol {a}=(a_1, a_2, a_3, a_4, a_6)$ to be

$$ \begin{align*} \operatorname{\mathrm{height}}(\boldsymbol{a}) = \max_i \{|{a_i}|^{1/i}\}. \end{align*} $$

Let S be a set of Weierstrass equations with integer coefficients $\boldsymbol {a}$ that are ordered by height. The proportion of Weierstrass equations that lie in the set S is defined as

(3-3) $$ \begin{align} \lim_{X\rightarrow \infty}\frac{\#\{ \boldsymbol{a}\in S : \operatorname{\mathrm{height}}(\boldsymbol{a}) <X\}}{\#\{ \boldsymbol{a}\in {\mathbb Z}^5 : \operatorname{\mathrm{height}}(\boldsymbol{a}) <X\}}. \end{align} $$

For our purposes, we restrict to Weierstrass equations that are globally minimal.

Henceforth, we assume that the rank of the elliptic curve and the reduction type at $\ell $ are independent of each other. In other words, we assume that even if we only order the rank 0 elliptic curves by height, the proportion of elliptic curves with split multiplicative reduction or good reduction is the same as that in Lemma 3.18.

We know from [Reference Cremona and SadekCS21, Section 3] that the local conditions such as the reduction type of elliptic curves at distinct primes are independent.

We now record the assumption we have made.

What we mean is that density results from Lemma 3.18 hold even when we restrict to rank 0 elliptic curves defined over ${\mathbb Q}$ . We can now prove the main result in this section.

Theorem 3.20. Let $p\geq 5$ be a fixed odd prime. Suppose that Hypothesis 3.19 holds. Varying over all rank 0 elliptic curves defined over ${\mathbb Q}$ and ordered by height, the proportion with

1. (i) good reduction at p;

2. (ii) any reduction type at $q\not \equiv 1 \ \pmod {\,p}$ ; and

3. (iii) at least one prime of bad reduction where the reduction type is not split multiplicative at a prime $q\equiv 1\ \pmod {\,p}$ is given by

(3-4) $$ \begin{align} \bigg(1 - \frac{1}{p} \bigg) \bigg( 1 - \prod_{q \equiv 1 \ \pmod{\,p}} \bigg( \frac{q-1}{2q^2} + 1 - \frac{1}{q}\bigg) \bigg). \end{align} $$

Further, suppose that the Shafarevich–Tate group is finite for all rank 0 elliptic curves defined over ${\mathbb Q}$ . There is a positive proportion of rank 0 elliptic curves for which there is at least one degree-p cyclic extension over ${\mathbb Q}$ such that $\operatorname {\mathrm {Sel}}_{p^{\infty }}(E/L)=0$ upon base-change.

Proof. Let $q \equiv 1\ \pmod {\,p}$ . We require that among all such primes, there is ‘at least one prime of bad reduction where the reduction type is not split multiplicative’. Equivalently, among all primes of the form $1\ \pmod {\,p}$ , there is ‘at least one prime of additive or nonsplit multiplicative reduction’. In other words, at $q\equiv 1 \ \pmod {\,p}$ , we want the negation of ‘all primes have either good or split multiplicative reduction’. This gives Equation (3-4) from Lemma 3.18.

From our earlier discussion, to prove the second assertion, it remains to show that Equation (3-4) is strictly positive. For any fixed prime p, note that

$$ \begin{align*} \prod_{q \equiv 1 (\textrm{mod }{p})} \bigg( \frac{q-1}{2q^2} + 1 - \frac{1}{q}\bigg) = \prod_{q \equiv 1 (\textrm{mod }{p})} \bigg( 1 - \bigg(\frac{q+1}{2q^2}\bigg)\bigg)< 1. \end{align*} $$

The inequality follows from the fact that each term in the product is $<1$ . Therefore,

$$ \begin{align*} \bigg( 1 - \prod_{q \equiv 1 (\textrm{mod }{p})} \bigg( 1 - \bigg(\frac{q+1}{2q^2}\bigg)\bigg)\bigg)>0. \end{align*} $$

The claim follows.

In Table 1, we compute Equation (3-4) for $3\leq p < 50$ and $q \leq 179\,424\,673$ (that is, the first 10 million primes).

A reasonable question to ask at this point is the following.

In this direction, we can provide partial answers. Given a number field $L/{\mathbb Q}$ , we can find a lower bound for the proportion of elliptic curves (defined over ${\mathbb Q}$ ) for which Equation (3-1) holds.

Proposition 3.22. Let p be a fixed odd prime. Let $L/{\mathbb Q}$ be a fixed ${\mathbb Z}/p{\mathbb Z}$ -extension that is tamely ramified at primes $q_1, \ldots , q_r$ . The proportion of elliptic curves defined over ${\mathbb Q}$ (ordered by height) such that Equation (3-1) holds has a lower bound of

$$ \begin{align*} \prod_{q_i=1}^r \bigg( \frac{q_i+1}{2q_i^2}\bigg). \end{align*} $$

Proof. Note that for Equation (3-1) to hold, the reduction type at p does not matter. To find a lower bound, it suffices that all the ramified primes (that is, the $q_i$ terms) are primes of bad reduction of nonsplit multiplicative or additive reduction type. Indeed, this would ensure $P_1 =\emptyset $ and the primes of good reduction are not ramified. Equivalently, each $q_i$ is such that it does not have good ordinary or split multiplicative reduction. By Lemma 3.18, the lower bound is

$$ \begin{align*} \prod_{q_i=1}^r \bigg( 1 - \bigg(\frac{q_i -1}{2q_i^2} + 1 - \frac{1}{q_i}\bigg)\bigg) = \prod_{q_i=1}^r \bigg( \frac{q_i+1}{2q_i^2}\bigg).\\[-46pt] \end{align*} $$

4.1

Here, we first prove a criterion for nontriviality of the p-primary Selmer group.

To prove the above result, we briefly recall some key properties of the Euler characteristic associated to an elliptic curve. The reader is referred to [Reference Coates and SujathaCS00, Reference Ray and SujathaRS19, Reference Ray and SujathaRS20] for a more comprehensive discussion of the topic.

Let $E_{/{\mathbb Q}}$ be an elliptic curve and $L/{\mathbb Q}$ a number field extension. Assume that the following conditions are satisfied:

1. (i) $\operatorname {\mathrm {rank}}_{{\mathbb Z}} E(L)=0$ ;

2. (ii) E has good ordinary reduction at p;

3. (iii) is finite.

Then, the cohomology groups $H^i(L_{\operatorname {\mathrm {cyc}}}/L, \operatorname {\mathrm {Sel}}_{p^{\infty }}(E/L_{\operatorname {\mathrm {cyc}}}))$ are finite and the Euler characteristic $\chi (L_{\operatorname {\mathrm {cyc}}}/L, E[p^{\infty }])$ is defined as follows:

$$ \begin{align*} \chi(L_{\operatorname{\mathrm{cyc}}}/L, E[p^{\infty}]):=\frac{\# H^0(L_{\operatorname{\mathrm{cyc}}}/L, \operatorname{\mathrm{Sel}}_{p^{\infty}}(E/L_{\operatorname{\mathrm{cyc}}}))}{\# H^1(L_{\operatorname{\mathrm{cyc}}}/L, \operatorname{\mathrm{Sel}}_{p^{\infty}}(E/L_{\operatorname{\mathrm{cyc}}}))}. \end{align*} $$

The Euler characteristic is an important invariant associated to the Selmer group, and captures its key Iwasawa theoretic properties. There is an explicit formula for this invariant, which we now describe. At each prime $v\nmid p$ of L, let $c_v(E/L)$ denote the Tamagawa number at v, and let $c_v^{(p)}(E/L)$ be the p-part, given by $c_v^{(p)}(E/L):=|c_v(E/L)|_p^{-1}$ . Here, $|{\cdot }|_p$ is the absolute value, normalized by $|{p}|_p=p^{-1}$ . At each prime v of L, let $k_v$ be the residue field at v and $\widetilde {E}(k_v)$ be the group of $k_v$ -valued points of the reduction of E at v.

The next result relates the Euler characteristic and Iwasawa invariants.

Proof. Recall the well-known short exact sequence (see Equation (2-2)),

Since $E({\mathbb Q})$ is finite and $E({\mathbb Q})[p^{\infty }]=0$ , it follows from the above that

It follows from Proposition 4.4 that $\chi ({\mathbb Q}_{\operatorname {\mathrm {cyc}}}/{\mathbb Q}, E[p^{\infty }])=1$ . However, by Equation (4-1),

As a result, it is indeed the case that

.

We now give a proof of Proposition 4.1.

Proof of Proposition 4.1. According to Lemma 4.5, we have that

Assume by way of contradiction that

Since $\operatorname {\mathrm {rank}}_{{\mathbb Z}} E(L)=0$ , it follows from Theorem 4.3 that the Euler characteristic $\chi (L_{\operatorname {\mathrm {cyc}}}/L, E[p^{\infty }])$ is defined and given by the formula

(4-2)

However, the Euler characteristic $\chi ({\mathbb Q}_{\operatorname {\mathrm {cyc}}}/{\mathbb Q}, E[p^{\infty }])$ is given by the formula

Note that $E({\mathbb Q})[p^{\infty }]$ does not contribute to the above formula since it is assumed to be trivial. Since it is assumed that $\mu _p(E/{\mathbb Q})=0$ and $\lambda _p(E/{\mathbb Q})=0$ , it follows from Proposition 4.4 that $\chi ({\mathbb Q}_{\operatorname {\mathrm {cyc}}}/{\mathbb Q}, E[p^{\infty }])=1$ .

We use this and the assumptions on E to show that $\chi (L_{\operatorname {\mathrm {cyc}}}/L, E[p^{\infty }])=1$ as well. Since $\chi ({\mathbb Q}_{\operatorname {\mathrm {cyc}}}/{\mathbb Q}, E[p^{\infty }])=1$ , it follows that

To show that $\chi (L_{\operatorname {\mathrm {cyc}}}/L, E[p^{\infty }])=1$ , we show that

By assumption,

, and hence,

. It follows from Lemma 4.6 that $\prod _{v\nmid p}c_v^{(p)}(E/L)=1$ . If p splits or ramifies in L, then $k_v={\mathbb F}_p$ for all primes $v|p$ . Since $\#\widetilde {E}({\mathbb F}_p)[p^{\infty }]=1$ , it follows that $\#\widetilde {E}(k_v)[p^{\infty }]=1$ as well. However, suppose p is inert in L and $v|p$ is the only prime above p in L. Then, since $k_v/{\mathbb F}_p$ is a p-extension, it follows from [Reference Neukirch, Schmidt and WingbergNSW13, Proposition 1.6.12] that

$$ \begin{align*} \#\widetilde{E}({\mathbb F}_p)[p^{\infty}]=1\Rightarrow \#\widetilde{E}(k_v)[p^{\infty}]=1. \end{align*} $$

Hence, the numerator of Equation (4-2) is $1$ . Theorem 4.3 asserts the Euler characteristic is an integer, and so, we deduce that $\chi (L_{\operatorname {\mathrm {cyc}}}/L, E[p^{\infty }])=1$ . We deduce from Proposition 4.4 that

$$ \begin{align*} \mu_p(E/L)=0 \quad\text{and}\quad\lambda_p(E/L)=0. \end{align*} $$

In particular, we find that

$$ \begin{align*} \lambda_p(E/L)=\lambda_p(E/{\mathbb Q})=0. \end{align*} $$

However, recall that according to Kida’s formula,

$$ \begin{align*} \lambda_p(E/L)=p\lambda_p(E/{\mathbb Q})+\sum_{w \in P_1} (e_{L/{\mathbb Q}}(w/\ell)-1)+\sum_{w \in P_2} 2(e_{L/{\mathbb Q}}(w/\ell)-1).\end{align*} $$

However, there is a prime $\ell \in \Sigma _L$ for which E has good reduction at $\ell $ and $p| \#\widetilde {E}({\mathbb F}_{\ell })$ , so we deduce that

$$ \begin{align*} \sum_{w \in P_2} 2(e_{L/{\mathbb Q}}(w/\ell)-1)>0. \end{align*} $$

Thus, the above formula shows that

$$ \begin{align*} \lambda_p(E/L)>\lambda_p(E/{\mathbb Q}). \end{align*} $$

In particular, $\lambda _p(E/L)>0$ , this is a contradiction. Therefore, we have shown that either $\operatorname {\mathrm {rank}}_{{\mathbb Z}} E(L)>0$ or

.

We illustrate Proposition 4.1 through an example in the case when $p=5$ .

4.2

We now come to an application to arithmetic statistics. Recall that we fix an elliptic curve $E_{/{\mathbb Q}}$ and a prime p for which the aforementioned conditions are satisfied. Consider the family of ${\mathbb Z}/p{\mathbb Z}$ -extensions L of ${\mathbb Q}$ not contained in ${\mathbb Q}_{\operatorname {\mathrm {cyc}}}$ and ramified at precisely one prime. For each prime $q\equiv 1\ \pmod {\,p}$ , there is exactly one such extension $L_q/{\mathbb Q}$ that is ramified at q (see Proposition 3.4). Note that $L_q$ is contained in ${\mathbb Q}(\mu _q)$ . For $X>0$ , let $\pi (X)$ be the prime counting function (that is, denote the number of primes $q\leq X$ ) and $\pi '(X)$ be the number of primes $q\leq X$ for which:

1. (i) $q\equiv 1\ \pmod {\,p}$ ; and

2. (ii) $\operatorname {\mathrm {Sel}}_{p^{\infty }}(E/L_q)\neq 0$ .

Note that since $L_q/{\mathbb Q}$ is a ${\mathbb Z}/p{\mathbb Z}$ -extension, we have the following implication:

$$ \begin{align*} E({\mathbb Q})[p^{\infty}]=0\Rightarrow E(L_q)[p^{\infty}]=0; \end{align*} $$

see [Reference Neukirch, Schmidt and WingbergNSW13, Proposition 1.6.12]. Therefore, the nonvanishing of $\operatorname {\mathrm {Sel}}_{p^{\infty }}(E/L_q)$ is equivalent to at least one of the following conditions being satisfied:

1. (1) $\operatorname {\mathrm {rank}}_{{\mathbb Z}}E(L_q)>0$ ;

2. (2) .

Thus, if the Selmer group becomes nonzero after base-change, then either there is a rank jump or the Shafarevich–Tate group witnesses growth. The main result of this section is Theorem 4.9, where it is shown that the set of primes $q\equiv 1\ \pmod {\,p}$ for which the above conditions are satisfied is cut out by explicit Chebotarev conditions. In other words, there is an explicit subset $\mathcal {S}\subset \operatorname {\mathrm {Gal}}({\mathbb Q}(E[p])/{\mathbb Q})$ such that for any prime $q\neq p$ coprime to the conductor of E,

$$ \begin{align*} \operatorname{\mathrm{Frob}}_q\in \mathcal{S}\Rightarrow \operatorname{\mathrm{Sel}}_{p^{\infty}}(E/L_q)\neq 0. \end{align*} $$

Therefore, if the Frobenius of a prime $q\nmid Np$ lies in the Chebotarev set $\mathcal {S}$ , then the Selmer group becomes nonzero when base changed to $L_q$ . We calculate the size of $\mathcal {S}$ and apply the Chebotarev density theorem to obtain a lower bound for $\limsup _{X\rightarrow \infty } \pi '(X)/\pi (X)$ . Stated differently, we are able to show there is growth in the Selmer group in $L_q$ for a positive density set of primes q. Let $\bar {\rho }:\operatorname {\mathrm {Gal}}(\bar {{\mathbb Q}}/{\mathbb Q})\rightarrow \operatorname {\mathrm {GL}}_2({\mathbb F}_p)$ be the Galois representation on $E[p]$ . We make the simplifying assumption that $\bar {\rho }$ is surjective. By the well-known open image theorem of Serre, this assumption is satisfied for all but finitely many primes p, as long as E does not have complex multiplication. Let ${\mathbb Q}(E[p])$ be the Galois extension of ${\mathbb Q}$ that is fixed by $\ker \bar {\rho }$ . We identify $\operatorname {\mathrm {Gal}}({\mathbb Q}(E[p])/{\mathbb Q})$ with its image under $\bar {\rho }$ , which, according to our assumption, is all of $\operatorname {\mathrm {GL}}_2({\mathbb F}_p)$ . Let $N=N_E$ be the conductor of E. If q is a prime that is coprime to $N p$ , then q is unramified in ${\mathbb Q}(E[p])$ . Set $a_q(E):=q+1-\# \widetilde {E}({\mathbb F}_q)$ ; then the characteristic polynomial of $\bar {\rho }(\operatorname {\mathrm {Frob}}_q)$ is $x^2-a_q x+q$ . Let $\mathcal {S}$ consist of elements $\sigma \in \operatorname {\mathrm {Gal}}({\mathbb Q}(E[p])/{\mathbb Q})$ such that

$$ \begin{align*} \operatorname{\mathrm{trace}}\bar{\rho}(\sigma)=2\quad\text{and}\quad\det\bar{\rho}(\sigma)=1.\end{align*} $$

We arrive at the following useful criterion for p to divide $\#\widetilde {E}({\mathbb F}_q)$ .

Proof. Since $\bar {\rho }$ is assumed to be surjective, we identify $\mathcal {S}$ with the set of all matrices in $\operatorname {\mathrm {GL}}_2({\mathbb F}_p)$ with trace $2$ and determinant $1$ . These matrices are all of the form $g=(\begin {smallmatrix} a & b \\ c & 2-a \end {smallmatrix})$ , where $a(2-a)-bc=1$ . We count the number of such matrices. Rewrite the equation as $bc=-1+a(2-a)=-(a-1)^2$ . For each choice of a such that $a\neq 1$ , the number of solutions is $(p-1)$ . For $a=1$ , either $b=0$ or $c=0$ , or both. Thus, the number of solutions for $a=1$ is $2p-1$ . Putting it all together,

$$ \begin{align*} \# \mathcal{S}=(p-1)^2+(2p-1)=p^2.\\[-34pt] \end{align*} $$

We now prove Theorem B, which is the main result of this section.

Theorem 4.9. Let $p\geq 5$ be a fixed prime and $E_{/{\mathbb Q}}$ an elliptic curve for which the following conditions are satisfied:

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

2. (ii) $\operatorname {\mathrm {rank}}_{{\mathbb Z}} E({\mathbb Q})=0$ and $E({\mathbb Q})[p^{\infty }]=0$ ;

3. (iii) $\mu _p(E/{\mathbb Q})=0$ and $\lambda _p(E/{\mathbb Q})=0$ ;

4. (iv) the image of the residual representation $\bar {\rho }$ is surjective.

For $X>0$ , let $\pi '(X)$ be the number of primes $q\equiv 1\ \pmod {\,p}$ for which the following equivalent conditions are satisfied:

1. (i) $\operatorname {\mathrm {Sel}}_{p^{\infty }}(E/L_q)\neq 0$ ;

2. (ii) either, $\operatorname {\mathrm {rank}}_{{\mathbb Z}} E(L_q)>0$ or (or both conditions are satisfied).

Then,

$$ \begin{align*} \limsup_{X\rightarrow \infty} \frac{\pi'(X)}{\pi(X)}\geq \frac{p}{(p-1)^2(p+1)}. \end{align*} $$

Proof. Let q be a prime such that $q\nmid N p$ . It follows from Lemma 4.7 that if $\operatorname {\mathrm {Frob}}_q\in \mathcal {S}$ , then $q\equiv 1\ \pmod {\,p}$ and $p|\#E({\mathbb F}_q)$ . Note that $\Sigma _{L_q}=\{q\}$ and E has good reduction at q. Since $p|\#E({\mathbb F}_q)$ , it follows from Proposition 4.1 that $\operatorname {\mathrm {Sel}}_{p^{\infty }}(E/L_q)\neq 0$ . Therefore, by the Chebotarev density theorem and Lemma 4.8,

$$ \begin{align*} \limsup_{X\rightarrow\infty} \frac{\pi'(X)}{\pi(X)}\geq \frac{\# \mathcal{S}}{\#(\operatorname{\mathrm{GL}}_2({\mathbb F}_p))}=\frac{p^2}{(p^2-1)(p^2-p)}=\frac{p}{(p-1)^2(p+1)}.\\[-42pt] \end{align*} $$

5.1 Large 2-rank of the Shafarevich–Tate group in quadratic extensions

Given an elliptic curve $E_{/{\mathbb Q}}$ , it is known that the 2-part of the Shafarevich–Tate group becomes arbitrarily large over some quadratic extension. More precisely, we get the following theorem.

A reasonable question to ask is whether the above result can be made effective.

5.1.1 Reviewing the proof of Matsuno’s construction

In this section, we briefly review the proof of Matsuno’s theorem. For details, we refer the reader to the original article [Reference MatsunoMat09]. Fix an elliptic curve $E_{/{\mathbb Q}}$ of conductor $N=N_E$ . Let $K/{\mathbb Q}$ be a quadratic extension. Let S be a finite set of primes in ${\mathbb Q}$ containing precisely the prime number 2, the primes of bad reduction of E, and the archimedean primes.

Let $\ell _1, \ldots , \ell _k$ be odd rational primes that are coprime to N and split completely in ${\mathbb Q}(E[2])/{\mathbb Q}$ . Recall that ${\mathbb Q}(E[2])/{\mathbb Q}$ is a Galois extension with Galois group isomorphic to either ${\mathbb Z}/2{\mathbb Z}$ or ${\mathbb Z}/3{\mathbb Z}$ or $S_3$ . By the Chebotarev density theorem, there is a positive proportion of primes that split completely in ${\mathbb Q}(E[2])/{\mathbb Q}$ . Having picked the primes $\ell _1, \ldots , \ell _k$ , it is clear that there exists a quadratic extension $K/{\mathbb Q}$ such that the chosen primes ramify. However, more is true. Results of Waldspurger (see [Reference Bump, Friedberg and HoffsteinBFH90, Theorem in Section 0]) and Kolyvagin (see [Reference KolyvaginKol89]) guarantee the existence of $K/{\mathbb Q}$ such that the chosen primes ramify and the Mordell–Weil rank of $E^{\prime }({\mathbb Q})$ is 0 where $E^{\prime }$ is the quadratic twist of E corresponding to K. It follows that

$$ \begin{align*} \operatorname{\mathrm{rank}}_{{\mathbb Z}}(E(K)) = \operatorname{\mathrm{rank}}_{{\mathbb Z}}(E({\mathbb Q})) + \operatorname{\mathrm{rank}}_{{\mathbb Z}} (E^{\prime}({\mathbb Q})) = \operatorname{\mathrm{rank}}_{{\mathbb Z}} (E({\mathbb Q})). \end{align*} $$

Observe that the primes $\ell _1, \ldots , \ell _k$ are primes of good ordinary reduction that lie in $T_{E/K}$ . Therefore, it follows from Lemma 5.3 that

$$ \begin{align*} \operatorname{\mathrm{rank}}_{2} \operatorname{\mathrm{Sel}}_2(E/K) \geq k - 4. \end{align*} $$

Using the Kummer sequence, we see that

Even though the results of Waldspurger and Kolyvagin guarantee that there exists a quadratic extension $K/{\mathbb Q}$ with the desired properties, it is not easy to determine all the primes that ramify in K. Using the proof of Matsuno as our inspiration, combined with more recent results of K. Ono, we prove an effective version of the theorem in Section 5.1.3. Assuming Goldfeld’s conjecture, we prove an effective version using simpler arguments in Section 5.1.2.

5.1.2 Effective version conditional on Goldfeld’s conjecture

Let us begin by reminding the reader of Goldfeld’s conjecture. This conjecture predicts that given an elliptic curve, $50\%$ of the quadratic twists have rank 0 and $50\%$ of the quadratic twists have rank 1. Therefore, if Goldfeld’s conjecture is true, then for $100\%$ of the time, the Mordell–Weil rank of $E^{\prime }({\mathbb Q})$ is either 0 or 1, where $E^{\prime }$ is the quadratic twist of E (corresponding to a quadratic field K). Since

$$ \begin{align*} \operatorname{\mathrm{rank}}_{{\mathbb Z}}(E(K)) = \operatorname{\mathrm{rank}}_{{\mathbb Z}}(E({\mathbb Q})) + \operatorname{\mathrm{rank}}_{{\mathbb Z}} (E^{\prime}({\mathbb Q})), \end{align*} $$

for $100\%$ of the time, $\operatorname {\mathrm {rank}}_{{\mathbb Z}}(E(K))=\operatorname {\mathrm {rank}}_{{\mathbb Z}}(E({\mathbb Q}))$ or $\operatorname {\mathrm {rank}}_{{\mathbb Z}}(E(K))=\operatorname {\mathrm {rank}}_{{\mathbb Z}}(E({\mathbb Q}))+1$ . Suppose that the primes $\ell _1, \ldots , \ell _k$ are chosen as in Matsuno’s theorem. These are primes of good ordinary reduction that lie in $T_{E/K}$ . Therefore, it follows from Lemma 5.3 that

$$ \begin{align*} \operatorname{\mathrm{rank}}_{2} \operatorname{\mathrm{Sel}}_2(E/K) \geq k - 4. \end{align*} $$

Using the Kummer sequence, we see that for $100\%$ of the quadratic fields K,

(5-1)

Given $E_{/{\mathbb Q}}$ of conductor N and $n\in {\mathbb Z}_{\geq 0}$ , we want to find an imaginary quadratic field $K/{\mathbb Q}$ with minimal conductor $\mathfrak {f}_{K}$ such that we can guarantee

. Let $\mathcal {P} = \{\ell _1, \ldots , \ell _k\}$ be a set of (distinct) rational primes not dividing N, with the additional property that they split completely in ${\mathbb Q}(E[2])/{\mathbb Q}$ and that precisely the primes in $\mathcal {P}$ ramify in K. From Equation (5-1), we need that

Equivalently,

$$ \begin{align*} k \geq n + \operatorname{\mathrm{rank}}_{{\mathbb Z}} (E({\mathbb Q})) + 7. \end{align*} $$

Note that this inequality ensures that k can take all but finitely many values. Therefore, in view of Goldfeld’s conjecture, there exists a nonnegative integer $\epsilon _E$ such that the set $\mathcal {P}$ contains $k+\epsilon _{E}$ (which we still call k by abuse of notation) many primes instead. To answer our question, we need to carefully pick the distinct primes $\ell _1, \ldots , \ell _k$ . Given any integer M, the number of distinct prime factors denoted by $\omega (M)$ is asymptotically $\log \log M$ . Recall that the prime number theorem asserts that the average gap between consecutive primes among the first x many primes is $\log x$ . Since $\ell _1$ is a prime of good reduction, $\ell _1\nmid N$ . The prime number theorem implies that $\ell _1 \sim \log ( \omega (N)) \sim \log _{(3)} (N)$ . However, we need to do more. We require that $\ell _1$ splits completely in ${\mathbb Q}(E[2])/{\mathbb Q}$ . Using the Chebotarev density theorem, we can conclude that $\ell _1 \sim c \cdot \log \, ( \omega (N))$ , where c is either $2$ , $3$ , or $6$ depending on the degree of the Galois extension ${\mathbb Q}(E[2])/{\mathbb Q}$ . Of course, asymptotically, $\ell _1 \sim \log \, ( \omega (N))$ . Next, $\ell _2 \nmid N\ell _1$ and splits completely in ${\mathbb Q}(E[2])/{\mathbb Q}$ . Note that the number of distinct prime divisors of $N\ell _1 \sim \omega (N)+ 1$ . Using the same argument, we see that $\ell _2 \sim \log \,( \omega (N) + 1)$ . Continuing this process,

(5-2) $$ \begin{align} \mathfrak{f} = \prod_{i=1}^k \ell_i &\sim \prod_{i=1}^k( \log (\omega(N) + (i-1))) \nonumber\\ &\sim \frac{(\log (\omega(N)-1+k))^{\omega(N)+2+k}}{\exp \operatorname{\mathrm{li}}(\omega(N)-1+k)} \nonumber\\ &\sim \frac{(\log n)^{n+c}}{\exp \operatorname{\mathrm{li}}(n)}, \end{align} $$

where $\operatorname {\mathrm {li}}(x) := \int _0^x ({dt}/{\log t})$ is the logarithmic integral and c is a constant depending on E. Let us explain the above estimates in greater detail. Setting

$$ \begin{align*} P(k):= \prod_{i=1}^k \log (\omega(N)+(i-1)), \end{align*} $$

we wish to estimate the sum

$$ \begin{align*} \log(P(k))= \sum_{i=1}^k \log (\log (\omega(N)+(i-1))). \end{align*} $$

Using Abel’s partial summation formula (see for example [Reference ApostolApo76, Theorem 4.2]) with $f(t) = \log (\log (\omega (N) - 1 + t))$ , $a(n) = 1$ , and $A(x) = \sum _{n\leq x}a(n)$ ,

$$ \begin{align*} &\log P(k)\\ &\quad= \sum_{i=1}^k \log\log(\omega(N) - 1 + i)\\&\quad= k \log\log(\omega(N) - 1 + k) - \int_0^k \frac{\lfloor x \rfloor}{(\omega(N) - 1 + x)\log(\omega(N) - 1 + x)}\, dx \\&\quad= k \log\log(\omega(N) - 1 + k) - \int_0^k \frac{x}{(\omega(N) - 1 + x)\log(\omega(N) - 1 + x)}\, dx \\& \qquad+ O\bigg( \int_0^k \frac{dx}{(\omega(N) - 1 + x)\log(\omega(N) - 1 + x)}\bigg) \\&\quad= k \log\log(\omega(N) - 1 + k) - ( \operatorname{\mathrm{li}}(\omega(N) - 1 + x) - (\omega(N) - 1)\log\log(\omega(N) - 1 + x) )|_0^k \\ & \qquad + O\bigg( \int_0^k \frac{dx}{(\omega(N) - 1 + x)\log(\omega(N) - 1 + x)}\bigg) \\ &\quad= k \log\log(\omega(N) - 1 + k) - \operatorname{\mathrm{li}}(\omega(N) - 1 + k) + (\omega(N) - 1)\log\log(\omega(N) - 1 + k) \\ & \qquad + O(\log\log(\omega(N) - 1 + k)) \\ &\quad= (k + \omega(N) + 1)\log\log(\omega(N) - 1 + k) - \operatorname{\mathrm{li}}(\omega(N) - 1 + k)\\ & \qquad + O(\log\log(\omega(N) - 1 + k)). \end{align*} $$

This tells us that

$$ \begin{align*} P(k) &= \exp((k + \omega(N) + 1)\log\log(\omega(N) - 1 + k) - \operatorname{\mathrm{li}}(\omega(N) - 1 + k)\\ & \quad + O(\log\log(\omega(N) - 1 + k)))\\ &= \frac{\exp((k + \omega(N) + 1)\log\log(\omega(N) - 1 + k))}{\exp( \operatorname{\mathrm{li}}(\omega(N) - 1 + k))}( \exp(O(\log\log(\omega(N) - 1 + k)))) \\ &= \frac{\exp((k + \omega(N) + 1)\log\log(\omega(N) - 1 + k))}{\exp( \operatorname{\mathrm{li}}(\omega(N) - 1 + k))}( O(\log(\omega(N) - 1 + k))) \\ &= \frac{\log(\omega(N) - 1 + k)^{k + \omega(N) + 2}}{\exp( \operatorname{\mathrm{li}}(\omega(N) - 1 + k))}. \end{align*} $$

To obtain Equation (5-2), we note that for the given n, the difference between n and k depends on the rank of E, and further $\omega (N)$ depends on E. Thus, we can rewrite $\omega (N)+2+k$ as $n+c$ , where c is a constant depending only on the elliptic curve E.

We have therefore proven the following result.

Theorem 5.6. Suppose that Goldfeld’s conjecture is true. Given an elliptic curve $E_{/{\mathbb Q}}$ and a positive integer n, there exists a quadratic extension $K/{\mathbb Q}$ with conductor $\mathfrak {f}_K \sim {(\log n)^{n+c}}/{\exp \operatorname {\mathrm {li}} (n)}$ such that . Here, c is an explicit constant depending only on E.

Using a result of Ono, we can prove an unconditional statement which we now explain.

5.1.3 An unconditional effective version

Given an elliptic curve $E_{/{\mathbb Q}}$ of conductor N and a positive integer n, we want to find an imaginary quadratic field $K/{\mathbb Q}$ with minimal conductor $\mathfrak {f}_{K}$ such that we can guarantee .

For this section, we consider elliptic curves $E_{/{\mathbb Q}}$ that have no exceptional primes p, that is, no primes p such that the $\mod p$ Galois representation attached to E is nonsurjective. This condition is a mild one: W. Duke has shown in [Reference DukeDuk97, Theorem 1] that almost all elliptic curves defined over ${\mathbb Q}$ have no exceptional primes.

For such elliptic curves $E_{/{\mathbb Q}}$ , [Reference OnoOno01, Theorem 1] asserts that there exists a fundamental discriminant $D_E$ such that the twisted curve $E^{d}$ has rank $0$ , where $d = D_E\ell _1\cdots \ell _{k}$ for some even integer k and where the primes $\ell _i$ are chosen from a set (of primes) with density $1$ . Indeed, the aforementioned theorem asserts (more generally) that there is a set of primes (call it S) with positive Frobenius density (that is, there exists a finite Galois extension $L/{\mathbb Q}$ such that for all but finitely many primes in this set, these primes represent a fixed Frobenius conjugacy class in $\operatorname {\mathrm {Gal}}(L/{\mathbb Q})$ ) such that $E^{d}$ has rank $0$ , where $d = D_E\ell _1\cdots \ell _{k}$ for some even integer k and the primes $\ell _i$ are chosen from S. However, it follows from [Reference OnoOno01, proof of Theorem 2.2] that the field $L/{\mathbb Q}$ in the definition of the Frobenius density is the field that contains the coefficients of the newform associated to E. Since the elliptic curves we are working with are defined over ${\mathbb Q}$ , it follows from the modularity theorem that the newform will have integral coefficients, that is, $L = {\mathbb Q}$ . Since there is only one Frobenius class in ${\mathbb Q}$ , namely the trivial class, [Reference OnoOno01, Theorem 1] says that the density of the set S is $1$ .

Let K be the quadratic field associated with the twist d. Since $E^d$ has rank 0, we know that $\operatorname {\mathrm {rank}}_{{\mathbb Z}}(E(K))=\operatorname {\mathrm {rank}}_{{\mathbb Z}}(E({\mathbb Q}))$ . It follows that

If K is a quadratic field arising from Ono’s theorem such that

, it is required that

$$ \begin{align*} k \geq n + \operatorname{\mathrm{rank}}_{{\mathbb Z}} (E({\mathbb Q})) + 6. \end{align*} $$

To answer our question, we pick the distinct primes $\ell _1, \ldots , \ell _k$ using the exact same process as before. We have that

$$ \begin{align*} \mathfrak{f} = D_E\prod_{i=1}^k \ell_i \sim \prod_{i=1}^k( \log (\omega(N) + (i-1))) \sim \frac{(\log (\omega(N)+k))^{\omega(N)+k}}{\exp \operatorname{\mathrm{li}}(\omega(N)+k)} \sim \frac{(\log n)^{n+c}}{\exp \operatorname{\mathrm{li}}(n)},\end{align*} $$

where $\operatorname {\mathrm {li}}(x) := \int _0^x ({dt}/{\log t})$ is the logarithmic integral and c is a constant depending on E.

We have therefore proven the following result.

Theorem 5.7. Given an elliptic curve $E_{/{\mathbb Q}}$ with no exceptional primes and a positive integer n, there exists a quadratic extension $K/{\mathbb Q}$ with conductor $\mathfrak {f}_K \sim {(\log n)^{n+c}}/{\exp \operatorname {\mathrm {li}} (n)}$ such that . Here, c is an explicit constant depending only on E.

5.2 Arbitrarily large in ${\mathbb Z}/p{\mathbb Z}$ -extensions

Let $E_{/{\mathbb Q}}$ be a fixed rank 0 elliptic curve of conductor N. Let p be a fixed odd prime. Let F be any number field. We have the obvious short exact sequence

It follows that

(5-3)

When $F/{\mathbb Q}$ is a cyclic degree-p Galois extension, we know from Lemma 5.3 that

$$ \begin{align*} \operatorname{\mathrm{rank}}_p \operatorname{\mathrm{Sel}}_p(E/F) \geq \# T_{E/F} -4. \end{align*} $$

Given an integer n, there exists a number field $F_{(n)}$ such that (see [Reference ČesnavičiusČes17, Theorem 1.2])

$$ \begin{align*} \operatorname{\mathrm{rank}}_{p}\operatorname{\mathrm{Sel}}_{p}(E/F_{(n)}) \geq \# T_{E/F} -4\geq n. \end{align*} $$

Denote the conductor of $F_{(n)}$ by $\mathfrak {f}(F_{(n)})$ . Varying over all ${\mathbb Z}/p{\mathbb Z}$ -extensions of ${\mathbb Q}$ , there are infinitely many number fields $L/{\mathbb Q}$ such that $T_{E/L} \supseteq T_{E/F_{(n)}}$ , that is, $\mathfrak {f}(F_{(n)})\mid \mathfrak {f}(L)$ . For each such L,

$$ \begin{align*} \operatorname{\mathrm{rank}}_{p}\operatorname{\mathrm{Sel}}_{p}(E/L) \geq n. \end{align*} $$

Recall the conjecture of David, Fearnley, and Kisilevsky from Section 2. If $p\geq 7$ , it predicts the boundedness of the set

$$ \begin{align*} N_{E,p}(X) := \{L/{\mathbb Q} \textrm{ cyclic of degree } p: \mathfrak{f}(L)< X \textrm{ and } \operatorname{\mathrm{rank}}_{{\mathbb Z}}(E/L)> \operatorname{\mathrm{rank}}_{{\mathbb Z}}(E/{\mathbb Q})\}. \end{align*} $$

If this conjecture is true, then varying over all ${\mathbb Z}/p{\mathbb Z}$ -extensions of ${\mathbb Q}$ , there are only finitely many cyclic p-extensions $L/{\mathbb Q}$ in which there is a rank jump upon base-change. Therefore, given an integer n, one can find an integer $M = M(n)$ and a number field $L/{\mathbb Q}$ such that:

1. (i) $\mathfrak {f}(L)> M(n)$ ;

2. (ii) $\mathfrak {f}(F_{(n)})\mid \mathfrak {f}(L)$ ;

3. (iii) $\operatorname {\mathrm {rank}}_{{\mathbb Z}}(E(L)) = \operatorname {\mathrm {rank}}_{{\mathbb Z}}(E({\mathbb Q}))$ .

Thus, given n, there exists a cyclic extension $L/{\mathbb Q}$ of degree $p\geq 7$ such that Equation (5-3) becomes

Since $\operatorname {\mathrm {rank}}_{{\mathbb Z}}(E({\mathbb Q}))$ is independent of L and $\operatorname {\mathrm {rank}}_p E(L)[p]$ is at most 2, it means that given n, there exists a ${\mathbb Z}/p{\mathbb Z}$ -extension $L/{\mathbb Q}$ such that

.

We feel that the full force of the conjecture of David, Fearnley, and Kisilevsky is required. In particular, we do not see if the result of Mazur and Rubin is sufficient. This is because it is not obvious to us as to why, even if there are infinitely many number fields $\mathcal {L}/{\mathbb Q}$ with $\operatorname {\mathrm {rank}}_{{\mathbb Z}}(E(\mathcal {L})) = \operatorname {\mathrm {rank}}_{{\mathbb Z}}(E(\mathcal {{\mathbb Q}}))$ , there should be any (of these) satisfying $\mathfrak {f}({F_{(n)}})\mid \mathfrak {f}(\mathcal {L})$ .