Iwasawa Invariants for elliptic curves over $\mathbb{Z}_{p}$-extensions and Kida’s Formula

The classical Riemann–Hurwitz formula describes the relationship of the Euler characteristics of two surfaces when one is a ramified covering of the other. Suppose $\pi\mathrel{\mathop{\ordinarycolon}}R_{1}\rightarrow R_{2}$ is an $n$-fold covering of compact, connected Riemann surfaces and $g_{1},\ g_{2}$ are their respective genus. The classical Riemann–Hurwitz formula is the statement

where the sum is over all points $P_{2}$ on $R_{2}$ and $e(P_{2})$ denotes the ramification index of $P_{2}$ for the covering $\pi$ (see [Sil09, Chapter II Theorem 5.9]). An analogue of the above formula for algebraic number fields was proven by Y. Kida in [Kid80]. Kida’s formula describes the change of (cyclotomic) Iwasawa $\lambda$-invariants in a $p$-extension in terms of the degree and the ramification index. In [Iwa81], K. Iwasawa proved this formula using the theory of Galois cohomology for extensions of $\mathbb{Q}$ which are not necessarily finite. More precisely,

In the study of rational points of elliptic curves, the Selmer group plays a crucial role. In [Maz72], B. Mazur initiated the study of the growth of the $p$-primary Selmer group in $\mathbb{Z}_{p}$-extension of number fields. In [HM99], Y. Hachimori and K. Matsuno proved an analogue of Theorem 1 for $p$-primary Selmer groups of elliptic curves in $p$-power extensions of a (fixed) number field over the cyclotomic $\mathbb{Z}_{p}$-extension. This result has been generalized in several directions. For modular forms over the cyclotomic $\mathbb{Z}_{p}$-extension, analogous results have been worked out for signed Selmer groups at non-ordinary primes by J. Hatley–A. Lei, see [HL19, Theorem 6.7]. It has been extended to a general class of Galois representations over the cyclotomic $\mathbb{Z}_{p}$-extension, including the case of $p$-ordinary Hilbert modular forms and $p$-supersingular modular forms by R. Pollack–T. Weston in [PW06]. The first named author studied Kida-type formula for fine Selmer groups of elliptic curves over the cyclotomic $\mathbb{Z}_{p}$-extension of totally real number fields in [Kun21]. Kida-type formulae for $p$-primary Selmer groups of elliptic curves have also been proven over special classes of non-abelian $p$-adic Lie extensions containing the cyclotomic $\mathbb{Z}_{p}$-extension by A. Bhave in [Bha07]. In the first part of this article, we prove a Kida-like formula for $p$-primary Selmer groups of elliptic curves in more general $\mathbb{Z}_{p}$-extensions (see Theorem 4.3).

For ease of exposition, in the introduction, we state our main result in a simplified setting. Let $p\geq 5$ be a fixed rational prime. Let $E_{/\mathbb{Q}}$ be an elliptic curve without complex multiplication and with good ordinary reduction at $p$. Let $K$ be a fixed imaginary quadratic field and $K^{\operatorname{ac}}$ denote the anticyclotomic $\mathbb{Z}_{p}$-extension of $K$. Let $L/K$ be a Galois extension of $p$-power degree disjoint from $K^{\operatorname{ac}}$ and $L_{\infty}$ be the compositum $L\cdot K^{\operatorname{ac}}$. Suppose that the $p$-primary Selmer group over $K^{\operatorname{ac}}$ (and $L_{\infty}$) is cofinitely generated as a $\mathbb{Z}_{p}$-module. Then our result relates their respective $\lambda$-invariants.

We remark that the $p$-primary Selmer group need not always be $\Lambda$-cotorsion over the anticyclotomic $\mathbb{Z}_{p}$-extension, $K^{\operatorname{ac}}$ (see [Ber95, Cor02, Vat03]). However, by the work of Vatsal [Vat02], M. Bertolini–H. Darmon [BD05], and Pollack–Weston [PW11, Theorem 5.3], it is known that if some natural conditions are satisfied, then the $p$-primary Selmer group is not only $\Lambda$-cotorsion, but also cofinitely generated as a $\mathbb{Z}_{p}$-module. We show (in Corollary 4.9) that this property propagates to all $p$-power extensions $L$ of $K$ (which are disjoint from $K^{\operatorname{ac}}$) satisfying the additional condition that if $E$ has split multiplicative reduction at $\ell$ and $v|\ell$ is inert in $K$, then $v$ splits completely in $L$. We expect that the aforementioned result can be extended to the case of plus/minus Selmer groups when $p$ is a prime of supersingular reduction under reasonable hypotheses introduced by B. D. Kim in [Kim13].

Two elliptic curves $E_{1}$ and $E_{2}$ over a number field $K$ are said to be $p$-congruent if their associated residual representations are isomorphic, i.e., $E_{1}[p]$ and $E_{2}[p]$ are isomorphic as $\operatorname{Gal}({}\mkern 3.0mu\overline{\mkern-3.0muK}/K)$-modules. There is much interest in investigating the behaviour of Iwasawa invariants for congruent Galois representations. Such investigations were initiated by Greenberg–Vatsal in [GV00]. They considered the case when both $E_{1}$ and $E_{2}$ are defined over $\mathbb{Q}$, and showed how the (cyclotomic) Iwasawa $\mu$ and $\lambda$-invariants for the $p$-primary Selmer groups of $E_{1}$ and $E_{2}$ are related. Let $\mathfrak{T}$ be the set of primes $v\nmid p$ of $K$ at which $E_{1}$ or $E_{2}$ has bad reduction. Over $K^{\operatorname{cyc}}$, Greenberg–Vatsal compare the $\mathfrak{T}$-imprimitive Selmer groups of $E_{1}$ and $E_{2}$. These results were conditionally generalized by Kidwell for $\mathbb{Z}_{p}$-extensions of number fields (see [Kid18, Theorem 6.1]). This requires a finiteness assumption, which is satisfied if primes in $\mathfrak{T}$ are finitely decomposed in the $\mathbb{Z}_{p}$-extension, $K_{\infty}$. We refine the results of Greenberg–Vatsal and Kidwell in two ways. We show that the set of primes $\mathfrak{T}$ can be replaced by a smaller set $\Omega_{0}$ (see Definition 6.1). Having a smaller set of primes $\Omega_{0}$ to consider makes for a more refined relationship between $\lambda$-invariants for the Selmer groups of $E_{1}$ and $E_{2}$. We show that

where $\sigma_{E_{i}}^{(v)}$ is the corank of a certain locally defined $\Lambda$-module (see Definition 6.16). We also clarify the assumptions on the splitting of primes in $\mathbb{Z}_{p}$-extensions. These assumptions apply only to the smaller set of primes $\Omega_{0}$ and not to the set $\mathfrak{T}$. The approach used here is similar to that of the second named author and R. Sujatha, who prove such results for the cyclotomic $\mathbb{Z}_{p}$-extension in [RS20].

As an application of our results on congruences, we provide an algorithm for constructing elliptic curves whose Selmer group has a large $\lambda$-invariant. Such elliptic curves are considered over the anticyclotomic $\mathbb{Z}_{p}$-extension of an imaginary quadratic field. In [Mat07], this question has been studied for some small primes (i.e., $p\leq 7$ or $p=13$) for elliptic curves with good ordinary reduction at $p$, over the cyclotomic $\mathbb{Z}_{p}$-extension (of $\mathbb{Q}$). In [Kim09], elliptic curves over $\mathbb{Q}$ with supersingular reduction at $p=3$ and arbitrarily large (plus/minus) $\lambda$-invariant are constructed. The latter method is more amenable for generalization to our setting. In Section 7, we fix the elliptic curve $E=32a2$ (Cremona label) and the prime $p=5$. Using the work of K. Rubin–A. Silverberg (see [RS95]), we have an algorithm to produce elliptic curves which are 5-congruent to $E$ with large anticyclotomic $\lambda$-invariant over $K=\mathbb{Q}(\sqrt{-1})$. Here, we crucially use the formula (1.1), which relates $\lambda$-invariants of the Selmer groups of elliptic curves in Rubin–Silverberg family. We illustrate our algorithm via explicit computation.

Throughout this article, let $p\geq 5$ be a fixed prime number and $K$ be a number field. Let $K_{\infty}$ be any $\mathbb{Z}_{p}$-extension of $K$. Two examples of interest are as follows:

1. (1)

   $K$ is a number field and $K_{\infty}=K^{\operatorname{cyc}}$ is the cyclotomic $\mathbb{Z}_{p}$-extension of $K$. This is the unique $\mathbb{Z}_{p}$-extension of $K$ contained in $K(\mu_{p^{\infty}})$.

2. (2)

   $K$ is an imaginary quadratic field and $K_{\infty}=K^{\operatorname{ac}}$ is an anticyclotomic $\mathbb{Z}_{p}$-extension of $K$. This is a $\mathbb{Z}_{p}$-extension of $K$ which is Galois and pro-dihedral over $\mathbb{Q}$. More generally, $K_{\infty}$ may be chosen to be an anticyclotomic $\mathbb{Z}_{p}$-extension over any CM field, $K$.

If $K$ is any number field with signature $(r_{1},r_{2})$, the Leopoldt’s conjecture predicts that the $\mathbb{Z}_{p}$-rank of the maximal abelian pro-$p$ extension of $K$ unramified away from primes above $p$ is $1+r_{2}$. In other words, it predicts that there are $1+r_{2}$ independent $\mathbb{Z}_{p}$-extensions of $K$. For an imaginary quadratic field, the cyclotomic and anticyclotomic extensions together generate a $\mathbb{Z}_{p}^{2}$-extension, containing infinitely many $\mathbb{Z}_{p}$-extensions that are unramified at all primes $v\nmid p$ and not Galois over $\mathbb{Q}$.

Set $\Gamma\mathrel{\mathop{\ordinarycolon}}=\operatorname{Gal}(K_{\infty}/K)\simeq\mathbb{Z}_{p}$ and $\Gamma_{n}\mathrel{\mathop{\ordinarycolon}}=\operatorname{Gal}(K_{\infty}/K_{n})$. The Iwasawa algebra $\Lambda$ is the completed group ring $\varprojlim_{n}\mathbb{Z}_{p}\llbracket\Gamma/\Gamma_{n}\rrbracket$. Choose a topological generator $\gamma\in\Gamma$. There is an isomorphism $\Lambda\simeq\mathbb{Z}_{p}\llbracket T\rrbracket$ associating the formal variable $T$ with $\gamma-1$.

For a discrete $p$-primary abelian group $M$, set $M^{\vee}\mathrel{\mathop{\ordinarycolon}}=\operatorname{Hom}_{\operatorname{cnts}}(M,\mathbb{Q}_{p}/\mathbb{Z}_{p})$ to denote its Pontryagin dual. We say that $M$ is a cofinitely generated (resp. cotorsion) $\Lambda$-module if $M^{\vee}$ is finitely generated (resp. torsion) as a $\Lambda$-module. For a cofinitely generated and cotorsion $\Lambda$-module $M$, the Structure Theorem of $\Lambda$-modules asserts that $M^{\vee}$ is pseudo-isomorphic to a finite direct sum of cyclic $\Lambda$-modules, i.e., there is a map of $\Lambda$-modules

with finite kernel and cokernel. Here, $m_{i}>0$ and $h_{j}(T)$ is a distinguished polynomial (i.e., a monic polynomial with non-leading coefficients divisible by $p$). The characteristic ideal of $M^{\vee}$ is (up to a unit) generated by the characteristic element,

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

The $\lambda$-invariant of $M$ is the degree of the characteristic element, i.e.

Henceforth, we fix a number field $K$ and a $\mathbb{Z}_{p}$-extension $K_{\infty}$ over $K$. Let $E_{/K}$ be an elliptic curve with good ordinary reduction at $p$. We now define our module of interest, the $p$-primary Selmer group over $K_{\infty}$. Let $S$ be a finite set of primes $v$ of $K$ containing the primes above $p$ and the primes of bad reduction of $E$. Denote by $K_{S}$ the maximal algebraic extension of $K$ which is unramified at the primes $v\notin S$. For any finite extension $F$ of $K$ such that $F\subseteq K_{S}$ and a prime $v\in S$, define

where the product is over all primes $w$ of $F$ lying above $v$. The $p$-primary Selmer group over $F$ is defined as follows

The $p$-primary Selmer group $\operatorname{Sel}(E/K_{\infty})$ is the direct limit of $\operatorname{Sel}(E/F)$ as $F$ ranges over all finite extensions of $K$ contained in $K_{\infty}$. The $\mu$ and $\lambda$-invariants of $\operatorname{Sel}(E/K_{\infty})$ are denoted by $\mu(E/K_{\infty})$ and $\lambda(E/K_{\infty})$. We make the following assumption.

As before, let $p\geq 5$ be a prime number and $K$ be a number field. Let $L$ be a finite Galois extension of $K$ with the order of the Galois group $G=\operatorname{Gal}(L/K)$ a power of $p$. Let $K_{\infty}$ be a $\mathbb{Z}_{p}$-extension of $K$ and $L_{\infty}=K_{\infty}\cdot L$. The field diagram is drawn below

In the two examples of special interest, we know the following.

1. (1)

   When $K$ is a number field and $K_{\infty}=K^{\operatorname{cyc}}$ is the cyclotomic $\mathbb{Z}_{p}$-extension of $K$, $L_{\infty}$ is identified with the cyclotomic $\mathbb{Z}_{p}$-extension of $L$, namely $L^{\operatorname{cyc}}$.

2. (2)

   When $K$ is an imaginary quadratic field and $K_{\infty}=K^{\operatorname{ac}}$ is an anticyclotomic $\mathbb{Z}_{p}$-extension of $K$, the $\mathbb{Z}_{p}$-extension, $L_{\infty}$ over $L$ is a non-cyclotomic extension (but it is not an anticyclotomic extension).

In this section, we record some preliminary results required in the proof of Theorem 2. For the remainder of this discussion, we assume the following.

The following lemma is an analogue of [HM99, Proposition 2.3] in the current setting (see also [How98, Proposition 5.30]). In proving the following lemma, we crucially use the hypothesis introduced in Assumption 3.1(1) that $p$ has good ordinary reduction at $p$. Note that Lemma 3.2 and Corollary 3.3 do not require this assumption.

In this section, we state a generalization of Kida’s formula which applies to more general $\mathbb{Z}_{p}$-extensions. Recall that $p\geq 5$ is a fixed prime number. Let $K$ be a number field and $K_{\infty}$ be any $\mathbb{Z}_{p}$-extension over $K$ such that all primes above $p$ are ramified in $K_{\infty}$. Let $L$ be a finite Galois extension of $K$ with Galois group $G=\operatorname{Gal}(L/K)$ of $p$-power order. Let $E_{/K}$ be an elliptic curve such that the Assumption 3.1 holds. Recall that $S$ is a finite set of primes of $K$ containing the primes above $p$ and the primes of bad reduction of $E$. For our discussion on generalized Kida-type formula, we choose the set $S$ to be the primes $v$ in $K$ satisfying at least one of the following conditions

1. (1)

   $v|p$,

2. (2)

   $E$ has bad reduction at $v$,

3. (3)

   $v$ ramifies in the extension $L/K$.

Let $L_{\infty}$ denote the composite $K_{\infty}\cdot L$. We define two sets of primes $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$ in $L_{\infty}$, which play a crucial role in the generalization of Kida’s formula.

Recall that $\Sigma$ denotes the set of primes of $K$ that are finitely decomposed in $K_{\infty}$. We introduce an important assumption.

By assumption, any prime $w^{\prime}\in\mathcal{P}_{1}\cup\mathcal{P}_{2}$ must lie above a prime $v\in S\cap\Sigma$. The above assumption guarantees that the sets $\mathcal{P}_{i}$ are finite for $i=1,2$. Our choice of $\mathcal{P}_{1}$ is the same as that in [HM99], and $\mathcal{P}_{2}$ bears the additional condition that the primes $w^{\prime}\in\mathcal{P}_{2}$ are ramified over $K_{\infty}$. This does not make an actual difference in terms in the formula relating the $\lambda$-invariants.

For any algebraic extension $\mathcal{L}$ of $K$, and a set of primes $T$ of $K$, we write $T(\mathcal{L})$ to denote the set of primes of $w$ of $\mathcal{L}$ such that $w|v$ for primes $v\in T$. For a prime $v$ of $K$, we write $v(\mathcal{L})$ for the set of primes $w|v$ of $\mathcal{L}$.

We record our first main result; its proof will occupy this section and the next.

Since $G$ is a $p$-group, it is solvable. Let $L_{0}\subset L_{1}\subset\cdots\subset L_{M}$ be a filtration of $L$ such that $L_{i+1}$ is Galois over $L_{i}$ with Galois group $\operatorname{Gal}(L_{i+1}/L_{i})\simeq\mathbb{Z}/p\mathbb{Z}$. It is easy to show that it suffices to prove Theorem 4.3 when $\#G=p$ (see for example [Mat00, Lemma 3.5]). Without loss of generality, we assume from here on that $G\simeq\mathbb{Z}/p\mathbb{Z}$.

We keep the notation introduced at the start of this section. Let $v$ (resp. $w$) be primes above $\ell$ in $K$ (resp. $L$) such that $w|v$. Let $w^{\prime}|w$ be a prime of $L_{\infty}$ and $v^{\prime}\mathrel{\mathop{\ordinarycolon}}=w^{\prime}_{\restriction K_{\infty}}$. The diagram below depicts the labelling of primes