2 Preliminaries

Throughout, we fix an odd prime number p and number field F. Let $F^{\operatorname {cyc}}$ be the cyclotomic $\mathbb {Z}_{p}$ -extension of F. Denote by $\Sigma _{p}$ the set of primes of F above p. Let E be an elliptic curve over F and denote by $\Sigma _{\operatorname {ss}}(E)=\{\mathfrak {p}_{1},\dots , \mathfrak {p}_{d}\}$ the set of primes in $\Sigma _{p}$ at which E has supersingular reduction. Let $\mathcal {F}_{\infty }/F$ be a Galois extension of F containing $F^{\operatorname {cyc}}$ such that $\operatorname {Gal}(\mathcal {F}_{\infty }/F)\simeq \mathbb {Z}_{p}^{m}$ for an integer $m\geq 1$ . Set $\operatorname {G}:=\operatorname {Gal}(\mathcal {F}_{\infty }/F)$ , $H:=\operatorname {Gal}(\mathcal {F}_{\infty }/F^{\operatorname {cyc}})$ and let $\Gamma $ be the Galois group $\operatorname {Gal}(F^{\operatorname {cyc}}/F)$ , which is identified with $\operatorname {G}/\operatorname {H}$ . Associated to any pro-p group $\mathcal {G}$ , the Iwasawa algebra $\mathbb {Z}_{p}[[\mathcal {G}]]$ is the inverse limit $\varprojlim _{U} \mathbb {Z}_{p}[\mathcal {G}/U]$ , where U ranges over all open normal subgroups of $\mathcal {G}$ . Set $\mathcal {F}_{0}:=F$ and for $n\geq 1$ , let $\mathcal {F}_{n}$ denote the unique subextension $F\subseteq \mathcal {F}_{n}\subset \mathcal {F}_{\infty }$ such that $\operatorname {Gal}(\mathcal {F}_{n}/F)\simeq (\mathbb {Z}/p^{n}\mathbb {Z})^{m}$ . We introduce the following hypothesis on the elliptic curve E.

Note that for $v\in \Sigma _{\operatorname {ss}}(E)$ , it follows from Hasse’s theorem that $|a_{v}(E)|< 2\sqrt {p}$ . Since E has supersingular reduction at v, we find that p divides $\# \widetilde {E}_{v}(\mathbb {F}_{p})$ . Therefore, condition (4) above is satisfied for all primes $p\geq 7$ .

Part (2) of Lemma 2.1 ensures that the signed norm groups of E over $F_{n,\eta }$ may be defined. This is used in Definition 2.2, where the multisigned Selmer groups associated to E over $\mathcal {F}_{\infty }$ are defined. We shall now define local conditions at each of the primes $\mathfrak {p}_{i}\in \Sigma _{\operatorname {ss}}(E)$ . By Hypothesis (2.1), the local field $F_{\mathfrak {p}_{i}}$ is isomorphic to $\mathbb {Q}_{p}$ . Denote by $\widehat {E}_{\mathfrak {p}_{i}}$ the formal group of E over $F_{\mathfrak {p}_{i}}$ . For any algebraic extension L of $F_{\mathfrak {p}_{i}}\simeq \mathbb {Q}_{p}$ , denote by $\widehat {E}_{\mathfrak {p}_{i}}(L):=\widehat {E}_{\mathfrak {p}_{i}}(\mathfrak {m}_{L})$ , where $\mathfrak {m}_{L}$ is the maximal ideal of $\mathcal {O}_{L}$ . Let K be a finite unramified extension of $\mathbb {Q}_{p}$ and $K^{\operatorname {cyc}}$ the cyclotomic $\mathbb {Z}_{p}$ -extension of K. Denote by $K_{n}$ the unique subextension of $K^{\operatorname {cyc}}/K$ of degree $p^{n}$ . Kobayashi introduced plus and minus norm groups:

$$ \begin{align*} \widehat{E}_{\mathfrak{p}_{i}}^{+}(K_{n}) := \left\{P\in \widehat{E}_{\mathfrak{p}_{i}}(K_{n}) \mid \operatorname{tr}_{n/m+1} (P)\in \widehat{E}_{\mathfrak{p}_{i}}(K_{m}), \text{ for }0\leq m < n\text{ and }m \text{ even}\right\}\kern-1.2pt,\end{align*} $$

$$ \begin{align*}\widehat{E}_{\mathfrak{p}_{i}}^{-}(K_{n}):=\left\{P\in \widehat{E}_{\mathfrak{p}_{i}}(K_{n}) \mid \operatorname{tr}_{n/m+1} (P)\in \widehat{E}_{\mathfrak{p}_{i}}(K_{m}),\text{ for }0\leq m < n\text{ and }m \text{ odd}\right\}\kern-1.2pt,\end{align*} $$

where $\operatorname {tr}_{n/m+1}:\widehat {E}_{\mathfrak {p}_{i}}(K_{n})\rightarrow \widehat {E}_{\mathfrak {p}_{i}}(K_{m+1})$ denotes the trace map with respect to the formal group law on $\widehat {E}_{\mathfrak {p}_{i}}$ .

Let $\ddagger =(\ddagger _{1},\dots , \ddagger _{d})\in \{+,-\}^{d}$ be a multisigned vector, so that each component $\ddagger _{i}$ is either a $+$ or $-$ sign. For each finite prime $v\notin \Sigma _{\operatorname {ss}}(E)$ , and each integer $n\geq 1$ , set

(2.1) $$ \begin{align}\mathcal{H}_{v}(\mathcal{F}_{n}, E[p^{\infty}]):=\prod_{\eta|v} \frac{H^{1}(\mathcal{F}_{n,\eta}, E[p^{\infty}])}{E(\mathcal{F}_{n,\eta})\otimes \mathbb{Q}_{p}/\mathbb{Z}_{p}},\end{align} $$

where $\eta $ runs through the primes of $\mathcal {F}_{n}$ above v. For each prime $\mathfrak {p}_{i}\in \Sigma _{\operatorname {ss}}(E)$ , set

(2.2) $$ \begin{align}\mathcal{H}_{\mathfrak{p}_{i}}^{\pm}(\mathcal{F}_{n}, E[p^{\infty}]):=\prod_{\eta|\mathfrak{p}_{i}} \frac{H^{1}(\mathcal{F}_{n,\eta}, E[p^{\infty}])}{\widehat{E}_{\mathfrak{p}_{i}}^{\pm}(\mathcal{F}_{n,\eta})\otimes \mathbb{Q}_{p}/\mathbb{Z}_{p}}.\end{align} $$

Let $\eta $ be a prime of $\mathcal {F}_{\infty }$ above $\mathfrak {p}_{i}$ . By (2) of Lemma 2.1, $\mathcal {F}_{n,\eta }$ is isomorphic to $K_{n}$ for some finite unramified extension K over $\mathbb {Q}_{p}$ . Thus the norm groups $\widehat {E}_{\mathfrak {p}_{i}}^{\pm } (\mathcal {F}_{n,\eta })$ are defined. We now come to the definition of the multisigned Selmer groups.

Definition 2.2 Let d be the cardinality of $\Sigma _{\operatorname {ss}}(E)$ and let $\ddagger =(\ddagger _{1},\dots , \ddagger _{d})$ be a multi-signed vector, so that each component $\ddagger _{i}$ is either a $+$ sign or $-$ sign. Let $\Sigma $ be a finite primes of F containing $\Sigma _{p}$ and the primes at which E has bad reduction. Let $\mathfrak {p}_{1},\dots , \mathfrak {p}_{d}$ be the primes in $\Sigma _{\operatorname {ss}}(E)$ . For $i=1,\dots , d$ , let $\mathcal {H}_{\mathfrak {p}_{i}}^{\ddagger _{i}}(\mathcal {F}_{n}, E[p^{\infty }])$ be the group defined above, see (2.2). The multisigned Selmer group $\operatorname {Sel}^{\ddagger }(E/\mathcal {F}_{n})$ is the kernel of the map:

$$ \begin{align*}\Phi_{E,\mathcal{F}_{n}}^{\ddagger}:H^{1}(\mathcal{F}_{n},E[p^{\infty}])\rightarrow \prod_{v\in \Sigma\backslash \Sigma_{\operatorname{ss}}(E)} \mathcal{H}_{v}(\mathcal{F}_{n}, E[p^{\infty}])\times \prod_{i=1}^{d} \mathcal{H}_{\mathfrak{p}_{i}}^{\ddagger_{i}}(\mathcal{F}_{n}, E[p^{\infty}]).\end{align*} $$

Let $\operatorname {Sel}^{\ddagger }(E/\mathcal {F}_{\infty })$ be the direct limit $\varinjlim _{n} \operatorname {Sel}^{\ddagger }(E/\mathcal {F}_{n})$ and write $\operatorname {X}^{\ddagger }(E/\mathcal {F}_{\infty })$ for its Pontryagin-dual.

For practical purposes, it is convenient to work with an alternative description of $\operatorname {Sel}^{\ddagger }(E/F^{\operatorname {cyc}})$ . For $v\in \Sigma _{p}\backslash \Sigma _{\operatorname {ss}}(E)$ denote by $E_{v}$ the curve over the local field $F_{v}$ . Since $E_{v}$ has good ordinary or multiplicative reduction, it fits in a short exact sequence of $\operatorname {Gal}(\overline {F_{v}}/F_{v})$ -modules:

(2.3) $$ \begin{align} 0\rightarrow C_{v}\rightarrow E_{v}[p^{\infty}]\rightarrow D_{v}\rightarrow 0. \end{align} $$

Here, $C_{v}$ and $D_{v}$ are both of corank one over $\mathbb {Z}_{p}$ with the property that $C_{v}$ is a divisible subgroup and $D_{v}$ is the maximal subgroup on which $\operatorname {I}_{v}:=\operatorname {Gal}(\overline {F_{v}}/F_{v}^{\operatorname {nr}})$ acts via a finite order quotient. In fact, $D_{v}$ is specified as follows:

(2.4) $$ \begin{align}D_{v}=\begin{cases} \widetilde{E_{v}}[p^{\infty}] \quad \text{if }E_{v} \text{ has good ordinary reduction},\\ \mathbb{Q}_{p}/\mathbb{Z}_{p}(\phi) \quad \text{if }E_{v} \text{ has multiplicative reduction,} \end{cases}\end{align} $$

where $\phi $ is an unramified quadratic character, which is trivial if and only if $E_{v}$ has split multiplicative reduction. For further details, the reader is referred to the discussions in [Reference Coates and Greenberg2, p. 150] and [Reference Greenberg and Vatsal8, section 2].

Note that above each prime v of F, there are finitely many primes $\eta |v$ of $F^{\operatorname {cyc}}$ . For $\mathfrak {p}_{i}\in \Sigma _{\operatorname {ss}}(E)$ , set

$$ \begin{align*}\mathcal{H}_{\mathfrak{p}_{i}}^{\pm}(F^{\operatorname{cyc}}, E[p^{\infty}]):=\prod_{\eta|\mathfrak{p}_{i}} \frac{H^{1}(F_{\eta}^{\operatorname{cyc}}, E[p^{\infty}])}{\widehat{E}^{\pm}(F^{\operatorname{cyc}}_{\eta})\otimes \mathbb{Q}_{p}/\mathbb{Z}_{p}},\end{align*} $$

where $\eta $ runs through the primes of $F^{\operatorname {cyc}}$ above $\mathfrak {p}_{i}$ . For $v\in \Sigma \backslash \Sigma _{\operatorname {ss}}(E)$ , set

(2.5) $$ \begin{align} \mathcal{H}_{v}(F^{\operatorname{cyc}}, E[p^{\infty}]):=\begin{cases}\prod_{\eta|v} \operatorname{im}\left(H^{1}(F_{\eta}^{\operatorname{cyc}}, E[p^{\infty}])\rightarrow H^{1}(\operatorname{I}_{\eta}, D_{v})\right)\text{ if }v\in \Sigma_{p}\backslash \Sigma_{\operatorname{ss}}(E),\\ \prod_{\eta|v} H^{1}(F_{\eta}^{\operatorname{cyc}}, E[p^{\infty}])\text{ if }v\in \Sigma\backslash \Sigma_{p}, \end{cases} \end{align} $$

where $\operatorname {I}_{\eta }$ is the inertia subgroup of $\operatorname {Gal}(\overline {F_{\eta }^{\operatorname {cyc}}}/F_{\eta }^{\operatorname {cyc}})$ . The Selmer group $\operatorname {Sel}^{\ddagger }(E/F^{\operatorname {cyc}})$ coincides with the kernel of the restriction map

$$ \begin{align*} \Phi_{E,F^{\operatorname{cyc}}}^{\ddagger}:H^{1}(F^{\operatorname{cyc}},E[p^{\infty}])\rightarrow \prod_{v\in \Sigma\backslash \Sigma_{\operatorname{ss}}(E)} \mathcal{H}_{v}(F^{\operatorname{cyc}}, E[p^{\infty}])\times \prod_{i=1}^{d} \mathcal{H}_{\mathfrak{p}_{i}}^{\ddagger_{i}}(F^{\operatorname{cyc}}, E[p^{\infty}]). \end{align*} $$

For further details, see [Reference Greenberg and Vatsal8, pp. 32, 42] and the discussion in [Reference Greenberg6, Section 5]. Let $\Sigma _{0}$ be a finite set of primes of F which does not contain any prime $v\in \Sigma _{p}$ . The $\Sigma _{0}$ -imprimitive Selmer group $\operatorname {Sel}^{\Sigma _{0},\ddagger }(E/F^{\operatorname {cyc}})$ is the kernel of the restriction map

$$ \begin{align*} \Phi_{E,F^{\operatorname{cyc}}}^{\Sigma_{0},\ddagger}:H^{1}(F^{\operatorname{cyc}},E[p^{\infty}])\rightarrow\! \prod_{v\in \Sigma\backslash (\Sigma_{\operatorname{ss}}(E)\cup \Sigma_{0}\kern-1pt)} \!\mathcal{H}_{v}(F^{\operatorname{cyc}}, E[p^{\kern-1pt\infty}])\kern1pt{\times}\kern1pt \prod_{i=1}^{d} \mathcal{H}_{\mathfrak{p}_{i}}^{\ddagger_{i}}(F^{\operatorname{cyc}}, E[p^{\infty}]). \end{align*} $$

The Pontryagin dual of $\operatorname {Sel}^{\Sigma _{0}, \ddagger }(E/F^{\operatorname {cyc}})$ is denoted by $\operatorname {X}^{\Sigma _{0}, \ddagger }(E/F^{\operatorname {cyc}})$ . The next Proposition follows from [Reference Greenberg and Vatsal8, Proposition 2.1] and [Reference Lei and Lim21, Proposition 4.6].

Thus we have a short exact sequence relating the $\Sigma _{0}$ -imprimitive Selmer group $\operatorname {Sel}^{\Sigma _{0},\ddagger }(E/F^{\operatorname {cyc}})$ with the Selmer group $\operatorname {Sel}^{\ddagger }(E/F^{\operatorname {cyc}})$

$$ \begin{align*} 0\rightarrow \operatorname{Sel}^{\ddagger}(E/F^{\operatorname{cyc}})\rightarrow \operatorname{Sel}^{\Sigma_0,\ddagger}(E/F^{\operatorname{cyc}})\xrightarrow{\Phi_{E,F^{\operatorname{cyc}}}^{\ddagger, \Sigma_0}}\prod_{v\in \Sigma_0} \mathcal{H}_v(F^{\operatorname{cyc}},E[p^{\infty}])\rightarrow 0.\end{align*} $$

Choose a topological generator $\gamma $ of $\Gamma $ and identify $\gamma -1$ with T in choosing an isomorphism $\mathbb {Z}_{p}[[\Gamma ]]\simeq \mathbb {Z}_{p}[[T]]$ . A polynomial $f(T)\in \mathbb {Z}_{p}[T]$ is said to be distinguished if it is a monic polynomial and all nonleading coefficients are divisible by p. By the structure theorem of finitely generated torsion $\mathbb {Z}_{p}[[T]]$ -modules, there is a pseudo-isomorphism

$$ \begin{align*}\operatorname{X}^{\ddagger}(E/F^{cyc})\sim \left(\bigoplus_{i=1}^{n} \mathbb{Z}_{p}[[T]]/(f_{i}(T))\right)\oplus \left(\bigoplus_{j=1}^{m} \mathbb{Z}_{p}[[T]]/(p^{\mu_{j}}) \right).\end{align*} $$

For, $i=1,\dots , n$ , the elements $f_{i}(T)$ above are distinguished polynomials and the product $f_{E}^{\ddagger }(T):=p^{\sum _{j} \mu _{j}} \prod _{i} f_{i}(T)$ is called the characteristic polynomial. The signed Iwasawa invariants are defined by $\lambda _{E}^{\ddagger }:=\operatorname {deg} f_{E}^{\ddagger }(T)$ and $\mu _{E}^{\ddagger }:=\sum _{j=1}^{m} \mu _{j}$ . Denote by $f_{E}^{\Sigma _{0},\ddagger }(T)$ , $\lambda _{E}^{\Sigma _{0},\ddagger }$ and $\mu _{E}^{\Sigma _{0},\ddagger }$ the characteristic polynomial, $\lambda $ -invariant and $\mu $ -invariant of the imprimitive Selmer group $\operatorname {X}^{\Sigma _{0},\ddagger }(E/F^{\operatorname {cyc}})$ .

3 The truncated Euler characteristic

In this section, we recall the notion of the truncated Euler characteristic, which is a generalization of the usual Euler characteristic. We then discuss explicit formulas for the truncated Euler characteristic, as predicted by the p-adic Birch and Swinnerton-Dyer conjecture. For a more detailed exposition, the reader may refer to [Reference Coates, Schneider and Sujatha3, Section 3] and [Reference Zerbes37]. For a discrete p-primary cofinitely generated $\mathbb {Z}_{p}[[\Gamma ]]$ -module M for which the cohomology groups $H^{i}(\Gamma , M)$ have finite order, the Euler characteristic is defined to be the quotient

$$ \begin{align*} \chi(\Gamma, M):=\frac{\# H^{0}(\Gamma, M)}{\# H^{1}(\Gamma, M)}. \end{align*} $$

For an elliptic curve E with potentially good ordinary reduction at all primes above p, the Euler characteristic of the Selmer group is defined, provided the Selmer group of E over the base field is finite. This definition does not extend to the case when the Selmer group of E is infinite. The natural substitute is the truncated Euler characteristic.

Definition 3.1 Let M be a discrete p-primary $\Gamma $ -module let $\phi _{M}$ be the natural map

$$ \begin{align*}\phi_{M}: H^{0}\left(\Gamma, M\right)=M^{\Gamma}\rightarrow M_{\Gamma}\simeq H^{1}\left(\Gamma,M\right)\end{align*} $$

for which $\phi _{M}(x)$ is the residue class of x in $M_{\Gamma }$ . The truncated Euler characteristic $\chi _{t}\left (\Gamma , M\right )$ is defined if both $\operatorname {ker}(\phi _{M})$ and $\operatorname {cok}(\phi _{M})$ are finite. In this case, $\chi _{t}(\Gamma , M)$ is defined by

$$ \begin{align*}\chi_{t}\left(\Gamma, M\right):=\frac{\#\operatorname{ker}(\phi_{M})}{\#\operatorname{cok}(\phi_{M})}.\end{align*} $$

For a discrete $\mathbb {Z}_{p}[[\Gamma ]]$ -module M, set $f_{M}(T)$ to be the characteristic polynomial of the Pontryagin dual of M and write $f_{M}(T)= T^{r_{M}} g_{M}(T)$ , where $g_{M}(0)\neq 0$ . In other words, $r_{M}$ is the order of vanishing of $f_{M}(T)$ at $T=0$ .

Proof Let X be the Pontryagin dual of M. The invariant submodule $M^{\Gamma }$ is dual to $X_{\Gamma }$ and the quotient $M_{\Gamma }$ is dual to $X^{\Gamma }$ . By an application of the structure theorem for finitely generated torsion $\mathbb {Z}_{p}[[\Gamma ]]$ -modules, there is a pseudoisomorphism

$$ \begin{align*}X\sim\bigoplus_{i=1}^{m} \mathbb{Z}_{p}[[T]]/(f_{i}(T))\end{align*} $$

for some elements $f_{i}(T)\in \mathbb {Z}_{p}[[\Gamma ]]$ . The module $X_{\Gamma }$ may be identified with $X/TX$ and $X^{\Gamma }$ is the kernel of the multiplication by T map $\times T: X\rightarrow X$ . It is easy to see that the groups $X_{\Gamma }$ and $X^{\Gamma }$ are finite precisely when $T\nmid f_{i}(T)$ for $i=1,\dots , m$ . Therefore, $X_{\Gamma }$ and $X^{\Gamma }$ are finite precisely when $r_{M}=0$ .▪

The following lemma is a criterion for the truncated Euler characteristic to be well defined.

Let $|\cdot |_{p}$ denote the absolute value on $\mathbb {Q}_{p}$ normalized by $|p|_{p}=p^{-1}$ . When both $\operatorname {ker}(\phi _{M})$ and $\operatorname {cok}(\phi _{M})$ are finite, the truncated Euler characteristic $\chi _{t}(\Gamma , M)$ is related to the quantity $|g_{M}(0)|_{p}$ .

Lemma 3.4 Let M be a discrete $\mathbb {Z}_{p}[[\Gamma ]]$ module which is cofinitely generated and cotorsion. If the kernel and cokernel of $\phi _{M}$ are finite, then

$$ \begin{align*} \chi_{t}(\Gamma, M)=|g_{M}(0)|_{p}^{-1},\end{align*} $$

and further, $r_{M}=\operatorname {cork}_{\mathbb {Z}_{p}} M^{\Gamma }=\operatorname {cork}_{\mathbb {Z}_{p}} M_{\Gamma }$ .

Evidently, it follows that $\chi _{t}(\Gamma , M)=p^{N}$ , where $N\in \mathbb {Z}_{\geq 0}$ . Let $\mu _{M}$ (resp. $\lambda _{M}$ ) denote its $\mu $ -invariant (resp. $\lambda $ -invariant) of $M^{\vee }$ as a $\mathbb {Z}_{p}[[\Gamma ]]$ -module.

Proof Suppose that $\chi _{t}(\Gamma , M)=1$ . Recall that $g_{M}(T)$ is a polynomial such that $f_{M}(T)=T^{r_{M}} g_{M}(T)$ and $T\nmid g_{M}(T)$ . By Lemma 3.4,

$$ \begin{align*} |g_{M}(0)|_{p}^{-1}=\chi_{t}(\Gamma, M)=1. \end{align*} $$

As a result, $f_{M}(T)$ and $g_{M}(T)$ are distinguished polynomials. Since $g_{M}(0)$ is a unit, it follows that $g_{M}(T)$ is a unit. Since $g_{M}(T)$ is a distinguished polynomial, it follows that

$$ \begin{align*} g_{M}(T)=1\text{ and }f_{M}(T)=T^{r_{M}}. \end{align*} $$

As a result, $\mu _{M}=1$ and $\lambda _{M}=\operatorname {deg} f_{M}(T)=r_{M}$ .

Conversely, suppose that $\mu _{M}=0$ and $\lambda _{M}=r_{M}$ . Since $\mu _{M}=0$ , it follows that $f_{M}(T)$ and $g_{M}(T)$ are distinguished polynomials. The degree of $f_{M}(T)$ is $\lambda _{M}=r_{M}$ , it follows that $g_{M}(T)$ is a constant polynomial and hence, $g_{M}(T)=1$ . By Lemma 3.4,

$$ \begin{align*} \chi_{t}(\Gamma, M)=|g_{M}(0)|_{p}^{-1}=1. \\[-34pt]\end{align*} $$

▪

Let $r_{E}^{\ddagger }$ denote the order of vanishing of $f_{E}^{\ddagger }(T)$ at $T=0$ , and write

$$ \begin{align*}f_{E}^{\ddagger}(T)=p^{r_{E}^{\ddagger}} g_{E}^{\ddagger}(T).\end{align*} $$

Note that $g_{E}^{\ddagger }(0)\neq 0$ . According to Lemma 3.4, the truncated Euler characteristic $\chi _{t}^{\ddagger }(\Gamma , E):=\chi _{t}(\Gamma , \operatorname {Sel}^{\ddagger }(E/F^{\operatorname {cyc}}))$ is determined by the constant term of $g_{E}^{\ddagger }(T)$ . By Lemma 3.5, if the truncated Euler characteristic is defined (in the sense of Definition 3.1), then $\chi _{t}^{\ddagger }(\Gamma , E)=1$ if and only if $\mu _{E}^{\ddagger }=0$ and $\lambda _{E}^{\ddagger }=r_{E}^{\ddagger }$ .

We next discuss the p-adic Birch and Swinnerton-Dyer conjecture and its relationship with explicit formulas for truncated Euler characteristics. Note that there are formulations of the p-adic Birch and Swinnerton-Dyer conjecture in very general contexts (see for instance [Reference Rubin31, p. 6]). For ease of exposition, we restrict ourselves to the case where the elliptic curves E are defined over $\mathbb {Q}$ . For elliptic curves with good ordinary or multiplicative reduction, the p-adic Birch and Swinnerton-Dyer conjecture in its current form was formulated by Mazur et al. [Reference Mazur, Tate and Teitelbaum24, p. 38]. This is a p-adic analog of the classical Birch and Swinnerton-Dyer conjecture which predicts the order of vanishing of the Mazur and Swinnerton-Dyer p-adic L-function $\mathcal {L}(E/\mathbb {Q},T)$ at $T=0$ , and postulates an explicit formula for the leading term (see also [Reference Bernardi and Perrin-Riou1]). When E has good supersingular reduction at p, a version of the p-adic Birch and Swinnerton-Dyer conjecture for signed p-adic L-functions was formulated by Sprung [Reference Sprung35]. The conjecture of loc. cit. is equivalent to that of Bernardi and Perrin-Riou [Reference Bernardi and Perrin-Riou1]. Lemma 3.4 asserts that the truncated Euler-characteristic is related to the leading coefficient of the characteristic element of the Selmer group. The main conjecture and the p-adic Birch and Swinnerton-Dyer conjecture together predict precise formulas for the truncated Euler characteristic.

Assume that E has either good ordinary or multiplicative reduction at p. When E has split multiplicative reduction at p, set $\mathcal {L}_{p}(E)$ to denote the $\mathcal {L}$ -invariant associated to the Galois representation on the p-adic Tate module of E (see [Reference Greenberg and Vatsal8, p. 407]). The p-adic height pairing (cf. [Reference Schneider32] and [Reference Schneider33]) is a p-adic analog of the usual height pairing. This pairing is conjectured to be non-degenerate (cf. [Reference Schneider33]) and the p-adic regulator $R_{p}(E/\mathbb {Q})$ is defined to be the determinant of this pairing. Let $\kappa $ denote the p-adic cyclotomic character. Fix a branch of the p-adic logarithm and set $\mathcal {R}_{\gamma }(E/\mathbb {Q})$ to denote the normalized height pairing $(\log _{p}\kappa (\gamma ))^{-r} R_{p}(E/\mathbb {Q})$ , where r denotes the rank of the Mordell Weil group $E(\mathbb {Q})$ . Let $E_{0}(\mathbb {Q}_{l})\subset E(\mathbb {Q}_{l})$ be the subgroup of l-adic points with nonsingular reduction modulo l. Denote by $\mathcal {\tau }(E)$ the Tamagawa product $\prod _{l} c_{l}$ , where $c_{l}$ is the index of $E_{0}(\mathbb {Q}_{l})$ in $E(\mathbb {Q}_{l})$ . Let $a_{l}(E)$ be the lth coefficient of the normalized eigenform associated to E. For p-adic numbers a and b, write $a\sim b$ if $a=ub$ for a p-adic unit u. Let $r^{\prime }$ be the order of vanishing at $T=0$ of the characteristic element of the Selmer group $\operatorname {Sel}(E/\mathbb {Q}_{\operatorname {cyc}})$ . The following result does not assume the main conjecture.

4 Iwasawa invariants of congruent elliptic curves

In this section, we show that the imprimitive Iwasawa-invariants associated to congruent elliptic curves satisfy certain relations. Throughout, E is an elliptic curve over F which satisfies Hypothesis (2.1). Denote by $T_{p}(E):=\varprojlim _{n} E[p^{n}]$ the p-adic Tate-module equipped with natural $\operatorname {Gal}(\overline {F}/F)$ action and set $V_{p}(E):=T_{p} (E)\otimes \mathbb {Q}_{p}$ . At a prime $v\in \Sigma _{p}\backslash \Sigma _{\operatorname {ss}}(E)$ , recall that there are corank one $\mathbb {Z}_{p}$ -modules which fit into a short exact sequence

$$ \begin{align*} 0\rightarrow C_v\rightarrow E_v[p^{\infty}]\rightarrow D_v\rightarrow 0. \end{align*} $$

When E has good ordinary reduction at v, the quotient $D_{v}$ may be identified with $\widetilde {E}_{v}[p^{\infty }]$ , where $\widetilde {E}_{v}$ is the reduction of E at v. On the other hand, when E has multiplicative reduction at v, identify $D_{v}$ with a twist of $\mathbb {Q}_{p}/\mathbb {Z}_{p}$ by an unramified quadratic character.

Let $E_{1}$ and $E_{2}$ be p-congruent elliptic curves over F. It follows from the proof of [Reference Nuccio and Sujatha25, Proposition 3.9] that $\Sigma _{\operatorname {ss}}(E_{1})$ is equal to $\Sigma _{\operatorname {ss}}(E_{2})$ . Set $\Sigma _{\operatorname {ss}}=\{\mathfrak {p}_{1},\dots , \mathfrak {p}_{d}\}$ to denote the set of supersingular primes $v|p$ of $E_{1}$ and $E_{2}$ , and let $\ddagger \in \{+,-\}^{d}$ be a signed vector. We introduce the following hypothesis on the triple $(E_{1},E_{2},\ddagger )$ .

We now define the imprimitive Selmer group associated to the residual Galois representation $E_{i}[p]$ . Let $\Sigma $ be a set of finite primes of F containing $\Sigma _{p}$ and the primes at which $E_{1}$ or $E_{2}$ has bad reduction. For a prime $v\in \Sigma _{p}$ let $\eta _{v}$ be the unique prime of $F^{\operatorname {cyc}}$ above v. Let $\operatorname {I}_{\eta _{v}}$ be the inertia subgroup of $\operatorname {Gal}(\overline {F_{\eta _{v}}^{\operatorname {cyc}}}/F_{\eta _{v}}^{\operatorname {cyc}})$ . For a prime $v\in \Sigma _{\operatorname {ss}}$ , and $i=1,2$ , define

$$ \begin{align*} \mathcal{H}_{v}^{\pm}(F^{\operatorname{cyc}},E_{i}[p]):=\frac{H^{1}(F^{\operatorname{cyc}}_{\eta_{v}}, E_{i}[p])}{\widehat{E}_{i}^{\pm}(F^{\operatorname{cyc}}_{\eta_{v}})/p\widehat{E}_{i}^{\pm}(F^{\operatorname{cyc}}_{\eta_{v}})}. \end{align*} $$

For $v\in \Sigma \backslash \Sigma _{\operatorname {ss}}$ , set

$$ \begin{align*}\mathcal{H}_{v}(F^{\operatorname{cyc}},E_{i}[p]):=\begin{cases}\prod_{\eta| v} H^{1}(F^{\operatorname{cyc}}_{\eta}, E_{i}[p])\quad \text{if }v\in \Sigma\backslash \Sigma_{p},\\ H^{1}(\operatorname{I}_{\eta_{v}}, D_{v}(E_{i})[p])\quad \text{if }v\in \Sigma_{p}\backslash \Sigma_{\operatorname{ss}}. \end{cases}\end{align*} $$

For an elliptic curve E over F, denote by $\mathcal {N}_{E}$ the conductor of E and $\overline {\mathcal {N}}_{E}$ the prime to p part of the Artin conductor of the residual representation $E[p]$ . Let $v\nmid p$ be a finite prime of F. Note that v divides $\mathcal {N}_{E}$ (resp. $\overline {\mathcal {N}}_{E}$ ) is and only if it is a bad reduction prime of E (resp. the residual Galois representation $E[p]$ is ramified at v). To ease notation, let $\mathcal {N}_{1}$ and $\mathcal {N}_{2}$ denote the conductors of $E_{1}$ and $E_{2}$ , respectively. Denote by $\overline {\mathcal {N}}$ the prime to p part of the Artin conductor of $E_{1}[p]$ . Note that since $E_{1}[p]$ is isomorphic to $E_{2}[p]$ , $\overline {\mathcal {N}}$ is the conductor of $E_{2}[p]$ .

The $\Sigma _{1}$ -imprimitive mod-p Selmer group $\operatorname {Sel}^{\Sigma _{1},\ddagger }(E_{i}[p]/F^{\operatorname {cyc}})$ is defined to be the kernel of the restriction map:

$$ \begin{align*}\overline{\Phi}_{E_{i}}^{\Sigma_{1},\ddagger}:H^{1}(F_{\Sigma}/F^{\operatorname{cyc}}, E_{i}[p])\rightarrow \prod_{\Sigma/(\Sigma_{\operatorname{ss}}\cup \Sigma_{1})} \mathcal{H}_{v}(F^{\operatorname{cyc}},E_{i}[p])\times \prod_{j=1}^{d}\mathcal{H}_{\mathfrak{p}_{j}}^{\ddagger_{j}}(F^{\operatorname{cyc}},E_{i}[p]).\end{align*} $$

Proof Let $\Phi :E_{1}[p]\xrightarrow {\sim } E_{2}[p]$ be a choice of isomorphism of Galois modules. Clearly, $\Phi $ induces an isomorphism $H^{1}(F_{\Sigma }/F^{\operatorname {cyc}}, E_{1}[p])\xrightarrow {\sim } H^{1}(F_{\Sigma }/F^{\operatorname {cyc}}, E_{2}[p])$ . It suffices to show that for $v\in \Sigma $ , the isomorphism $\Phi :E_{1}[p]\xrightarrow {\sim } E_{2}[p]$ induces an isomorphism

$$ \begin{align*}\mathcal{H}_{v}(F^{\operatorname{cyc}},E_{1}[p])\xrightarrow{\sim} \mathcal{H}_{v}(F^{\operatorname{cyc}},E_{2}[p]).\end{align*} $$

This is clear for $v\in \Sigma \backslash \Sigma _{\operatorname {ss}}$ . For $v\in \Sigma _{\operatorname {ss}}$ , this assertion follows from the arguments in [Reference Kim13, p. 186].▪

Proposition 4.3 Let E be an elliptic curve over F satisfying Hypothesis (2.1) and $\ddagger $ a signed vector. Let $\Sigma _{0}$ be a finite set of primes $v\nmid p$ containing $\Sigma _{1}(E)$ . Then, there is an isomorphism

$$ \begin{align*}\operatorname{Sel}^{\Sigma_{0}, \ddagger}(E/F^{\operatorname{cyc}})[p]\simeq \operatorname{Sel}^{\Sigma_{0}, \ddagger}(E[p]/F^{\operatorname{cyc}}).\end{align*} $$

Proof Let $\Sigma $ be a finite set of primes containing $\Sigma _{0}$ , the primes at which E has bad reduction and $\Sigma _{p}$ . We consider the diagram relating the two Selmer groups:

Since $\Gamma $ is pro-p and $E[p]$ is an irreducible Galois module, clearly

$$ \begin{align*} H^{0}(F,E[p])=H^{0}(F^{\operatorname{cyc}},E[p])^{\Gamma}=0. \end{align*} $$

Hence, we deduce that $H^{0}(F^{\operatorname {cyc}}, E[p^{\infty }])=0$ and have shown that g is an isomorphism.

It only remains to show that h is injective. For $v\in \Sigma /(\Sigma _{\operatorname {ss}}(E)\cup \Sigma _{0})$ denote by $h_{v}$ the natural map

$$ \begin{align*} h_{v}: \mathcal{H}_{v}(F^{\operatorname{cyc}},E[p])\rightarrow \mathcal{H}_{v}(F^{\operatorname{cyc}},E[p^{\infty}]) \end{align*} $$

and for $v\in \Sigma _{\operatorname {ss}}(E)$ ,

$$ \begin{align*} h_{v}: \mathcal{H}_{v}^{\dagger_{v}}(F^{\operatorname{cyc}},E[p])\rightarrow \mathcal{H}_{v}^{\dagger_{v}}(F^{\operatorname{cyc}},E[p^{\infty}]). \end{align*} $$

We show that the maps $h_{v}$ are injective for $v\in \Sigma \backslash \Sigma _{0}$ . This has been shown in the proof of [Reference Greenberg and Vatsal8, Proposition 2.8] for $v\in \Sigma _{p}\backslash \Sigma _{\operatorname {ss}}$ and in [Reference Kim13, Proposition 2.10] for $v\in \Sigma _{\operatorname {ss}}$ . Therefore, it remains to consider primes $v\in \Sigma \backslash (\Sigma _{0}\cup \Sigma _{p})$ .

First consider the case when $p\geq 5$ . Recall that $\Sigma _{1}(E)$ consists of primes of F such that $(\mathrm {i}) \ v\nmid p$ , $(\mathrm {ii}) \ v|(\mathcal {N}_{E}/\overline {\mathcal {N}}_{E})$ , and $(\mathrm {iii})$ if $\mu _{p}$ is contained in $F_{v}$ , then E has split multiplicative reduction at v. Since $v\notin \Sigma _{0}$ , one of the above conditions is not satisfied. Since $v\notin \Sigma _{p}$ , $(\mathrm {i})$ is satisfied. Hence, there are two cases to consider:

1. (1) $(\mathrm {ii})$ is not satisfied and

2. (2) $(\mathrm {ii})$ is satisfied, but $(\mathrm {iii})$ is not.

First consider the case when $(\mathrm {ii})$ is not satisfied, i.e., $v\nmid (\mathcal {N}_{E}/\overline {\mathcal {N}}_{E})$ . In this case, the injectivity of $h_{v}$ follows from the proof of [Reference Emerton, Pollack and Weston5, Lemma 4.1.2]. Next, consider the case when $(\mathrm {ii})$ is satisfied, but $(\mathrm {iii})$ is not. Since $(\mathrm {ii})$ is satisfied, $v\mid \mathcal {N}_{E}$ , hence, E has bad reduction at v. Since $(\mathrm {iii})$ is not satisfied, it follows that $\mu _{p}$ is contained in $F_{v}$ and E has either non-split multiplicative reduction or additive reduction at v. In this case, the injectivity of $h_{v}$ follows from [Reference Hachimori and Matsuno9, Proposition 5.1]. When $p=3$ , the injectivity of $h_{v}$ follows from the same reasoning as above.▪

Proof We adapt the proof of [Reference Greenberg7, Proposition 4.14], which is due to Greenberg. Let $\Sigma $ be a finite set of primes containing $\Sigma _{0}\cup \Sigma _{p}$ and the primes of E at which E has bad reduction. Consider the $\Sigma _{0}\cup \Sigma _{p}$ -strict and relaxed Selmer groups:

$$ \begin{align*} \operatorname{Sel}^{\operatorname{rel}}(E/F^{\operatorname{cyc}}):=\operatorname{ker}\left\{H^{1} (F_{\Sigma}/F^{\operatorname{cyc}},E[p^{\infty}])\rightarrow \prod_{v\in \Sigma\backslash (\Sigma_{0}\cup \Sigma_{p})} \mathcal{H}_{v}(F^{\operatorname{cyc}}, E[p^{\infty}])\right\}, \end{align*} $$

$$ \begin{align*} \operatorname{Sel}^{\operatorname{str}}(E/F^{\operatorname{cyc}}):= \operatorname{ker}\left\{\operatorname{Sel}^{\operatorname{rel}}(E/F^{\operatorname{cyc}})\rightarrow \prod_{v\in \Sigma_{0}\cup \Sigma_{p}} \prod_{\eta|v} H^{1}(F_{\eta}^{\operatorname{cyc}}, E[p^{\infty}])\right\}. \end{align*} $$

By Proposition 2.3, it follows that

$$ \begin{align*} \operatorname{Sel}^{\operatorname{rel}}(E/F^{\operatorname{cyc}})/\operatorname{Sel}^{\Sigma_{0}, \ddagger} (E/F^{\operatorname{cyc}})\simeq \prod_{i=1}^{d} \mathcal{H}_{\mathfrak{p}_{i}}^{\ddagger_{i}}(F^{\operatorname{cyc}}, E[p^{\infty}]). \end{align*} $$

For $i=1,\dots , d$ , it follows from standard arguments (see the proof of [Reference Kim13, Proposition 2.11]) that $ \mathcal {H}_{\mathfrak {p}_{i}}^{\ddagger _{i}}(F^{\operatorname {cyc}}, E[p^{\infty }])^{\vee }$ is isomorphic to $\mathbb {Z}_{p}[[\Gamma ]]$ . By [Reference Greenberg and Vatsal8, Lemma 2.6], it suffices to show that $\operatorname {Sel}^{\operatorname {rel}} (E/F^{\operatorname {cyc}})$ has no proper finite index $\mathbb {Z}_{p}[[\Gamma ]]$ -submodules. Since $\operatorname {Sel}^{\operatorname {str}}(E/F^{\operatorname {cyc}})^{\vee }$ is a quotient of $\operatorname {Sel}^{\Sigma _{0}, \ddagger } (E/F^{\operatorname {cyc}})^{\vee }$ , it is $\mathbb {Z}_{p}[[\Gamma ]]$ -torsion.

Recall that $\kappa $ denotes the p-adic cyclotomic character. For $s\in \mathbb {Z}$ , let $A_{s}$ denote the twisted Galois module $E[p^{\infty }]\otimes \kappa ^{s} $ . Since $E(F)[p]=0$ and $\Gamma $ is pro-p, it follows that $E(F^{\operatorname {cyc}})[p]=0$ , and as a result, $H^{0}(F^{\operatorname {cyc}}, A_{s})=0$ . Since $\Gamma $ is pro-p it follows from standard arguments that $H^{0}(F, A_{s})=0$ for all s. For a subfield K of $F^{\operatorname {cyc}}$ , and v a prime of F which does not divide p, set $\mathcal {H}_{v}(K, A_{s})$ to be the product $\prod _{\eta |v} H^{1}(K_{\eta }, A_{s})/(A_{s}(K_{\eta })\otimes \mathbb {Q}_{p}/\mathbb {Z}_{p})$ , where $\eta $ ranges the finitely many primes of K above v. Set $P^{\Sigma , \operatorname {rel}}(K, A_{s})$ to be the product

$$ \begin{align*} P^{\Sigma, \operatorname{rel}}(K, A_{s}):=\prod_{v\in \Sigma\backslash (\Sigma_{0}\cup \Sigma_{p})} \mathcal{H}_{v}(K, A_{s}) \end{align*} $$

and set $P^{\Sigma , \operatorname {str}}(K, A_{s})$ to be the product

$$ \begin{align*} P^{\Sigma, \operatorname{str}}(K, A_{s}):=\prod_{v\in \Sigma\backslash (\Sigma_{0}\cup \Sigma_{p})} \mathcal{H}_{v}(K, A_{s})\times \prod_{v\in \Sigma_{0}\cup \Sigma_{p}} H^{1}(K, A_{s}). \end{align*} $$

Let $S_{A_{s}}^{\operatorname {rel}}(K)$ and $S_{A_{s}}^{\operatorname {str}}(K)$ be the Selmer groups defined as follows

$$ \begin{align*} S_{A_{s}}^{\operatorname{rel}}(K):=\ker\left(H^{1}(F_{\Sigma}/F, A_{s})\rightarrow P^{\Sigma, \operatorname{rel}}(K, A_{s})\right)\kern-1.5pt, \end{align*} $$

$$ \begin{align*} S_{A_{s}}^{\operatorname{str}}(K):=\ker\left(H^{1}(F_{\Sigma}/F, A_{s})\rightarrow P^{\Sigma, \operatorname{str}}(K, A_{s})\right)\kern-1.5pt. \end{align*} $$

Since $\operatorname {Sel}^{\operatorname {str}}(E/F^{\operatorname {cyc}})$ is $\mathbb {Z}_{p}[[\Gamma ]]$ -cotorsion, we have that $S_{A_{s}}^{\operatorname {str}}(F^{\operatorname {cyc}})^{\Gamma }$ is finite for all but finitely many values of s. Hence, $S_{A_{s}}^{\operatorname {str}}(F)$ is finite for all but finitely many values of s. We set $M=A_{s}$ , and in accordance with the proof of [Reference Greenberg7, Proposition 4.14], $M^{*}=A_{-s}$ . Denote by $S_{M}(F)$ the Selmer group defined by the relaxed conditions $S_{A_{s}}^{\operatorname {rel}}(F)$ and in accordance with the discussion on [Reference Greenberg7, p. 100], $S_{M^{*}}(F)$ is the strict Selmer group $S_{A_{-s}}^{\operatorname {str}}(F)$ . Let s be such that $S_{M^{*}}(F)$ is finite. Since $S_{M^{*}}(F)$ is finite and $M^{*}(F)=0$ , it follows that the map $H^{1}(F_{\Sigma }/F, M)\rightarrow P^{\Sigma , \operatorname {rel}}(K, M)$ is surjective (see [Reference Greenberg7, Proposition 4.13]). It follows from the proof of [Reference Greenberg7, Proposition 4.14] that $\operatorname {Sel}^{\operatorname {rel}} (E/F^{\operatorname {cyc}})$ has no proper finite index $\mathbb {Z}_{p}[[\Gamma ]]$ -submodules. This completes the proof.▪

Recall that for $i=1,2$ , the $\mu $ -invariant (resp. $\lambda $ -invariant) of the Selmer group $\operatorname {Sel}^{\Sigma _{1},\ddagger }(E_{i}/F^{\operatorname {cyc}})$ is denoted by $\mu _{E_{i}}^{\Sigma _{1},\ddagger }$ (resp. $\lambda _{E_{i}}^{\Sigma _{1},\ddagger }$ ).

5 Congruences for Euler characteristics

Consider elliptic curves $E_{1}$ and $E_{2}$ that are p-congruent and let $\Sigma ^{\operatorname {ss}}=\{\mathfrak {p}_{1},\dots , \mathfrak {p}_{d}\}$ be the set of supersingular primes v of $E_{1}$ such that $v|p$ . As noted earlier, these are also the supersingular primes v of $E_{2}$ such that $v|p$ . Let $\ddagger \in \{+,-\}^{d}$ be a signed vector and assume that Hypothesis (4.2) is satisfied for the triple $(E_{1}, E_{2}, \ddagger )$ . Associated to $E_{1}$ and $E_{2}$ is the set of primes $\Sigma _{1}$ , see Definition 4.1. In this section, it is shown that there is an explicit relationship between the multisigned Euler characteristics $\chi _{t}^{\ddagger }(\Gamma , E_{1})$ and $\chi _{t}^{\ddagger }(\Gamma , E_{2})$ .

Let E be an elliptic curve satisfying Hypothesis (2.1) and $\Sigma _{0}$ a finite set of primes $v\nmid p$ . Recall that $r_{E}^{\ddagger }$ is the order of vanishing of the characteristic polynomial $f_{E}^{\ddagger }(T)$ at $T=0$ . The following Proposition shows that the quantity $r_{E}^{\ddagger }$ is related to the Mordell Weil rank of E and the reduction type of E at the primes $v|p$ . The proof of the following result is based on an adaptation of the control theorem from the ordinary setting, see [Reference Mazur23].

Proof Let $\Sigma $ be a finite set of primes containing $\Sigma _{p}$ and the primes at which E has bad reduction. By Lemma 3.4, the multisigned rank $r_{E}^{\ddagger }$ is equal to the $\mathbb {Z}_{p}$ -corank of $\operatorname {Sel}^{\ddagger }(E/F^{\operatorname {cyc}})^{\Gamma }$ . We compare $\operatorname {Sel}^{\ddagger }(E/F^{\operatorname {cyc}})^{\Gamma }$ with the usual Selmer group $\operatorname {Sel}(E/F)$ . Let ${\Phi }_{E,F}: H^{1}(F_{\Sigma }/F, E[p^{\infty }])\rightarrow \prod _{v\in \Sigma } \mathcal {H}_{v}(F, E[p^{\infty }])$ be the defining map for the Selmer group $\operatorname {Sel}(E/F)$ . Consider the fundamental diagram

where the vertical maps are induced by restriction. Since $E(F)[p]=0$ and $\Gamma $ is pro-p, it follows that $H^{0}(F^{\operatorname {cyc}}, E[p^{\infty }])=0$ . It follows from the inflation-restriction sequence that g is an isomorphism. Thus, we have the following short exact sequence:

(5.1) $$ \begin{align} 0\rightarrow \operatorname{Sel}(E/F)\rightarrow \operatorname{Sel}^{ \ddagger}(E/F^{\operatorname{cyc}})^{\Gamma}\rightarrow \operatorname{ker}h\rightarrow 0. \end{align} $$

It suffices to show that the kernel of h has corank less than or equal to $\operatorname {sp}_{E}$ . For $v\in \Sigma \backslash \Sigma _{\operatorname {ss}}$ , let $h_{v}^{\prime }$ denote the natural map

$$ \begin{align*} h_{v}^{\prime}: \mathcal{H}_{v}(F, E[p^{\infty}])\rightarrow \mathcal{H}_{v}(F^{\operatorname{cyc}}, E[p^{\infty}]). \end{align*} $$

For $\mathfrak {p}_{i}\in \Sigma _{\operatorname {ss}}$ , set $h_{\mathfrak {p}_{i}}^{\prime }$ denotes the map

$$ \begin{align*} h_{\mathfrak{p}_{i}}^{\prime}: \mathcal{H}_{\mathfrak{p}_{i}}^{\ddagger_{i}}(F, E[p^{\infty}])\rightarrow \mathcal{H}_{\mathfrak{p}_{i}}^{\ddagger_{i}}(F^{\operatorname{cyc}}, E[p^{\infty}]). \end{align*} $$

Let $h^{\prime }$ be the product of the maps $h_{v}^{\prime }$ as v ranges over $\Sigma $ . Note that the kernel of h is contained in the kernel of $h^{\prime }$ . Let $\Sigma _{\operatorname {sp}}(E)$ be the set of primes $v|p$ at which E has split multiplicative reduction. We show that $\operatorname {ker} h_{v}^{\prime }$ is finite if $v\notin \Sigma _{\operatorname {sp}}(E)$ and of corank one for $v\in \Sigma _{\operatorname {sp}}(E)$ . For $v\in \Sigma \backslash \Sigma _{p}$ , this follows from [Reference Coates and Sujatha4, Lemma 3.4], for $v\in \Sigma _{p}$ at which E has good ordinary reduction, it follows from [Reference Coates and Sujatha4, Proposition 3.5]. Next consider the case where $v|p$ and E has multiplicative reduction at v.

The Galois representation of the local Galois group $\operatorname {G}_{F_{v}}$ on $E[p]$ is of the form ${ \left (\!\!\! {\begin {array}{cc} {\varphi _{v} \kappa } & {\ast } \\ {} & {\varphi _{v}^{-1}} \\ \end {array} } \!\!\!\right )}$ , where $\varphi _{v}$ is an unramified character. Let w be a prime of $F^{\operatorname {cyc}}$ above v and let $\Gamma _{v}$ denote $\operatorname {Gal}(F_{w}^{\operatorname {cyc}}/F_{v})$ . By Shapiro’s Lemma,

$$ \begin{align*} H^{0}\left(\Gamma, \prod_{\eta|v} H^{1}(F_{\eta}^{\operatorname{cyc}}, D_{v})\right)\simeq H^{0}(\Gamma_{v}, H^{1}(F_{w}^{\operatorname{cyc}}, D_{v})). \end{align*} $$

The kernel of $h_{v}^{\prime }$ is contained in the kernel of the restriction map

$$ \begin{align*} H^{1}(F_{v}, D_{v})\rightarrow H^{1}(F_{w}^{\operatorname{cyc}}, D_{v}) \end{align*} $$

and by the inflation restriction sequence, the kernel of $h_{v}^{\prime }$ is contained in $H^{1}(\Gamma _{v}, H^{0}(F_{w}^{\operatorname {cyc}},D_{v}))$ . When E has nonsplit multiplicative reduction at v, the character $\varphi _{v}$ is nontrivial and thus $H^{0}(F_{v}, D_{v})=0$ . Since $\Gamma _{v}$ is pro-p, it follows that $H^{0}(F_{w}^{\operatorname {cyc}},D_{v})=0$ when $\varphi _{v}\neq 1$ . On the other hand, when $\varphi _{v}=1$ , the Galois action on $H^{0}(F_{w}^{\operatorname {cyc}}, D_{v})\simeq \mathbb {Q}_{p}/\mathbb {Z}_{p}$ is trivial. As a result, the corank of $H^{1}(\Gamma _{v}, H^{0}(F_{w}^{\operatorname {cyc}}, D_{v}))$ is one when E has split multiplicative reduction at v.

Finally, consider the case when $v|p$ and E has supersingular reduction at v. In this case, $h_{v}^{\prime }$ is injective, as shown in the proof of [Reference Lei and Lim21, Theorem 5.3].

Putting all this together, we have that $\operatorname {corank}_{\mathbb {Z}_{p}} \operatorname {ker} h\leq \operatorname {sp}_{E}$ , and thus, from the short exact sequence (5.1), we have

Since it is assumed that

is finite, the result follows.▪

We now study congruences for truncated Euler characteristics. For $v\in \Sigma _{0}$ , set $h_{E}^{(v)}(T)$ to be the characteristic polynomial of the Pontryagin dual of $\mathcal {H}_{v}(F^{\operatorname {cyc}},E[p^{\infty }])$ . Since $ \operatorname {Sel}^{\ddagger }(E/F^{\operatorname {cyc}})$ is $\mathbb {Z}_{p}[[\Gamma ]]$ -cotorsion, it follows from Proposition 2.3 that there is a short exact sequence:

$$ \begin{align*} 0\rightarrow \operatorname{Sel}^{\ddagger}(E/F^{\operatorname{cyc}})\rightarrow \operatorname{Sel}^{\ddagger,\Sigma_0}(E/F^{\operatorname{cyc}})\rightarrow \prod_{v\in \Sigma_0} \mathcal{H}_v(F^{\operatorname{cyc}}, E[p^{\infty}])\rightarrow 0.\end{align*} $$

As a result, we arrive at the following relation:

(5.2) $$ \begin{align} f_{E}^{\Sigma_{0}, \ddagger}(T)=f_{E}^{\ddagger}(T)\prod_{l\in \Sigma_{0}} h_{E}^{(v)}(T). \end{align} $$

Recall that $V_{p}(E)$ is the p-adic vector space $T_{p}(E)\otimes \mathbb {Q}_{p}$ , equipped with $\operatorname {Gal}(\overline {F}/F)$ -action. Let $P_{v}(E,T)$ denote the characteristic polynomial

(5.3) $$ \begin{align} P_{v}(E,T):=\operatorname{det}\left((\operatorname{Id}-\operatorname{Frob}_{v} X)_{|V_{p}(E)^{\operatorname{I}_{v}}}\kern-2pt\right), \end{align} $$

where $\operatorname {I}_{v}$ is the inertia group at v. For $s\in \mathbb {C}$ , set $L_{v}(E,s)$ to denote $P_{v}(E,Nv^{-s})^{-1}$ , where $Nv$ denotes the norm $N_{L/\mathbb {Q}}v$ . Recall that $\gamma $ is a topological generator of $\Gamma $ . Let $\rho :\Gamma \rightarrow \mu _{p^{\infty }}$ be a finite order character and let $\sigma _{v}$ denote the Frobenius at v. Let $\mathcal {P}_{v}(E,T)$ be the element in $\mathbb {Z}_{p}[[\Gamma ]]$ defined by the relation $\mathcal {P}_{v}(E,\rho (\gamma )-1)=P_{v}(E,\rho (\sigma _{v}) Nv^{-1})$ . According to [Reference Greenberg and Vatsal8, Proposition 2.4], the polynomial $\mathcal {P}_{v}(E,T)$ generates the Pontryagin dual of $\mathcal {H}_{v}(F^{\operatorname {cyc}}, E[p^{\infty }])$ and thus coincides with $h^{(v)}(T)$ up to a unit in $\mathbb {Z}_{p}[[\Gamma ]]$ .

Write

$$ \begin{align*} f_{E}^{\Sigma_{0},\ddagger}(T)= T^{r_{E}^{\ddagger}} g_{E}^{\Sigma_{0},\ddagger}(T), \end{align*} $$

and set $|g_{E,\Sigma _{0}}(0)|_{p}^{-1}$ to be zero if $g_{E,\Sigma _{0}}(0)$ equals zero. It follows from the relation (5.2) that

(5.4) $$ \begin{align} g_{E}^{\Sigma_{0}, \ddagger}(T)=g_{E}^{\ddagger}(T)\prod_{l\in \Sigma_{0}} h_{E}^{(v)}(T). \end{align} $$

One has the following result.

Lemma 5.3 Let E be an elliptic curve satisfying Hypothesis (2.1) and let $\ddagger $ be a signed vector, for a finite set $\Sigma _{0}$ consisting of primes $v\nmid p$ . Then we have the following equality:

$$ \begin{align*} \delta_{E,\Sigma_{0}}\times \chi_{t}^{\ddagger}(\Gamma, E)=|g_{E}^{\Sigma_{0}, \ddagger}(0)|_{p}^{-1}. \end{align*} $$

Proof Note that $\mathcal {P}_{v}(E,0)$ equals $P_{v}(E,l^{-1})=L_{v}(E,1)^{-1}$ . Therefore, it follows that $h^{(v)}(0)$ coincides with $|L_{v}(E,1)|_{p}$ up to a unit in $\mathbb {Z}_{p}$ . From the relation (5.4), we have that

$$ \begin{align*} |g_{E}^{\Sigma_{0}, \ddagger}(0)|_{p}^{-1}= |g_{E}^{\ddagger}(0)|_{p}^{-1}\prod_{l\in \Sigma_{0}} |L_{v}(E,1)|_{p}=|g_{E}^{\ddagger}(0)|_{p}^{-1}\times \delta_{E,\Sigma_{0}}. \end{align*} $$

Lemma 3.4 asserts that $\chi _{t}^{\ddagger }(\Gamma , E)$ is equal to $|g_{E}^{\ddagger }(0)|_{p}^{-1}$ , and this completes the proof. ▪

Denote the $\mu $ -invariants of $\operatorname {Sel}^{\ddagger }(E/F^{\operatorname {cyc}})$ and $\operatorname {Sel}^{\Sigma _{0},\ddagger }(E/F^{\operatorname {cyc}})$ by $\mu _{E}^{ \ddagger }$ and $\mu _{E}^{\Sigma _{0}, \ddagger }$ , respectively. Denote the $\lambda $ -invariants by $\lambda _{E}^{ \ddagger }$ and $\lambda _{E}^{\Sigma _{0}, \ddagger }$ respectively. It is easy to show that for $v\in \Sigma _{0}$ , the $\mu $ -invariant of $h_{E}^{(v)}(T)$ is equal to zero, and hence $\mu _{E}^{\Sigma _{0}, \ddagger }$ is equal to $\mu _{E}^{\ddagger }$ .

Lemma 5.4 Let E be an elliptic curve satisfying Hypothesis (2.1) and let $\ddagger $ be a signed vector. For a finite set $\Sigma _{0}$ consisting of primes $v\nmid p$ , we have that

$$ \begin{align*} \delta_{E,\Sigma_{0}}\times \chi_{t}^{\ddagger}(\Gamma, E)=1\Leftrightarrow \mu_{E}^{\ddagger}=0 \text{ and }\lambda_{E}^{\Sigma_{0}, \ddagger}=r_{E}^{\ddagger}. \end{align*} $$

Proof First assume that $\delta _{E,\Sigma _{0}}\times \chi _{t}^{\ddagger }(\Gamma , E)=1$ , it is then shown that $\mu _{E}^{\ddagger }=0$ and $\lambda _{E}^{\Sigma _{0}, \ddagger }=r_{E}^{\ddagger }$ . Recall that $f_{E}^{\Sigma _{0},\ddagger }(T)$ is the characteristic polynomial for the Selmer group $\operatorname {Sel}^{\Sigma _{0},\ddagger }(E/F^{\operatorname {cyc}})$ and is written as $T^{r_{E}^{\ddagger }}g_{E}^{\Sigma _{0},\ddagger }(T)$ . It follows from Lemma 5.3 that $g_{E}^{\Sigma _{0},\ddagger }(0)$ is a unit in $\mathbb {Z}_{p}$ , and thus $g_{E}^{\Sigma _{0},\ddagger }(T)$ is a unit in $\mathbb {Z}_{p}[[\Gamma ]]$ . It follows from this that $\mu _{E,\Sigma _{0}}^{\ddagger }=0$ and $\lambda _{E,\Sigma _{0}}^{\ddagger }=r_{E}^{\ddagger }$ . The following relation is satisfied:

$$ \begin{align*} f_{E,\Sigma_{0}}^{\ddagger}(T)=f_{E}^{\ddagger}(T)\times \prod_{v\in \Sigma_{0}} h^{(v)}_{E}(T) \end{align*} $$

(see (5.2)). According to [Reference Greenberg and Vatsal8, Proposition 2.4], the polynomial $\mathcal {P}_{v}(E,T)$ generates the Pontryagin dual of $\mathcal {H}_{v}(F^{\operatorname {cyc}}, E[p^{\infty }])$ and thus coincides with $h^{(v)}(T)$ up to a unit in $\mathbb {Z}_{p}[[\Gamma ]]$ . It is easy to see that $\mathcal {P}_{v}(E,T)$ is equal to a monic polynomial, up to a unit in $\mathbb {Z}_{p}$ . As a result, p does not divide $h^{(v)}(T)$ . It follows that $\mu _{E}^{\ddagger }=\mu _{E,\Sigma _{0}}^{\ddagger }=0$ .

Conversely, suppose that $\mu _{E}^{\ddagger }=0$ and that $\lambda _{E,\Sigma _{0}}^{\ddagger }=r_{E}^{\ddagger }$ . The degree of $f_{E,\Sigma _{0}}^{\ddagger }(T)$ is equal to $\lambda _{E,\Sigma _{0}}^{\ddagger }=r_{E}^{\ddagger }$ . Since $f_{E,\Sigma _{0}}^{\ddagger }(T)$ is expressed as $T^{r_{E}^{\ddagger }} g_{E,\Sigma _{0}}(T)$ , the degree of $g_{E,\Sigma _{0}}(T)$ is equal to zero. It has been shown that $\mu _{E,\Sigma _{0}}^{\ddagger }=\mu _{E}^{\ddagger }=0$ . It follows that $g_{E,\Sigma _{0}}(T)$ is a unit in $\mathbb {Z}_{p}$ . Lemma 5.3 asserts that $\delta _{E,\Sigma _{0}}\times \chi _{t}^{\ddagger }(\Gamma , E)$ is equal to $|g_{E,\Sigma _{0}}(0)|_{p}^{-1}$ and therefore equal to $1$ .▪

Next, we come to the main theorem of the section. It is shown that the truncated Euler characteristic of $E_{1}$ is related to that of $E_{2}$ after one accounts for the auxiliary set of primes $\Sigma _{1}$ . The smaller the set of primes $\Sigma _{1}$ , the more refined the congruence relation between truncated Euler characteristics is. This is why it is of considerable importance that the set of primes $\Sigma _{1}$ be carefully chosen to be as small as possible. We note that in the statement of [Reference Ray and Sujatha30, Theorem 3.3], the elliptic curves $E_{1}$ and $E_{2}$ were defined over $\mathbb {Q}$ with good ordinary reduction at p. Furthermore, the set of auxiliary primes $\Sigma _{0}$ was the full set of primes $v\nmid p$ at which $E_{1}$ or $E_{2}$ has bad reduction. The set of primes $\Sigma _{1}$ is smaller and hence, when $F=\mathbb {Q}$ the result below refines the result [Reference Ray and Sujatha30, Theorem 3.3].

Consider the case when $r_{E_{1}}^{\ddagger }=r_{E_{2}}^{\ddagger }$ . It is natural to ask if $\chi _{t}^{\ddagger }(\Gamma ,E_{1})=1$ implies that $\chi _{t}^{\ddagger }(\Gamma , E_{2})=1$ ? This statement is false, as is shown in [Reference Ray and Sujatha30, Example 5.2], where it is shown that there are $5$ -congruent elliptic curves $E_{1}$ and $E_{2}$ over $\mathbb {Q}$ which both have Mordell-Weil rank $1$ and good ordinary reduction at $5$ , such that $\chi _{t}(\Gamma , E_{1})=1$ and $\chi _{t}(\Gamma , E_{2})=5^{2}$ . Therefore it is necessary to account the factors $\delta _{E_{i}, \Sigma _{1}}$ , i.e.,

$$ \begin{align*} \delta_{E_{1}, \Sigma_{1}}\times \chi_{t}(\Gamma, E_{1})=1\Leftrightarrow\delta_{E_{2}, \Sigma_{1}}\times \chi_{t}(\Gamma, E_{2})=1. \end{align*} $$

7 Examples

We discuss two concrete examples which illustrate our results for $p=5$ . Our computations are aided by Sage. The reader is referred to [Reference Ray and Sujatha30, Section 5] for examples when both elliptic curves are defined over $\mathbb {Q}$ and have good ordinary reduction at $p=5$ .

7.1 Example 1

Consider elliptic curves $E_{1}=66a1$ and $E_{2}=462d1$ . The elliptic curves $E_{1}$ and $E_{2}$ are $5$ -congruent and supersingular at $5$ . Both elliptic curves $E_{1}$ and $E_{2}$ have Mordell-Weil rank zero, hence the Euler characteristic $\chi (\Gamma , E_{i})$ is well defined for $i=1,2$ . Hypothesis (2.1) is satisfied for the elliptic curves $E_{1}$ and $E_{2}$ . Let $\ddagger \in \{+,-\}$ be a choice of sign. Assume that for $i=1,2$ , the Selmer group $\operatorname {Sel}^{\ddagger }(E_{i}/\mathbb {Q}^{\operatorname {cyc}})$ is $\mathbb {Z}_{p}[[\Gamma ]]$ -cotorsion. Therefore, Hypothesis (4.2) is satisfied. We show that $\delta _{E_{i}, \Sigma _{1}}=1$ and $\chi _{t}^{\ddagger }(\Gamma , E_{i})=1$ for $i=1,2$ . This demonstrates Theorem 5.5 which asserts that

$$ \begin{align*} \delta_{E_{1}, \Sigma_{1}}\times \chi_{t}^{\ddagger}(\Gamma, E_{1})=1\text{ if and only if }\delta_{E_{2}, \Sigma_{1}}\times \chi_{t}^{\ddagger}(\Gamma, E_{2})=1. \end{align*} $$

The conductor $N_{1}$ of $E_{1}$ (resp. $N_{2}$ of $E_{2}$ ) is $66=2\times 3\times 11$ (resp. $462=2\times 3\times 7\times 11$ ). The set $\Sigma _{0}$ consists of the primes $v\neq 5$ at which $E_{1}$ or $E_{2}$ has bad reduction. We have that $\Sigma _{0}=\{2,3,7,11\}$ . We calculate the optimal set of primes $\Sigma _{1}\subseteq \Sigma _{0}$ . Let $\overline {N}$ denote the prime to $5$ Artin conductor of the residual representation. Recall that $\Sigma _{1}$ is the set of primes $\Sigma (E_{1})\cup \Sigma (E_{2})$ , where $\Sigma (E_{i})$ consists of the primes $v\neq 5$ such that $v|(N_{i}/\overline {N})$ , and if $\mu _{5}$ is contained in $\mathbb {Q}_{v}$ , then E has split multiplicative reduction at v. Note that $\mathbb {Q}_{11}$ contains $\mu _{5}$ and both elliptic curves $E_{1}$ and $E_{2}$ have nonsplit multiplicative reduction at $11$ . Hence, $\Sigma _{1}\subseteq \{2,3,7\}$ is strictly smaller than $\Sigma _{0}=\{2,3,7,11\}$ . Recall that $\delta _{E,\Sigma _{1}}$ is the product $\prod _{v\in \Sigma _{1}} |L_{v}(E, 1)|_{5}$ . From the discussion in [Reference Ray and Sujatha30, p. 7], $L_{v}(E_{i},1)^{-1}$ is equal to $v^{-1}(l+\beta _{v}(E_{i})-a_{l}(E))$ , where $\beta _{v}(E_{i})$ is $1$ when $E_{i}$ has good reduction at v and is $0$ otherwise. These values are calculated from the following:

$$ \begin{align*} \begin{split} &2+\beta_{2}(E_{1})-a_{2}(E_{1})=3, 3+\beta_{3}(E_{1})-a_{3}(E_{1})=2, 7+\beta_{7}(E_{1})-a_{7}(E_{1})=6,\\ &2+\beta_{2}(E_{2})-a_{2}(E_{2})=3, 3+\beta_{3}(E_{2})-a_{3}(E_{2})=3, 7+\beta_{7}(E_{2})-a_{7}(E_{2})=8.\\ \end{split} \end{align*} $$

None of the above values are not divisible by $5$ and hence, $\delta _{E_{1},\Sigma _{1}}$ and $\delta _{E_{2}, \Sigma _{1}}$ are both $1$ . The Euler characteristics $\chi ^{\pm }(\Gamma , E_{i})$ may be calculated explicitly. The Euler characteristic $\chi (\Gamma , E_{i})$ is given up to $5$ -adic unit by

$$ \begin{align*} \chi^{\pm}(\Gamma, E_{i})\sim \# \operatorname{Sel}(E/\mathbb{Q}) \times \prod c_{v}(E_{i}), \end{align*} $$

see [Reference Kim14, Theorem 1.2]. For both elliptic curves, and $E_{i}(\mathbb {Q})$ has no $5$ -torsion. It follows that $\#\operatorname {Sel}(E/\mathbb {Q})\sim 1$ . Furthermore, the Tamagawa products $\prod c_{v}(E_{1})=6$ and $\prod c_{v}(E_{1})=16$ . It thus follows that both Euler characteristics $\chi ^{\pm }(\Gamma , E_{i})=1$ for $i=1,2$ . We have thus demonstrated the relationship between Euler characteristics via explicit computation.