Remarks on Kato's Euler systems for elliptic curves with additive reduction

Abstract

Extending the former work for the good reduction case, we provide a numerical criterion to verify a large portion of the “Iwasawa main conjecture without p-adic L-functions” for elliptic curves with additive reduction at an odd prime p over the cyclotomic ${\mathbb{Z}}_p$-extension. We also deduce the corresponding p-part of the Birch and Swinnerton-Dyer formula for elliptic curves of rank zero from the same numerical criterion. We give some explicit examples at the end and specify our choice of Kato's Euler system in the appendix.

Introduction

This article is a generalization of the numerical criterion for the verification of the Iwasawa main conjecture for modular forms at a good prime [KKS] to the additive reduction case. For the multiplicative reduction case, the main conjecture follows from the good ordinary case and the use of Hida theory ([Ski16]). This criterion also has an application to the p-part of the Birch and Swinnerton-Dyer (BSD) formula for elliptic curves of rank zero (cf. [JSW17, Theorem 7.2.1]). Since there are only finitely many bad reduction primes for elliptic curves, the criterion can be practically used to check the full Birch and Swinnerton-Dyer formula for an elliptic curve of rank zero and not necessarily square-free conductor. All the known non-CM examples have square-free conductors ([Wan18, Appendix]). We put some numerical examples of elliptic curves with additive reduction at the end.

Let p be an odd prime and E be an elliptic curve of conductor N over $\mathbb{Q}$. Throughout this article, we assume that E has additive reduction at p. In other words, $p^2$ divides N.

The construction of Kato's Euler systems [Kat04] and the formulation of the Iwasawa main conjecture without p-adic L-functions [Kat04, Conjecture 12.10] are insensitive to the reduction type of elliptic curves. See [Jac18] for the ${\mathbf{D}}_{\mathrm{dR}}$-valued p-adic L-function of elliptic curves with additive reduction and also [Del98] and [Del02] for different attempts to understand the additive reduction case.

We expect that the reader has some familiarity with [Rub00] and [MR04].

We briefly explain the synopsis of the proof. One can see the detail in [KKS].

Let c be the Kato's Euler system for E following the convention described in Appendix A, κ the Kolyvagin system associated to c, and ${\mathit{\kappa }}^{\infty }$ the Λ-adic Kolyvagin system associated to c where Λ is the Iwasawa algebra. We say that

• κ is primitive if κ does not vanish modulo p;

• ${\mathit{\kappa }}^{\infty }$ is Λ-primitive if ${\mathit{\kappa }}^{\infty }$ does not vanish modulo any height one prime of Λ.

In the theory of Kolyvagin systems, the concept of primitivity and Λ-primitivity play the central role to establish the equality of the p-part of the BSD formula for the rank zero case and the Iwasawa main conjecture, respectively. Roughly, we expect the following statements in general.

• If κ is primitive and E has rank zero, then the p-part of the Birch and Swinnerton-Dyer formula for E holds.

• If ${\mathit{\kappa }}^{\infty }$ is Λ-primitive, then the “Iwasawa main conjecture for E” holds.

It is also known that the primitivity of κ implies the Λ-adic primitivity of ${\mathit{\kappa }}^{\infty }$. In the original argument [MR04, Theorems 6.2.4 and 6.2.7], the good ordinary case is only treated. We extend both statements to the additive reduction case. See §3 for the main conjecture and §4 for the p-part of the BSD formula.

Now the problem reduces to how to check κ is primitive. In order to check this, we compute the mod p reduction of the image of Kolyvagin derivatives of Kato's Euler system under the dual exponential map, which are called Kurihara numbers. In order to consider the mod p reduction in the image of the dual exponential map, we need to compute the integral image of the dual exponential map. This computation is done in §2.

In the rest of this section, we describe the working assumptions and Kurihara numbers, review Kato's main conjecture, and then state the main theorems.

Let

• $\mathrm{Tam}(E)$ be the Tamagawa number of E,

• $N_{\mathrm{st}}$ be the product of split multiplicative reduction primes of E, and

• $N_{\mathrm{ns}}$ be the product of non-split multiplicative reduction primes of E.

We assume the following conditions throughout this article.

Assumption 1.1 Working assumptions

• p does not divide $\mathrm{Tam}(E)\cdot \prod _{\ell |N_{\mathrm{st}}}(\ell -1)\cdot \prod _{\ell |N_{\mathrm{ns}}}(\ell +1)$ (cf. Remark 1.8.(4)).

• The mod p Galois representation $\bar{\rho }:G_{\mathbb{Q}}\to {\mathrm{Aut}}_{{\mathbb{F}}_p}(E[p])$ is surjective. (Thus, E is non-CM.)

• The Manin constant is prime to p. (It is expected to be true always.)

Let $f\in S_2({\mathrm{\Gamma }}_0(N))$ be the newform attached to E by [BCDT01, Theorem A]. For $\frac{a}{b}\in \mathbb{Q}$, we define ${\left[\frac{a}{b}\right]}^{+}$ by

$$2\pi \underset{0}{\overset{\infty }{\int }}f(\frac{a}{b}+iy)dy={\left[\frac{a}{b}\right]}^{+}\cdot {\mathrm{\Omega }}_E^{+}+{\left[\frac{a}{b}\right]}^{-}\cdot {\mathrm{\Omega }}_E^{-}$$

where ${\mathrm{\Omega }}_E^{\pm }$ are the Néron periods of E. Then it is well-known that ${\left[\frac{a}{b}\right]}^{+}\in \mathbb{Q}$. The following theorem due to G. Stevens yields the p-integrality of the value.

Theorem 1.2 [Ste89, §3]

Under (2) and (3) of Assumption 1.1, we have ${\left[\frac{a}{b}\right]}^{+}\in {\mathbb{Z}}_{(p)}$ for $\frac{a}{b}\in \mathbb{Q}$.

A prime ℓ is a Kolyvagin prime for $(E,p)$ if $(\ell ,Np)=1$, $\ell \equiv 1(\text{mod}p)$, and $a_{\ell }(E)\equiv \ell +1(\text{mod}p)$. We define the Kurihara number for $(E,p)$ at n by

$${\tilde{\delta }}_n:=\sum _{a\in {(\mathbb{Z}/n\mathbb{Z})}^{\times }}(\overline{{\left[\frac{a}{n}\right]}^{+}}\cdot \prod _{\ell |n}\stackrel{‾}{{\log }_{{\mathbb{F}}_{\ell }}(a)})\in {\mathbb{F}}_p$$

where n is the square-free product of Kolyvagin primes, $\overline{{\left[\frac{a}{n}\right]}^{+}}$ is the mod p reduction of ${\left[\frac{a}{n}\right]}^{+}$, and $\stackrel{‾}{{\log }_{{\mathbb{F}}_{\ell }}(a)}$ is the mod p reduction of the discrete logarithm of a modulo ℓ with a fixed primitive root modulo ℓ. The number ${\tilde{\delta }}_n$ itself is not well-defined, but its non-vanishing question is well-defined.

Let ${\mathbb{Q}}_{\infty }$ be the cyclotomic ${\mathbb{Z}}_p$-extension of $\mathbb{Q}$ and ${\mathbb{Q}}_n$ be the subextension of $\mathbb{Q}$ in ${\mathbb{Q}}_{\infty }$ of degree $p^n$. Let $\mathrm{\Lambda }:={\mathbb{Z}}_p〚\mathrm{Gal}({\mathbb{Q}}_{\infty }/\mathbb{Q})〛$ be the Iwasawa algebra. Let $T={\mathrm{Ta}}_{p}E$ be the p-adic Tate module of E and $j:\mathrm{Spec}({\mathbb{Q}}_n)\to \mathrm{Spec}({\mathcal{O}}_{{\mathbb{Q}}_n}[1/p])$ be the natural map. Then we define the i-th Iwasawa cohomology of E by

$${\mathbb{H}}^i(T):=\underset{n}{\underleftarrow{\lim }}{\mathrm{H}}_{\acute{\mathrm{e}}\mathrm{t}}^i(\mathrm{Spec}({\mathcal{O}}_{{\mathbb{Q}}_n}[1/p]),j_{⁎}T)$$

where ${\mathrm{H}}_{\acute{\mathrm{e}}\mathrm{t}}^i(\mathrm{Spec}({\mathcal{O}}_{{\mathbb{Q}}_n}[1/p]),j_{⁎}T)$ is the étale cohomology group. See Appendix A for the full cyclotomic extension.

Theorem 1.3 [Kat04, Theorem 12.4.(1) and (3)]

The following statements hold.

• ${\mathbb{H}}^2(T)$ is a finitely generated torsion module over Λ.

• ${\mathbb{H}}^1(T)$ is free of rank one over Λ under Assumption 1.1.(2).

We recall the Iwasawa main conjecture without p-adic zeta functions à la Kato.

Conjecture 1.4 Kato's IMC, [Kat04, Conjecture 12.10], [Kur02, Conjecture 6.1]

$${\mathrm{char}}_{\mathit{\Lambda }}\left(\frac{{\mathbb{H}}^1(T)}{\mathit{\Lambda }{\mathbf{z}}_{\mathrm{Kato}}}\right)={\mathrm{char}}_{\mathit{\Lambda }}({\mathbb{H}}^2(T))$$

where ${\mathbf{z}}_{\mathrm{Kato}}$ is Kato's zeta element (Definition A.13).

Remark 1.5

• Following the argument of [Kur02, §6], [Kob03, Theorem 7.1.ii)], if $E({\mathbb{Q}}_{\infty ,p})[p^{\infty }]$ is finite, then ${\mathrm{Sel}}_{\mathrm{str}}{({\mathbb{Q}}_{\infty },E[p^{\infty }])}^{\vee }$ and ${\mathbb{H}}^2(T)$ are pseudo-isomorphic as Λ-modules where ${\mathrm{Sel}}_{\mathrm{str}}({\mathbb{Q}}_{\infty },E[p^{\infty }])$ is the p-strict (“fine”) Selmer group of E over ${\mathbb{Q}}_{\infty }$.

• Indeed, due to the argument of [Kat04, §13.13], if E has potentially good reduction at p, then $E({\mathbb{Q}}_{\infty ,p})[p^{\infty }]$ is finite.

• The following statements are equivalent:

  • E has potentially good reduction at p.

  • The corresponding local automorphic representation at p is not (a twist of) Steinberg. (See [Roh94, §15].)

  • The j-invariant of E is p-integral. (See [Liu06, Proposition 10.2.33].)

Theorem 1.6 The main conjecture

Let E be an elliptic curve with additive reduction at $p>7$ satisfying Assumption 1.1. If

$${\tilde{\delta }}_n\ne 0\in {\mathbb{F}}_p$$

for some square-free product of Kolyvagin primes n, then we have

$${\mathrm{char}}_{\mathit{\Lambda }}\left(\frac{{\mathbb{H}}^1(T)}{\mathit{\Lambda }{\mathbf{z}}_{\mathrm{Kato}}}\right)={\mathrm{char}}_{\mathit{\Lambda }}\left({\mathrm{Sel}}_{\mathrm{str}}{({\mathbb{Q}}_{\infty },E[p^{\infty }])}^{\vee }\right).$$

If we further assume that E has potentially good reduction at p, then Kato's IMC (Conjecture 1.4) holds, i.e.

$${\mathrm{char}}_{\mathit{\Lambda }}\left(\frac{{\mathbb{H}}^1(T)}{\mathit{\Lambda }{\mathbf{z}}_{\mathrm{Kato}}}\right)={\mathrm{char}}_{\mathit{\Lambda }}({\mathbb{H}}^2(T)).$$

Theorem 1.7

Let E be an elliptic curve with additive reduction at $p>7$ satisfying Assumption 1.1. Suppose that $L(E,1)\ne 0$. If

$${\tilde{\delta }}_n\ne 0\in {\mathbb{F}}_p$$

for some square-free product of Kolyvagin primes n, then the p-part of Birch and Swinnerton-Dyer formula for E holds, i.e.

$${\mathrm{ord}}_p(\mathrm{\#}\mathrm{\text{СХ}}(E/\mathbb{Q})[p^{\infty }])={\mathrm{ord}}_p\left(\frac{L(E,1)}{{\mathit{\Omega }}_E^{+}}\right).$$

Remark 1.8

• Even in the $p\le 7$ case, Theorem 1.6 and Theorem 1.7 hold if $(E,p)$ does not satisfy Assumption 2.5 below.

• It is expected that there always exists a square-free product of Kolyvagin primes n such that ${\tilde{\delta }}_n\ne 0\in {\mathbb{F}}_p$. Practically, it is easy to find such n's (cf. [Kur14]).

• We do not know whether Theorem 1.6 directly implies Theorem 1.7 or not since there is neither a Mazur–Greenberg style main conjecture nor a control theorem for the additive reduction case.

• If we replace $L(E,1)$ by the N-imprimitive L-value $L^{(N)}(E,1)$ in Theorem 1.7, then we can weaken Assumption 1.1.(1) by $p\nmid \mathrm{Tam}(E)$ (cf. [MR04, Theorem 6.2.4]).