General SectionRemarks on Kato's Euler systems for elliptic curves with additive reduction
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 . Throughout this article, we assume that E has additive reduction at p. In other words, 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 -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 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;
- •
is Λ-primitive if does not vanish modulo any height one prime of Λ.
- (1)
If κ is primitive and E has rank zero, then the p-part of the Birch and Swinnerton-Dyer formula for E holds.
- (2)
If is Λ-primitive, then the “Iwasawa main conjecture for E” holds.
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
- •
be the Tamagawa number of E,
- •
be the product of split multiplicative reduction primes of E, and
- •
be the product of non-split multiplicative reduction primes of E.
Assumption 1.1 Working assumptions
- (1)
p does not divide (cf. Remark 1.8.(4)).
- (2)
The mod p Galois representation is surjective. (Thus, E is non-CM.)
- (3)
The Manin constant is prime to p. (It is expected to be true always.)
Let be the newform attached to E by [BCDT01, Theorem A]. For , we define by where are the Néron periods of E. Then it is well-known that . 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 for .
A prime ℓ is a Kolyvagin prime for if , , and . We define the Kurihara number for at n by where n is the square-free product of Kolyvagin primes, is the mod p reduction of , and is the mod p reduction of the discrete logarithm of a modulo ℓ with a fixed primitive root modulo ℓ. The number itself is not well-defined, but its non-vanishing question is well-defined.
Let be the cyclotomic -extension of and be the subextension of in of degree . Let be the Iwasawa algebra. Let be the p-adic Tate module of E and be the natural map. Then we define the i-th Iwasawa cohomology of E by where 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. is a finitely generated torsion module over Λ. is free of rank one over Λ under Assumption 1.1.(2).
Conjecture 1.4 Kato's IMC, [Kat04, Conjecture 12.10], [Kur02, Conjecture 6.1]
where is Kato's zeta element (Definition A.13).
Remark 1.5
- (1)
Following the argument of [Kur02, §6], [Kob03, Theorem 7.1.ii)], if is finite, then and are pseudo-isomorphic as Λ-modules where is the p-strict (“fine”) Selmer group of E over .
- (2)
Indeed, due to the argument of [Kat04, §13.13], if E has potentially good reduction at p, then is finite.
- (3)
The following statements are equivalent:
- (a)
E has potentially good reduction at p.
- (b)
The corresponding local automorphic representation at p is not (a twist of) Steinberg. (See [Roh94, §15].)
- (c)
The j-invariant of E is p-integral. (See [Liu06, Proposition 10.2.33].)
- (a)
Theorem 1.6 The main conjecture Let E be an elliptic curve with additive reduction at satisfying Assumption 1.1. If for some square-free product of Kolyvagin primes n, then we have If we further assume that E has potentially good reduction at p, then Kato's IMC (Conjecture 1.4) holds, i.e.
Theorem 1.7 Let E be an elliptic curve with additive reduction at satisfying Assumption 1.1. Suppose that . If for some square-free product of Kolyvagin primes n, then the p-part of Birch and Swinnerton-Dyer formula for E holds, i.e.
Remark 1.8 Even in the case, Theorem 1.6 and Theorem 1.7 hold if does not satisfy Assumption 2.5 below. It is expected that there always exists a square-free product of Kolyvagin primes n such that . 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 by the N-imprimitive L-value in Theorem 1.7, then we can weaken Assumption 1.1.(1) by (cf. [MR04, Theorem 6.2.4]).
Section snippets
Computing the integral lattice
The goal of this section is to extend [Rub00, Proposition 3.5.1] to the additive reduction case over unramified extensions of . More precisely, we compute the image of the logarithm map of the unramified local points of an elliptic curve with additive reduction.
In §2.1, we compute the integral image of the dual exponential map involving the Néron differential (Corollary 2.4) by computing the image of the logarithm map of the unramified local points of an elliptic curve with additive reduction
The main conjecture: review of [KKS]
In this section, we present a slightly different computation from that of [KKS] due to a different normalization of Kato's Euler system. Let E be an elliptic curve over with additive reduction at p; especially, the Euler factor at p of L-function is 1.
The p-part of the Birch and Swinnerton-Dyer formula for the rank zero case
The goal of this section is to prove Theorem 1.7, which generalizes [MR04, Theorem 6.2.4] to the additive reduction case when the corresponding Kolyvagin system is primitive. Assumption 4.1 There exists an element such that τ acts trivially on , and is free of rank one over ., [Rub00, §2.1]
is irreducible.
We recall the “error term” in the Euler system argument. Let and be the smallest extension of where acts trivially on . Let
Examples
In this section, we provide three new examples of the main conjecture and the p-part of the BSD formula. The Sage code for an effective computation of Kurihara numbers due to Alexandru Ghitza is available at https://github.com/aghitza/kurihara_numbers. Although the original code is for good reduction, a slight modification allows us to compute the additive reduction case.
Acknowledgment
This project grew out from the discussion when we both visit Shanghai Center for Mathematical Sciences in March 2018. We deeply thank Shanwen Wang and the generous hospitality of Shanghai Center for Mathematical Sciences. We deeply appreciate Michiel Kosters to allow us to reproduce some material in [KP] in §2. C.K. also thanks Karl Rubin for helpful discussions. C.K. was partially supported by Basic Science Research Program through the National Research Foundation of Korea (NRF-2018R1C1B6007009
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