On the theory of Kolyvagin systems of rank <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>0</mml:mn></mml:math>

Sakamoto, Ryotaro

doi:10.5802/jtnb.1189

Cited by 2 publications

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“…In our case, the theory of Kolyvagin systems developed in [15] does not work when p = 3. In Appendix A, we discuss this problem and extend the theory of Kolyvagin systems so that it can be used even when p = 3.…”

Section: Introductionmentioning

confidence: 66%

“…In this section, we recall the theory of Kolyvagin systems. The contents of this section are based on [5,15].…”

Section: The Theory Of Kolyvagin Systemmentioning

confidence: 99%

“…Kolyvagin systems of rank 0. In this subsection, we recall the definition of Kolyvagin system of rank 0 in our previous paper [15]. Fix an isomorphism…”

Section: The Theory Of Kolyvagin Systemmentioning

confidence: 99%

“…Then Theorem 2.20 and [13, Lemma 3.25] (see [15,Theorem 6.3]) show that the canonical map KS 0 (T Q∞ , F cl ) −→ KS 0 (T Qn /p m , F cl ) is surjective and the Λ Q∞ -module KS 0 (T Q∞ , F cl ) is free of rank 1. By construction,…”

Section: 4mentioning

confidence: 99%

“…The proof of Theorem 1.5 is based on the theory of Kolyvagin systems of rank 0 developed in [15]. In §2, we introduce the theory of Kolyvagin systems.…”

Section: Introductionmentioning

confidence: 99%

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$p$-Selmer group and Modular symbols

Sakamoto¹

2021

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In this paper, we prove that the dimension of the p-Selmer group for an elliptic curve is controlled by certain analytic quantities associated with modular symbols, which is conjectured by Kurihara. Contents 1. Introduction 1 Acknowledgement 3 2. The theory of Kolyvagin system 4 2.1. Selmer structures 4 2.2. Structure of local points 6 2.3. Kolyvagin systems of rank 1 8 2.4. Kolyvagin systems of rank 0 8 2.5. Map from Kolyvagin systems of rank 1 to Kolyvagin systems of rank 0 10 3. Construction of the Kolyvagin system of rank 0 from modular symbols 11 3.1. Modular sysmbols 12 3.2. Euler systems 13 3.3. Construction of κ ξ,m,n 14 3.4. Properties of κ ξ,m,n 16 4. Main results 17 4.1. Proof of Theorem 1.2 17 4.2. Proof of Theorem 1.5 18 Appendix A. Remarks on p = 3 20 A.1. Application of the Chebotarev density theorem 21 A.2. Connectedness of the graph X 0 23 A.3. Kolyvagin systems 25 References 27 Kurihara proved in [4] that the natural homomorphism Selis injective (see Remark 4.5), and Kurihara conjectured in [4, Conjecture 2] that the homomorphism ( 1) is an isomorphism for any δ-minimalfor any δ-minimal integer d ∈ N 1,0 . Here ν(d) denotes the number of distinct prime divisors of d. Kurihara showed in [4, Theorem 4] that ( 1) is an isomorphism in some special cases. In the present paper, we solve this conjecture.Theorem 1.5. For any δ-minimal integer d ∈ N 1,0 , the homomorphism (1) is an isomorphism, and hence dim Fp (Sel(Q, E[p])) = ν(d).Remark 1.6. The analogue of Theorem 1.5 for ideal class groups does not hold as Kurihara gave in [4, §5.4] a counter-example. In Remark 4.9, we explain what is an important property in order to prove Theorem 1.5. By using the functional equation for modular symbols (see [7, (1.6.1)]), Kurihara showed in [4, Lemma 4] that w E = (−1) ν(d) for any δ-minimal integer d ∈ N 1,0 . Here w E denotes the (global) root number of E/Q. Hence, as an application of Theorem 1.5, we obtain the following result concerning the parity of the order of L-function L(E/Q, s) for E/Q: Corollary 1.7. Suppose that the Iwasawa main conjecture for E/Q holds true. Then we have dim Fp (Sel(Q, E[p])) ≡ ord s=1 (L(E/Q, s)) (mod 2).Moreover, if the p-primary part of the Tate-Shafarevich group for E/Q is finite, then we have rank Z (E(Q)) ≡ ord s=1 (L(E/Q, s)) (mod 2).

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Section: Introductionmentioning

confidence: 66%

“…In this section, we recall the theory of Kolyvagin systems. The contents of this section are based on [5,15].…”

Section: The Theory Of Kolyvagin Systemmentioning

confidence: 99%

“…Kolyvagin systems of rank 0. In this subsection, we recall the definition of Kolyvagin system of rank 0 in our previous paper [15]. Fix an isomorphism…”

Section: The Theory Of Kolyvagin Systemmentioning

confidence: 99%

Section: 4mentioning

confidence: 99%

“…The proof of Theorem 1.5 is based on the theory of Kolyvagin systems of rank 0 developed in [15]. In §2, we introduce the theory of Kolyvagin systems.…”

Section: Introductionmentioning

confidence: 99%

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