Constructing elements in Shafarevich–Tate groups of modular motives

Dummigan, Neil; Stein, William; Watkins, Marley W.

doi:10.1017/cbo9780511734946.007

Cited by 16 publications

(37 citation statements)

References 19 publications

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“…One should note that we could rephrase our results in terms of X Σ (W f,λ (κ − 2)) and L alg (κ, f ) by adding in the condition that for no prime p | M is f congruent modulo λ to a newform of weight 2κ − 2, trivial character, and level dividing M/p. For the arguments needed to change our results to align with this, one can see [6,15]. We chose to phrase our results as above to avoid this extra congruence condition.…”

Section: Then There Existsmentioning

confidence: 99%

On the Bloch–Kato conjecture for elliptic modular forms of square-free level

Agarwal

Brown

2013

Math. Z.

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Abstract. Let κ ≥ 6 be an even integer, M an odd square-free integer, and f ∈ S2κ−2(Γ0(M )) a newform. We prove that under some reasonable assumptions that half of the λ-part of the Bloch-Kato conjecture for the near central critical value L(κ, f ) is true. We do this by bounding the ℓ-valuation of the order of the appropriate Bloch-Kato Selmer group below by the ℓ-valuation of algebraic part of L(κ, f ). We prove this by constructing a congruence between the Saito-Kurokawa lift of f and a cuspidal Siegel modular form.

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Section: Then There Existsmentioning

confidence: 99%

On the Bloch–Kato conjecture for elliptic modular forms of square-free level

Agarwal

Brown

2013

Math. Z.

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“…The second part of Theorem 1.3 now follows from Theorem 6.1 of [DSW03], as we now indicate (for details of some of the definitions below, see [DSW03]). Let T q denote the q-adic Tate module of E ∨ = E. Let L q denote the quotient field of O q , let V q = T q ⊗ Oq L q , and let A q denote V q /T q .…”

Section: Proofsmentioning

confidence: 92%

“…We are grateful to Neil Dummigan for answering some questions regarding [DSW03]. We would also like to thank John Cremona and Mark Watkins for some numerical data that encouraged the author to pursue the investigations in this article.…”

Section: Acknowledgementsmentioning

confidence: 97%

A visible factor of the Heegner index

Agashe

2010

Mathematical Research Letters

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Abstract. Let E be an optimal elliptic curve over Q of conductor N , such that the L-function of E vanishes to order one at s = 1. Let K be a quadratic imaginary field in which all the primes dividing N are split and such that the L-function of E over K also vanishes to order one at s = 1. In view of the Gross-Zagier theorem, the Birch and Swinnerton-Dyer conjecture says that the index in E(K) of the subgroup generated by the Heegner point is equal to the product of the Manin constant of E, the Tamagawa numbers of E, and the square root of the order of the Shafarevich-Tate group of E (over K). We extract an integer factor from the index mentioned above and relate this factor to certain congruences of the newform associated to E with eigenforms of analytic rank bigger than one. We use the theory of visibility to show that, under certain hypotheses (which includes the first part of the Birch and Swinnerton-Dyer conjecture on rank), if an odd prime q divides this factor, then q divides the order of the Shafarevich-Tate group, as predicted by the Birch and Swinnerton-Dyer conjecture.

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“…Let X f (L, j) denote the Shafarevich-Tate group associated to T f (j) over L. For the definition of Shafarevich-Tate group associated to T f (j), we refer the reader to [3]. By assumption, the Selmer group Sel BK (A f (j)/L) is finite and Sel BK (A f (j)/L) ∼ = X f (L, j).…”

Section: Theorem 38 Suppose That the Assumptions Of Theorem 31 Holmentioning

confidence: 99%