Mean Values of Derivatives of Modular L-Series

Murty, M. Ram; Murty, V. Kumar

doi:10.2307/2944316

Cited by 116 publications

(73 citation statements)

References 5 publications

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“…A similar result for linear families, i.e. E u,v : l(u, v)y 2 = f (x) with l a linear form, has been proved by Goldfeld-Hoffstein-Patterson [4] for CM elliptic curves, and by Murty-Murty [11] and Bump-Friedberg-Hoffstein [2] for general elliptic curves. The situation in the present paper is much more delicate owing to the fact that the quadratic family is too sparse for any direct application of harmonic analysis.…”

Section: Introductionsupporting

confidence: 59%

“…Indeed using genus theory, as explained above, a square-free positive integer d is represented by a form in the genus of Q if any only if (10) is nonzero. We set (11) and note that it is a multiplicative function on the set of square-free integers.…”

Section: Preliminariesmentioning

confidence: 99%

“…To derive the last equality we used (11). The Dirichlet series D n (s) converges absolutely for σ > 1/3, and in the domain σ ≥ 1/2 it satisfies the bound |D n (s)| n φ(n) , where the implied constant depends only on q, D. Observe that the inner sum over n in (18) converges absolutely and uniformly in any given compact domain for s, away from the possible poles coming from the L-function (i.e.…”

Section: Now the Inverse Mellin Transform Yieldsmentioning

confidence: 99%

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On quadratic families of CM elliptic curves

Munshi

2011

Trans. Amer. Math. Soc.

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Abstract. Given a CM elliptic curve with Weierstrass equation y 2 = f (x), and a positive definite binary quadratic form Q(u, v), we show that there are infinitely many reduced integer pairs (u, v) such that the twisted elliptic curve Q(u, v)y 2 = f (x) has analytic rank (and consequently Mordell-Weil rank) one. In fact it follows that the number of such pairs with |u|, |v| ≤ X is at least X 2−ε for any ε > 0.

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Section: Introductionsupporting

confidence: 59%

Section: Preliminariesmentioning

confidence: 99%

Section: Now the Inverse Mellin Transform Yieldsmentioning

confidence: 99%

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On quadratic families of CM elliptic curves

Munshi

2011

Trans. Amer. Math. Soc.

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“…By the results of Bump-Friedberg-Hoffstein-Murty-Murty-Waldspurger [4], [27], [37], there is an imaginary quadratic field M 0 where all bad primes of E split, and such that the quadratic twist of E by M 0 has analytic rank at most 1. By Kolyvagin's theorem [20], the parity conjecture holds for the twist, so it suffices to prove it for E=M 0 .…”

Section: Tim Dokchitser and Vladimir Dokchitsermentioning

confidence: 99%

On the Birch-Swinnerton-Dyer quotients modulo squares

Dokchitser

2010

Ann. Math.

140

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Let A be an abelian variety over a number field K. An identity between the L-functions L.A=K i ; s/ for extensions K i of K induces a conjectural relation between the Birch-Swinnerton-Dyer quotients. We prove these relations modulo finiteness of X, and give an analogous statement for Selmer groups. Based on this, we develop a method for determining the parity of various combinations of ranks of A over extensions of K. As one of the applications, we establish the parity conjecture for elliptic curves assuming finiteness of X.E=K.EOE2//OE6 1 and some restrictions on the reduction at primes above 2 and 3: the parity of the Mordell-Weil rank of E=K agrees with the parity of the analytic rank, as determined by the root number. We also prove the p-parity conjecture for all elliptic curves over ‫ޑ‬ and all primes p: the parities of the p 1 -Selmer rank and the analytic rank agree.

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“…More precisely, assume that the order of vanishing of TOME 56 (2006), FASCICULE 3 L(φ, 1) is 0 or 1. In this case, it is possible to choose an extension K/F so that all primes dividing N are split in K and L K (φ, 1) = 0 (if L(φ, 1) = 0 the existence of such a field K follows from the work of Bump-FriedbergHoffstein [9] and Murty-Murty [37], while for L(φ, 1) = 0 this is a result of Waldspurger [48]). Then Kolyvagin's result on the rank of E(K) imply the BSD conjecture over Q.…”

mentioning

confidence: 99%

On the Birch and Swinnerton-Dyer conjecture for modular elliptic curves over totally real fields

Longo¹

2006

Ann. inst. Fourier

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Let E/F be a modular elliptic curve defined over a totally real number field F and let f be its associated eigenform. This article presents a new method, inspired by a recent work of Bertolini and Darmon, to control the rank of E over suitable quadratic imaginary extensions K/F. In particular, this argument can also be applied to the cases not covered by the work of Kolyvagin and Logachev, that is, when [F : Q] is even and phi not new at any prime

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Mean Values of Derivatives of Modular L-Series

Cited by 116 publications

References 5 publications

On quadratic families of CM elliptic curves

On quadratic families of CM elliptic curves

On the Birch-Swinnerton-Dyer quotients modulo squares

On the Birch and Swinnerton-Dyer conjecture for modular elliptic curves over totally real fields

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