Nonvanishing of Quadratic Dirichlet L-Functions at S = &amp;#189

Soundararajan, K.

doi:10.2307/2661390

Cited by 197 publications

(278 citation statements)

References 17 publications

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“…Their conjectures agree with the known results for the first few integral moments of the Riemann zeta-function (see [12,13]), and with the known results the first few integral moments of twists by quadratic Dirichlet characters (see [10,15,26]). They also agree with the number theoretic heuristics of [3,4].…”

Section: Momentssupporting

confidence: 79%

On the Vanishing of TwistedL-Functions of Elliptic Curves

David¹,

Fearnley²,

Kisilevsky³

2004

Experimental Mathematics

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Abstract. Let E be an elliptic curve over Q with L-function L E (s). We use the random matrix model of Katz and Sarnak to develop a heuristic for the frequency of vanishing of the twisted L-functions L E (1, χ), as χ runs over the Dirichlet characters of order 3 (cubic twists). The heuristic suggests that the number of cubic twists of conductor less than X for which L E (1, χ) vanishes is asymptotic to b E X 1/2 log e E X for some constants b E , e E depending only on E. We also compute explicitely the conjecture of Keating and Snaith about the moments of the special values L E (1, χ) in the family of cubic twists. Finally, we present experimental data which is consistent with the conjectures for the moments and for the vanishing in the family of cubic twists of L E (s).

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Section: Momentssupporting

confidence: 79%

On the Vanishing of TwistedL-Functions of Elliptic Curves

David¹,

Fearnley²,

Kisilevsky³

2004

Experimental Mathematics

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“…Moreover for primes p which are inert in K we have λ(p) = 0, and for the remaining primes the angles θ p , where λ(p) = 2 cos θ p , are uniformly distributed over [0, π] (see [3]). We also define the two set of primes (12) Note that the first set of primes has density 1/2, and the second set of primes has density 1/3. This special property of CM elliptic curves will play a crucial role in this paper.…”

Section: Preliminariesmentioning

confidence: 99%

“…Our goal is to show, following the arguments given in [8] (the techniques are already present in [12]), that the sequence in question has level of distribution at least onehalf. Any improvement of this will be of substantial interest.…”

Section: The Level Of Distributionmentioning

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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“…In the same paper, Jutila also obtained a second moment for this family of L-function. Soundararajan [22] in a tour de force established an asymptotic formula for the third moment for this family of L-functions. It is conjectured that 9) where the sum is over all fundamental discriminants.…”

Section: Theorem 13 (Goldfeld and Hoffsteinmentioning

confidence: 99%

Mean value of the class number in function fields revisited

Andrade

Jung

2018

Monatsh Math

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In this paper an asymptotic formula for the sum L(1, χ) is established for the family of quadratic Dirichlet L-functions over the rational function field over a finite field F q with q fixed. Using the recent techniques developed by Florea we obtain an extra lower order terms that was never been predicted in number fields and function fields. As a corollary, we obtain a formula for the average of the class number over function fields which also contains strenuous lower order terms and so improving on previous results of Hoffstein and Rosen.

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Nonvanishing of Quadratic Dirichlet L-Functions at S = &#189

Cited by 197 publications

References 17 publications

On the Vanishing of TwistedL-Functions of Elliptic Curves

On the Vanishing of TwistedL-Functions of Elliptic Curves

On quadratic families of CM elliptic curves

Mean value of the class number in function fields revisited

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