Moments of quadratic twists of elliptic curve L-functions over function fields

Hung Bui; A. Florea; J. P. Keating; E. Roditty-Gershon

doi:10.2140/ant.2020.14.1853

Moments of quadratic twists of elliptic curve L-functions over function fields

Hung Bui
, A. Florea
, J. P. Keating
, E. Roditty-Gershon

Pure Mathematics

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Abstract

We calculate the ﬁrst and second moments of L–functions in the family of quadratic twists of a ﬁxed elliptic curve E over Fq[x], asymptotically in the limit as the degree of the twists tends to inﬁnity. We also compute moments involving derivatives of L–functions over quadratic twists, enabling us to deduce lower bounds on the correlations between the analytic ranks of the twists of two distinct curves.

Original language	English
Pages (from-to)	1853-1893
Number of pages	41
Journal	Algebra and Number Theory
Volume	14
Issue number	7
DOIs	https://doi.org/10.2140/ant.2020.14.1853
Publication status	Published - Aug 2020

Keywords

Correlation
Elliptic curve
Finite field
L-function
Rank

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title = "Moments of quadratic twists of elliptic curve L-functions over function fields",

abstract = "We calculate the ﬁrst and second moments of L–functions in the family of quadratic twists of a ﬁxed elliptic curve E over Fq[x], asymptotically in the limit as the degree of the twists tends to inﬁnity. We also compute moments involving derivatives of L–functions over quadratic twists, enabling us to deduce lower bounds on the correlations between the analytic ranks of the twists of two distinct curves.",

keywords = "Correlation, Elliptic curve, Finite field, L-function, Rank",

author = "Hung Bui and A. Florea and Keating, \{J. P.\} and E. Roditty-Gershon",

note = "Funding Information: A. Florea gratefully acknowledges the support of an NSF Postdoctoral Fellowship during part of the research which led to this paper. J.P. Keating is supported by a Royal Society Wolfson Research Merit Award, EPSRC Programme Grant EP/K034383/1 LMF: L-Functions and Modular Forms, and by ERC Advanced Grant 740900 (LogCorRM). The authors would also like to thank Chantal David, Matilde Lalin and Zeev Rudnick and the anonymous referee for useful comments on the paper. Publisher Copyright: {\textcopyright} 2020 Mathematical Sciences Publishers. Copyright: Copyright 2020 Elsevier B.V., All rights reserved.",

year = "2020",

month = aug,

doi = "10.2140/ant.2020.14.1853",

language = "English",

volume = "14",

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AU - Keating, J. P.

AU - Roditty-Gershon, E.

N1 - Funding Information: A. Florea gratefully acknowledges the support of an NSF Postdoctoral Fellowship during part of the research which led to this paper. J.P. Keating is supported by a Royal Society Wolfson Research Merit Award, EPSRC Programme Grant EP/K034383/1 LMF: L-Functions and Modular Forms, and by ERC Advanced Grant 740900 (LogCorRM). The authors would also like to thank Chantal David, Matilde Lalin and Zeev Rudnick and the anonymous referee for useful comments on the paper. Publisher Copyright: © 2020 Mathematical Sciences Publishers. Copyright: Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2020/8

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N2 - We calculate the ﬁrst and second moments of L–functions in the family of quadratic twists of a ﬁxed elliptic curve E over Fq[x], asymptotically in the limit as the degree of the twists tends to inﬁnity. We also compute moments involving derivatives of L–functions over quadratic twists, enabling us to deduce lower bounds on the correlations between the analytic ranks of the twists of two distinct curves.

AB - We calculate the ﬁrst and second moments of L–functions in the family of quadratic twists of a ﬁxed elliptic curve E over Fq[x], asymptotically in the limit as the degree of the twists tends to inﬁnity. We also compute moments involving derivatives of L–functions over quadratic twists, enabling us to deduce lower bounds on the correlations between the analytic ranks of the twists of two distinct curves.

KW - Correlation

KW - Elliptic curve

KW - Finite field

KW - L-function

KW - Rank

UR - https://www.scopus.com/pages/publications/85080981282

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DO - 10.2140/ant.2020.14.1853

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VL - 14

SP - 1853

EP - 1893

JO - Algebra and Number Theory

JF - Algebra and Number Theory

IS - 7

ER -

Moments of quadratic twists of elliptic curve L-functions over function fields

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