Moments of zeta functions associated to hyperelliptic curves over finite fields

Michael Rubinstein, Kaiyu Wu

Type: Article

Publication Date: 2015-03-24

Citations: 15

DOI: https://doi.org/10.1098/rsta.2014.0307

Abstract

Let q be an odd prime power, and denote the set of square-free monic polynomials D ( x )∈ F q [ x ] of degree d . Katz and Sarnak showed that the moments, over , of the zeta functions associated to the curves y 2 = D ( x ), evaluated at the central point, tend, as , to the moments of characteristic polynomials, evaluated at the central point, of matrices in USp (2⌊( d −1)/2⌋). Using techniques that were originally developed for studying moments of L -functions over number fields, Andrade and Keating conjectured an asymptotic formula for the moments for q fixed and . We provide theoretical and numerical evidence in favour of their conjecture. In some cases, we are able to work out exact formulae for the moments and use these to precisely determine the size of the remainder term in the predicted moments.

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Works Cited by This (28)

Action	Title	Year	Authors
+	FLINT : Fast library for number theory	2013	William B. Hart
+	Random Matrices, Frobenius Eigenvalues, and Monodromy	1998	Nicholas M. Katz Peter Sarnak
+	Average values of L-series in function fields.	1992	Jeffrey Hoffstein Michael K. Rosen
+	Counting Points on Hyperelliptic Curves using Monsky-Washnitzer Cohomology	2001	Kiran S. Kedlaya
+	Random Matrix Theory and L-Functions at s = 1/2	2000	Jonathan P. Keating N. C. Snaith
+	Conjectures and Experiments Concerning the Moments of<i>L</i>(1/2, χ<sub><i>d</i></sub>)	2012	Matthew W. Alderson Michael Rubinstein
+	The mean value of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mi>L</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mo>,</mml:mo><mml:mi>χ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>in the hyperelliptic ensemble	2012	Julio Andrade Jonathan P. Keating
+ PDF Chat	Counting Points in Medium Characteristic Using Kedlaya's Algorithm	2003	Pierrick Gaudry Nicolas Gürel
+	Lower order terms in the full moment conjecture for the Riemann zeta function	2007	J. Brian Conrey David W. Farmer Jonathan P. Keating Michael Rubinstein N. C. Snaith
+	On the Mean Value of L(1/2, χ ) FW Real Characters	1981	Matti Jutila