TY - JOUR

T1 - Statistics of the zeros of zeta functions in families of hyperelliptic curves over a finite field

AU - Faifman, Dmitry

AU - Rudnick, Zeév

PY - 2010/1

Y1 - 2010/1

N2 - We study the fluctuations in the distribution of zeros of zeta functions of a family of hyperelliptic curves defined over a fixed finite field, in the limit of large genus. According to the Riemann hypothesis for curves, the zeros all lie on a circle. Their angles are uniformly distributed, so for a curve of genus g a fixed interval will contain asymptotically 2g|I| angles as the genus grows. We show that for the variance of number of angles in is asymptotically (2/π2)log (2g) and prove a central limit theorem: the normalized fluctuations are Gaussian. These results continue to hold for shrinking intervals as long as the expected number of angles 2g|I| tends to infinity.

AB - We study the fluctuations in the distribution of zeros of zeta functions of a family of hyperelliptic curves defined over a fixed finite field, in the limit of large genus. According to the Riemann hypothesis for curves, the zeros all lie on a circle. Their angles are uniformly distributed, so for a curve of genus g a fixed interval will contain asymptotically 2g|I| angles as the genus grows. We show that for the variance of number of angles in is asymptotically (2/π2)log (2g) and prove a central limit theorem: the normalized fluctuations are Gaussian. These results continue to hold for shrinking intervals as long as the expected number of angles 2g|I| tends to infinity.

KW - central limit theorem

KW - hyperelliptic curve

KW - random matrix theory

KW - zeros of L-functions

UR - http://www.scopus.com/inward/record.url?scp=77957269308&partnerID=8YFLogxK

U2 - 10.1112/S0010437X09004308

DO - 10.1112/S0010437X09004308

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AN - SCOPUS:77957269308

SN - 0010-437X

VL - 146

SP - 81

EP - 101

JO - Compositio Mathematica

JF - Compositio Mathematica

IS - 1

ER -