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 -