TY - JOUR

T1 - Statistics of the Jacobians of hyperelliptic curves over finite fields

AU - Xiong, Maosheng

AU - Zaharescu, Alexandru

N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 2012

Y1 - 2012

N2 - Let C be a smooth projective curve of genus g ≥ 1 over a finite field Fq of cardinality q. Denote by # JC the size of the Jacobian of C over Fq. We first obtain an estimate on # JC when Fq(C)/Fq(X) is a geometric Galois extension, which improves a general result of Shparlinski [19]. Then we study the behavior of the quantity #JC as C varies over a large family of hyperelliptic curves of genus g. When g is fixed and q → ∞, its limiting distribution is given by the powerful theorem of Katz and Sarnak in terms of the trace of a random matrix. When q is fixed and the genus g → ∞, we also find explicitly the limiting distribution and show that the result is consistent with that of Katz and Sarnak when both q, g → ∞.

AB - Let C be a smooth projective curve of genus g ≥ 1 over a finite field Fq of cardinality q. Denote by # JC the size of the Jacobian of C over Fq. We first obtain an estimate on # JC when Fq(C)/Fq(X) is a geometric Galois extension, which improves a general result of Shparlinski [19]. Then we study the behavior of the quantity #JC as C varies over a large family of hyperelliptic curves of genus g. When g is fixed and q → ∞, its limiting distribution is given by the powerful theorem of Katz and Sarnak in terms of the trace of a random matrix. When q is fixed and the genus g → ∞, we also find explicitly the limiting distribution and show that the result is consistent with that of Katz and Sarnak when both q, g → ∞.

KW - Class number

KW - Gaussian distribution

KW - Jacobian

KW - Zeta functions of curves

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U2 - 10.4310/MRL.2012.v19.n2.a1

DO - 10.4310/MRL.2012.v19.n2.a1

M3 - Article

AN - SCOPUS:84866727042

SN - 1073-2780

VL - 19

SP - 255

EP - 272

JO - Mathematical Research Letters

JF - Mathematical Research Letters

IS - 2

ER -