Statistics for biquadratic covers of the projective line over finite fields

Abstract

We study the distribution of the traces of the Frobenius endomorphism of genus g curves which are quartic non-cyclic covers of P 1 Fq , as the curve varies in an irreducible component of the moduli space. We show that for q fixed, the limiting distribution of the trace of Frobenius equals the sum of q + 1 independent random discrete variables. We also show that when both g and q go to infinity, the normalized trace has a standard complex Gaussian distribution. Finally, we extend these computations to the general case of arbitrary covers of P 1 Fq with Galois group isomorphic to r copies of Z 2Z. For r = 1, we recover the already known hyperelliptic case.

Expected value of high powers of trace of frobenius of biquadratic curves over a finite field

“…Since F 1 and F 2 are square-free, we get that f 1 , f 2 , f 3 are square-free and pairwise coprime. [9] and [10]. The same argument works for N n (C).…”

“…Equation (1.3) shows that knowing Tr(Θ n C ) is equivalent to knowing N n (C), the number of F q n -points on C. A formula for N 1 (C) is developed in [9] and [10]. The same argument works for N n (C).…”

“…This was first discussed by Kurlberg and Rudnick in [8] in which they considered the distribution of Tr( C ) over the family of hyperelliptic curves. This was extended by various authors to several different families [2][3][4][9][10][11][12]. For all cases, the statistics can be given as a sum of q + 1 random variables.…”

Distribution of points on Abelian covers over finite fields

“…This was extended by Bucur, David, Feigon and Lalin [2], [3] for prime cyclic curves (G = Z/pZ, p a prime). Lorenzo, Meleleo and Milione [8] then determined this for n-quadratic curves (G = (Z/2Z) n ). More recently the author [9] extended the work of Bucur, David, Feigon and Lalin to the case of arbitrary cyclic curves (G = Z/rZ, r not necessarily a prime).…”

Distribution of points on cyclic curves over finite fields

“…Hyperelliptic curves are in one-to-one correspondence with Galois extensions of F q (X) with Galois group Z/2Z. Bucur, David, Feigon and Lalin [1], [2] extended this result to smooth projective curves that are in one-to-one correspondence with Galois extensions of F q (X) with Galois group Z/pZ, where p is a prime such that q ≡ 1 mod p. Recently Lorenzo, Milione and Meleleo [5] determined the case for Galois group (Z/2Z) n . In this paper we determine the case for cyclic Galois groups Z/rZ for any q ≡ 1 mod r where r is not necessarily a prime.…”