Traces, high powers and one level density for families of curves over finite fields

Abstract

Abstract. The zeta function of a curve C over a finite field may be expressed in terms of the characteristic polynomial of a unitary matrix Θ C . We develop and present a new technique to compute the expected value of tr(Θ n C ) for various moduli spaces of curves of genus g over a fixed finite field in the limit as g is large, generalizing and extending the work of Rudnick [Rud10] and Chinis [Chi15]. This is achieved by using function field zeta functions, explicit formulae, and the densities of prime polynomials with prescribed ramification types at certain places as given in [BDF + 16] and [Zha]. We extend [BDF + 16] by describing explicit dependence on the place and give an explicit proof of the Lindelöf bound for function field Dirichlet L-functions L(1/2 + it, χ). As applications, we compute the one-level density for hyperelliptic curves, cyclic ℓ-covers, and cubic non-Galois covers.

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

“…A similar theorem to Theorem 1.5 is proved for the space of cubic, non-Galois extensions in [1]. The symmetry type they determine is U Sp(2g).…”

“…Bucur, Costa, David, Guerreiro and Lowry-Duda [1] used a method different than Rudnick and Chinis to study the moduli space of cyclic ℓ covers of genus g, denoted H g,ℓ , when ℓ is prime and q ≡ 1 mod ℓ. That is, they considered curves with affine model Y ℓ = F (X) of genus g. In the case where ℓ = 2, they obtain the same results as Rudnick and Chinis only in a smaller range of n: 4 log q (g) < n < 2g(1 − ǫ), for some ǫ > 0.…”

“…Bucur, Costa, David, Guerreiro and Lowry-Duda [1] used a method different than Rudnick and Chinis to study the moduli space of cyclic covers of genus g, denoted H g, , when is prime and q ≡ 1 mod . That is, they considered curves with affine model…”

One Level Density for Cubic Galois Number Fields

“…This is to be expected as all these curves correspond to quadratic extensions and Theorem 1.2 show that quadratic extensions in the number field setting have symmetry type Sp. Now, Bucur, Costa, David, Guerreiro and Lowry-Duda [1] prove the following.…”

“…Moreover, Katz and Sarnak predicted that W (G)(t) would fall into one of these five categories Tel Aviv University E-mail address: meisner@mail.tau.ac.il. 1 where δ 0 is the Dirac distribution and U, Sp, O, SO(even), SO(odd) are the groups of unitary, symplectic, orthogonal, even orthogonal and odd orthogonal matrices, respectively.…”