Low-lying zeros of cubic Dirichlet L-functions and the Ratios conjecture

Abstract

We compute the one-level density for the family of cubic Dirichlet L-functions when the support of the Fourier transform of the associated test function is in (−1, 1). We also establish the Ratios conjecture prediction for the one-level density for this family, and confirm that it agrees with the one-level density we obtain.In this work, we study the low-lying zeros of cubic Dirichlet L-functions. Let φ be an even Schwartz function whose Fourier transform is compactly supported. For a cubic Dirichlet character χ, let ρ denote the nontrivial zeros of L(χ, s) in the critical strip. Define

Moments of traces of Frobenius of higher order Dirichlet $L$-functions over $\mathbb F_q[T]$

“…The statistics for families of higher order characters (r > 2) are less well known. However, there has been some recent progress in studying them [3,7,10,15]. The author has already considered simpler statistics for a similar, more geometric family in [21], extending the results of [3].…”

Low-lying zeros in families of elliptic curve $L$-functions over function fields

“…where L P (u, sym m E) is a polynomial of degree at most m + 1 with bounded coefficients. 7 We refer the reader to [6, Section 1.2] for more information on symmetric power L-functions, and the references therein (specifically [16] and [32]) for more general statements and proofs. See also [26] for symmetric power L-functions of elliptic curves defined over Q.…”

“…In recent years, there has been an increased interest in a variety of different aspects of higher order characters and twists; see, e.g., [1,7,8,9,10,11,12,13,27,28]. Motivated by this development, we investigate expected values of traces of high powers of the Frobenius class and the one-level density of families of cubic twists of elliptic curves of the form y 2 = x 3 + B defined over F q (t).…”

“…It is not clear that Theorem 1. 7 gives us what we were hoping for. That is, a term of size q −n/3 that is related to the Frobenius of L(u, sym 3 E) in some way which would lead to a deviation term for the one-level density that involves the logarithmic derivative of L(u, sym 3 E).…”