Zeros of quadratic Dirichlet $L$-functions in the hyperelliptic ensemble

Abstract

We study the 1-level density and the pair correlation of zeros of quadratic Dirichlet L-functions in function fields, as we average over the ensemble H 2g+1 of monic, square-free polynomials with coefficients in F q [x]. In the case of the 1-level density, when the Fourier transform of the test function is supported in the restricted interval ( 1 3 , 1), we compute a secondary term of size q − 4g 3 /g, which is not predicted by the Ratios Conjecture. Moreover, when the support is even more restricted, we obtain several lower order terms. For example, if the Fourier transform is supported in ( 1 3 , 1 2 ), we identify another lower order term of size q − 8g 5 /g. We also compute the pair correlation, and as for the 1-level density, we detect lower order terms under certain restrictions; for example, we see a term of size q −g /g 2 when the Fourier transform is supported in ( 1 4 , 1 2 ). The 1-level density and the pair correlation allow us to obtain non-vanishing results for L( 1 2 , χ D ), as well as lower bounds for the proportion of simple zeros of this family of L-functions.

Hybrid Euler-Hadamard product for quadratic Dirichlet L-functions in function fields

“…Throughout the paper we assume is fixed and . All theorems still hold for all odd by using the modified auxiliary lemmas in function fields as in , but we shall keep the assumption for simplicity. Let be the set of monic polynomials in , and be the sets of those of degree and degree at most , respectively.…”

“…Proof We first prove by induction on the number of prime factors of that where . The base case was proved in [, Lemma 3.7]. Now we assume holds for and we want to prove it for , where .…”

“…We first note that the sum over C in (6) can be extended to all C|(f ) ∞ with the cost of an error of size…”

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

“…The difference is explained by the fact that the infinite place behaves differently in F 2g+1 and F 2g+2 . work of The results of Rudnick were vastly generalized in a recent paper of Bui and Florea [BF14], which give formulas for the one-level density which are uniform in q and d, and they can then identify lower order terms when the support of the test function holds in various ranges. For the one-level density of classical Dirichlet L-functions associated to quadratic characters, some recent work of Fiorilli, Parks and Sodergren [FPS16] exhibits all the lower order terms which are descending powers of log X.…”

“…In Section 2, we first present a new proof for Theorem 1.2 using function field zeta functions and explicit formulae, specifically relying on densities of prime polynomials of different ramification types, as described in [BDF + 16]. Our technique is much simpler than what is used in [Rud10] and [BF14], and the result presented in Section 2 is weaker than the results of Rudnick and Chinis (as our result holds for a more limited range of n), but it has the benefit of having clear generalization to many families of curves. We present two such generalizations here.…”