Statistics of traces of high powers of the Frobenius for biquadratic function fields over finite fields

We investigate statistics of trace of high powers of the Frobenius class for biquadratic function fields over finite fields, which generalizes the result of Lorenzo, Meleleo and Milione on the trace of the Frobenius class for biquadratic covers of the projective line. Then we compare our result with Meisner’s result on the expected value of high powers of trace of Frobenius of biquadratic curves.

Computing the Trace of Frobenius

This chapter aims to compute the trace Tr(Frob-1 ¦H* (BunG(X);Zℓ)), where ℓ is a prime number which is invertible in F q. It follows the strategy outlined in Chapter 1. If X is an algebraic curve over the field C of complex numbers and G is a smooth affine group scheme over X whose fibers are semisimple and simply connected, then Theorem 1.5.4.10 (and Example 1.5.4.15) supply a quasi-isomorphism whose right-hand side is the continuous tensor product of Construction 1.5.4.8. The remainder of this chapter is devoted to explaining how Theorem 4.1.2.1 can be used to compute the trace of the arithmetic Frobenius automorphism on the ℓ-adic cohomology of BunG(X).

Supercharacters and mixed moments of Kloosterman sums

Recent work has realized Kloosterman sums as supercharacter values of a supercharacter theory on [Formula: see text]. We use this realization to express fourth degree mixed power moments of Kloosterman sums in terms of the trace of Frobenius of a certain elliptic curve.

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

Denote ΘCas the Frobenius class of a curveCover the finite field 𝔽q. In this paper we determine the expected value of Tr(ΘCn) whereCruns over all biquadratic curves whenqis fixed andgtends to infinity. This extends work done by Rudnick [15] and Chinis [5] who separately looked at hyperelliptic curves and Bucur, Costa, David, Guerreiro and Lowry-Duda [1] who looked at ℓ-cyclic curves, for ℓ a prime, as well as cubic non-Galois curves.

Elliptic curve variants of the least quadratic nonresidue problem and Linnik’s theorem

Let [Formula: see text] and [Formula: see text] be [Formula: see text]-nonisogenous, semistable elliptic curves over [Formula: see text], having respective conductors [Formula: see text] and [Formula: see text] and both without complex multiplication. For each prime [Formula: see text], denote by [Formula: see text] the trace of Frobenius. Assuming the Generalized Riemann Hypothesis (GRH) for the convolved symmetric power [Formula: see text]-functions [Formula: see text] where [Formula: see text], we prove an explicit result that can be stated succinctly as follows: there exists a prime [Formula: see text] such that [Formula: see text] and [Formula: see text] This improves and makes explicit a result of Bucur and Kedlaya. Now, if [Formula: see text] is a subinterval with Sato–Tate measure [Formula: see text] and if the symmetric power [Formula: see text]-functions [Formula: see text] are functorial and satisfy GRH for all [Formula: see text], we employ similar techniques to prove an explicit result that can be stated succinctly as follows: there exists a prime [Formula: see text] such that [Formula: see text] and [Formula: see text]

Hypergeometric functions and relations to Dwork hypersurfaces

We give an expression for number of points for the family of Dwork K3 surfaces over finite fields of order [Formula: see text] in terms of Greene’s finite field hypergeometric functions. We also develop hypergeometric point count formulas for all odd primes using McCarthy’s [Formula: see text]-adic hypergeometric function. Furthermore, we investigate the relationship between certain period integrals of these surfaces and the trace of Frobenius over finite fields. We extend this work to higher dimensional Dwork hypersurfaces.

Local models for Weil-restricted groups

We extend the group-theoretic construction of local models of Pappas and Zhu [Local models of Shimura varieties and a conjecture of Kottwitz, Invent. Math.194(2013), 147–254] to the case of groups obtained by Weil restriction along a possibly wildly ramified extension. This completes the construction of local models for all reductive groups when$p\geqslant 5$. We show that the local models are normal with special fiber reduced and study the monodromy action on the sheaves of nearby cycles. As a consequence, we prove a conjecture of Kottwitz that the semi-simple trace of Frobenius gives a central function in the parahoric Hecke algebra. We also introduce a notion of splitting model and use this to study the inertial action in the case of an unramified group.