We study the local statistics of zeros of L-functions attached to Artin-Scheier curves over finite fields. We consider three families of Artin-Schreier L-functions: the ordinary, polynomial (the p-rank 0 stratum) and odd-polynomial families. We compute the 1-level zero-density of the first and third families and the 2-level density of the second family for test functions with Fourier transform supported in a suitable interval. In each case we obtain agreement with a unitary or symplectic random matrix model.

We study the local statistics of zeros of L L -functions attached to Artin-Scheier curves over finite fields. We consider three families of Artin-Schreier L L -functions: the ordinary, polynomial (the p p -rank 0 stratum) and odd-polynomial families. We compute the 1-level zero-density of the first and third families and the 2-level density of the second family for test functions with Fourier transform supported in a suitable interval. In each case we obtain agreement with a unitary or symplectic random matrix model.

We prove that for any prime p > 2, q = p ν a power of p, n ≥ p and G = Sn or G = An (symmetric or alternating group) there exists a Galois extension K/Fq(T ) ramified only over ∞ with Gal(K/Fq(T )) = G, with the possible exception of G = S p+1 if Fq ⊃ F p 2 . This confirms a conjecture of Abhyankar for the case of symmetric and alternating groups over finite fields of odd characteristic (with the possible exception of S p+1 with Fq ⊃ F p 2 ).Definition 2.1. A finite group G is called L(q)-realizable if there exists a Galois extension K/F q (T ) with Gal(K/F q (T )) ∼ = G which is unramified outside of a single prime divisor of F q (T ). An extension with this property is called an L(q)-realization of G.Note that in Definition 2.1 we could require WLOG that the ramified point is ∞, since we can always move a given point of P 1 (F q ) to ∞ by an automorphism of P 1 Fq (equivalently, by a linear fractional transformation of the variable T with coefficients in F q ). Hence Conjecture 1.2 is equivalent to the assertion that G is L(q)-realizable iff it is cyclic-by-quasi-p. Next we introduce a closely related definition taken from [BSEF21, Definition 5.4].Definition 2.2. A finite group G is called L gt 1 (q)-realizable if there exists a geometric Galois extension K/F q (t) with Gal(K/F q (t)) ∼ = G, K/F q (T ) is unramified outside of two degree 1 prime divisors of F q (T ) and at most tamely ramified over one of them. An extension with this property is called an L gt 1 (q)-realization of G.