Let K be a number field and A/K be a polarized abelian variety with absolutely trivial endomorphism ring. We show that if the Néron model of A/K has at least one fiber with potential toric dimension 1, then, for almost all rational primes ℓ, the Galois group of the splitting field of the ℓ‐torsion of A is GSp2g(ℤ/ℓ).

Let k be a field not of characteristic two and Λ be a set consisting of almost all rational primes invertible in k. Suppose we have a variety X/k and strictly compatible system {M ℓ → X : ℓ ∈ Λ} of constructible F ℓ -sheaves. If the system is orthogonally or symplectically self-dual, then the geometric monodromy group of M ℓ is a subgroup of a corresponding isometry group Γ ℓ over F ℓ , and we say it has big monodromy if it contains the derived subgroup DΓ ℓ . We prove a theorem which gives sufficient conditions for M ℓ to have big monodromy. We apply the theorem to explicit systems arising from the middle cohomology of families of hyperelliptic curves and elliptic surfaces to show that the monodromy is uniformly big as we vary ℓ and the system. Theorem 1.1. Let V be an F ℓ -vector space together with a perfect pairing V × V → F ℓ and let G ≤ GL(V ) be an irreducible primitive subgroup which preserves the pairing. If the pairing is symmetric, G contains a reflection and an isotropic shear, and ℓ ≥ 5, then G is one of the following:2. the kernel of the spinor norm;3. the kernel of the product of the spinor norm and the determinant.If the pairing is alternating, G contains a transvection, and ℓ ≥ 3, then G is all of the symplectic group Sp(V ).For group-theoretic terms (e.g. transvection or isotropic shear) see section 3. Rather than assuming G is primitive, in which case we could appeal to [Wa1] or [Wa2] (cf. section 6 of [DR]), we make explicit assumptions about a set of elements generating G and show that they (essentially) imply G is primitive. In section 3 we also give a full statement of the theorem and its proof, and in the last two sections we give applications. Among those we single out (because it is easy to state and to prove) the following (unpublished) theorem of J-K. Yu.Theorem 1.2. The mod-ℓ monodromy of hyperelliptic curves is Sp(2g, F ℓ ) for ℓ > 2.See [Yu] for the preprint containing Yu's original proof or [AP] for another recent independent proof. The theorem has been used in several contexts. Yu originally proved his theorem in order to study the Cohen-Lenstra heuristics over function fields. Chavdarov [C] applied the theorem to study the irreducibility of numerators of zeta functions of families of curves over finite fields and Kowalski used his results to study the torsion fields of an abelian variety over a finite field [Kow2]. Achter applied Yu's theorem in [Ac] to prove a conjecture of Friedman and Washington on class groups of quadratic function fields. AcknowledgementsWe would like to thank D. Allcock, N.M. Katz, M. Olsson, D. Ulmer, and J.F. Voloch for several helpful conversations during the course of research and for their interest in this work, and we would like to thank the anonymous referree for carefully reading the paper and making several helpful suggestions for improving the exposition. We would also like to thank E. Kowalski for asking the question which motivated this paper, for suggesting that we could extend our results, originally for orthogonal monodromy, to symplectic mo...

Abstract. We show that families of coverings of an algebraic curve where the associated Cayley-Schreier graphs form an expander family exhibit strong forms of geometric growth. Combining this general result with finiteness statements for rational points under such conditions, we derive results concerning the variation of Galois representations in oneparameter families of abelian varieties.

Abstract. The Hardy-Littlewood prime k-tuples conjecture has long been thought to be completely unapproachable with current methods. While this sadly remains true, startling breakthroughs of Zhang, Maynard, and Tao have nevertheless made significant progress toward this problem. In this work, we extend the Maynard-Tao method to both number fields and the function field F q (t).

Let G be a finite connected graph, and let ρ be the spectral radius of its universal cover. For example, if G is k-regular then ρ = 2 √ k − 1. We show that for every r, there is an r-covering (a.k.a. an r-lift) of G where all the new eigenvalues are bounded from above by ρ. It follows that a bipartite Ramanujan graph has a Ramanujan r-covering for every r. This generalizes the r = 2 case due to Marcus, Spielman and Srivastava [MSS15a].Every r-covering of G corresponds to a labeling of the edges of G by elements of the symmetric group S r . We generalize this notion to labeling the edges by elements of various groups and present a broader scenario where Ramanujan coverings are guaranteed to exist.In particular, this shows the existence of richer families of bipartite Ramanujan graphs than was known before. Inspired by [MSS15a], a crucial component of our proof is the existence of interlacing families of polynomials for complex reflection groups. The core argument of this component is taken from [MSS15b].Another important ingredient of our proof is a new generalization of the matching polynomial of a graph. We define the r-th matching polynomial of G to be the average matching polynomial of all r-coverings of G. We show this polynomial shares many properties with the original matching polynomial. For example, it is real rooted with all its roots inside [−ρ, ρ].