For any sufficiently general family of curves over a finite field F q and any elementary abelian -group H with relatively prime to q, we give an explicit formula for the proportion of curves C for which Jac(C)[ ](F q )H . In doing so, we prove a conjecture of Friedman and Washington.

Recent studies have increasingly turned to graph theory to model more realistic contact structures that characterize disease spread. Because of the computational demands of these methods, many researchers have sought to use measures of network structure to modify analytically tractable differential equation models. Several of these studies have focused on the degree distribution of the contact network as the basis for their modifications. We show that although degree distribution is sufficient to predict disease behaviour on very sparse or very dense human contact networks, for intermediate density networks we must include information on clustering and path length to accurately predict disease behaviour. Using these three metrics, we were able to explain more than 98 per cent of the variation in endemic disease levels in our stochastic simulations.

We compute the Z/ℓ and Z ℓ monodromy of every irreducible component of the moduli spaces of hyperelliptic and trielliptic curves. In particular, we provide a proof that the Z/ℓ monodromy of the moduli space of hyperelliptic curves of genus g is the symplectic group Sp 2g (Z/ℓ). We prove that the Z/ℓ monodromy of the moduli space of trielliptic curves with signature (r, s) is the special unitary group SU (r,s) (Z/ℓ ⊗ Z[ζ 3 ]).

ABSTRACT. We determine the Z/ℓ-monodromy and Z ℓ -monodromy of every irreducible component of the stratum M f g of curves of genus g and p-rank f in characteristic p. In particular, we prove that the Z/ℓ-monodromy of every component of M f g is the symplectic group Sp 2g (Z/ℓ) if g ≥ 3 and if ℓ is a prime distinct from p. The method involves results on the intersection of M f g with the boundary of M g . We give applications to the generic behavior of automorphism groups, Jacobians, class groups, and zeta functions of curves of given genus and p-rank.

Abstract. We give an efficient, deterministic algorithm to decide if two abelian varieties over a number field are isogenous. From this, we derive an algorithm to compute the endomorphism ring of an elliptic curve over a number field.In this paper, we answer two fundamental decision problems about elliptic curves over number fields. Specifically, we explain how to detect whether two elliptic curves over a number field are isogenous, and how to decide whether an elliptic curve has complex multiplication. These algorithms rely on Lemma 1.2, which actually applies to abelian varieties of any dimension, and Proposition 2.2, respectively.In each case, we answer a question about a variety over a number field by examining its reduction at finitely many primes. At this level of generality, such a strategy is common in algorithmic number theory. For example, a common method for computing modular polynomials -that is, bivariate polynomials whose roots are j-invariants of elliptic curves related by an isogeny of fixed degree -is to perform the analogous computation over various finite fields, and then to lift the result using the Chinese remainder theorem. In contrast, we will see that to answer the decision problems posed here, one need not ever lift an object to characteristic zero.The engine driving the machines presented here is Faltings's paper on the Mordell conjecture. Milne observed in Mathematical Reviews that Faltings "seems to give an algorithm for deciding when two abelian varieties over a number field are isogenous." In this paper, we further refine the proof of [7, Theorem 5] to the point where it literally yields an efficient algorithm for the isogeny decision problem.At a crucial stage in that argument, Faltings shows that the isogeny class of X is determined by the action of Gal(L/K) on X[ℓ](K), where [L : K] has effectively bounded degree and ramification but is difficult to compute directly. He therefore works with L, the compositum of all possible such extensions of K, a large but still finite extension of K. An appeal to the Chebotarev density theorem guarantees that there is a finite set of primesTherefore, X and Y are isogenous if and only if the reductions X p and Y p are isogenous for each p ∈ T .We derive an algorithm for detecting isogeny by showing that it suffices to use a set of primes p with absolute norm smaller than some constant B. Effective Chebotarev-type theorems [3,8] let us calculate a suitable B solely in terms of the degree and ramification data of L, without requiring recourse to the compositum L.Subsequently, we show how to use this result to test the hypothesis that an elliptic curve E has complex multiplication by a field F . Briefly, after a finite extension of the base field, there exists an elliptic curve E ′ with complex multiplication by F . Even without computing E ′ explicitly, we can use Lemma 1.2 to detect whether 1