Let E be an elliptic curve defined over Q without complex multiplication. For each prime ℓ, there is a representation ρ E,ℓ : Gal(Q/Q) → GL 2 (Z/ℓZ) that describes the Galois action on the ℓ-torsion points of E. Building on recent work of Rouse-Zureick-Brown and Zywina, we find models for composite level modular curves whose rational points classify elliptic curves over Q with simultaneously non-surjective, composite image of Galois. We also provably determine the rational points on almost all of these curves. Finally, we give an application of our results to the study of entanglement fields. 7

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Let Γ be a finite graph and let Γ n be the "nth cone over Γ" (i.e., the join of Γ and the complete graph K n ). We study the asymptotic structure of the chip-firing group Pic 0 (Γ n ).

Let p be a rational prime, let K denote the completion of the maximal unramified extension of Q p , let K be a fixed algebraic closure of K, and let C p denote the completion of K. Let A be an abelian variety defined over Q p with good reduction and base change it to K. Classically, the Fontaine integral was seen as a Hodge-Tate comparison morphism, i.e. as a map ϕ, and as such it is surjective and has a large kernel.The present article starts with the observation that if we do not tensor T p (A) with C p , then the Fontaine integral is often injective. In particular, we conjecture that if T p (A) G K = 0, then ϕ A is injective. We prove this conjecture for elliptic curves, for abelian varieties with good, ordinary reduction, and for abelian varieties where the formal group associated to the Néron model of A satisfies a certain mod p condition. Furthermore, after extending the Fontaine integral to a perfectoid like universal cover of A, we show that if T p (A) G K = 0 and ϕ A is injective, then A(K) has a type of p-adic uniformization, which resembles the classical complex uniformization.Let us first suppose K is archimedean. In this case we have K = R or K = C, K = C = C, G K = {1, τ}, with τ the complex conjugation, or G K = {1}. The exponential map is everywhere defined and defines the exact sequence:From this, we obtain the complex uniformization of A, i.e. an isomorphism of complex Lie-groups:where Λ is the image of the lattice H 1 (A(C), Z).

We prove a non-archimedean analogue of the fact that a closed subvariety of a semiabelian variety is hyperbolic modulo its special locus, and thereby generalize a result of Cherry.