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In this paper, building on work of Wiles [Wi] and of Wiles and one of us (R.T.) [TW], we will prove the following two theorems (see §2.2). Theorem A. If E /Q is an elliptic curve, then E is modular. Theorem B. If ρ: Gal(Q/Q) → GL2(F5) is an irreducible continuous representation with cyclotomic

Abstract: We investigate integer solutions of the superelliptic equation (1) zm = F (x, y), where F is a homogenous polynomial with integer coefficients, and

It is well-known that an abelian variety is (absolutely) simple or is isogenous to a self-product of an (absolutely) simple abelian variety if and only if the center of its endomorphism algebra is a field. In this paper we prove that the center is a field if the field of definition of points of prime order ℓ is “big enough”. The paper is organized as follows. In §1 we discuss Galois properties of points of order ℓ on an abelian variety X that imply that its endomorphism algebra End 0 (X) is a central simple algebra over the field of rational numbers. In §2 we prove that similar Galois properties for two abelian varieties X and Y combined with the linear disjointness of the corresponding fields of definitions of points of order ℓ imply that X and Y are non-isogenous (and even Hom(X, Y) = 0). In §3 we give applications to endomorphism algebras of hyperelliptic jacobians. In §4 we prove that if X admits multiplications by a number field E and the dimension of the centralizer of E in End 0 (X) is “as large as possible ” then X is an abelian variety of CM-type isogenous to a self-product of an absolutely simple abelian variety. Throughout the paper we will freely use the following observation [21, p. 174]: if an abelian variety X is isogenous to a self-product Z d of an abelian variety Z then a choice of an isogeny between X and Z d defines an isomorphism between End 0 (X) and the algebra Md(End 0 (Z)) of d × d matrices over End 0 (Z). Since the center of End 0 (Z) coincides with the center of Md(End 0 (Z)), we get an isomorphism

Let E=K be an elliptic curve de ned over a number eld, let ^ h be the canonical height on E, and let K =K be the maximal abelian extension of K. Extending work of Baker [4], we prove that there is a constant C(E=K) > 0 so that every nontorsion ^ h(P ) > C(E=K).

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We prove that for almost all oeoe oe 2 G(Q) the field oe) has the following property: For each absolutely irreducible affine variety V of dimension r and each dominating . We then say that oe) is PAC over Z.

Abstract. We show that Mazur’s conjecture on the real topology of rational points on varieties implies that there is no diophantine model of the rational integers Z in the rational numbers Q, i.e., there is no diophantine set D in some cartesian power Q i such that there exist two binary relations S, P on D whose graphs are diophantine in Q 3i (via the inclusion D 3 ⊂ Q 3i), and such that for two specific elements d0, d1 ∈ D the structure (D, S, P, d0, d1) is a model for integer arithmetic (Z,+, ·,0, 1). Using a construction of Pheidas, we give a counterexample to the analogue of Mazur’s conjecture over a global function field, and prove that there is a diophantine model of the polynomial ring over a finite field in the ring of rational functions over a finite field. 1.

. This survey discusses algorithms and explicit calculations for curves of genus at least 2 and their Jacobians, mainly over number fields and finite fields. Miscellaneous examples and a list of possible future projects are given at the end. 1. Introduction An enormous number of people have performed an enormous number of computations on elliptic curves, as one can see from even a perfunctory glance at [29]. A few years ago, the same could not be said for curves of higher genus, even though the theory of such curves had been developed in detail. Now, however, polynomialtime algorithms and sometimes actual programs are available for solving a wide variety of problems associated with such curves. The genus 2 case especially is becoming accessible: in light of recent work, it seems reasonable to expect that within a few years, packages will be available for doing genus 2 computations analogous to the elliptic curve computations that are currently possible in PARI, MAGMA, SIMATH, apec...

Abstract. We conjecture that if C is a curve of genus> 1 over a number field k such that C(k) = ∅, then a method of Scharaschkin (equivalent to the Brauer-Manin obstruction in the context of curves) supplies a proof that C(k) = ∅. As evidence, we prove a corresponding statement in which C(Fv) is replaced by a random subset of the same size in J(Fv) for each residue field Fv at a place v of good reduction for C, and the orders of Jacobians over finite fields are assumed to be smooth (in the sense of having only small prime divisors) as often as random integers of the same size. If our conjecture holds, and if Shafarevich-Tate groups are finite, then there exists an algorithm to decide whether a curve over k has a k-point, and the Brauer-Manin obstruction to the Hasse principle for curves over the number fields is the only one. 1. Setup Let k be a number field. Fix an algebraic closure k of k, and let G = Gal(k/k). Let C be a curve of genus g over k. (In this paper, curves are assumed to be smooth, projective, and geometrically integral.) Let C = C ×k k. Let J be the Jacobian of C, which is an abelian variety of dimension g over k. Assume that C has a G-invariant line bundle of degree 1: this