this paper have been announced in [D3] where a short introduction to mixed motives can also be found. A much more thorough treatment of motives is given in [JKS]. For the Beilinson conjectures and related notions like Deligne cohomology we recommend the book [RSS]. Some observations of the present paper may be of interest to researchers in other fields of mathematics than arithmetic geometry e.g. in dynamical systems. For this reason I have occasionally recalled standard material in the first two sections. It is a pleasure for me to thank Y. Ihara for the invitation to Kyoto and the RIMS for support. I would also like to thank D.W. Boyd for intersting correspondence and for bringing the above mentioned polynomial to my attention.

The authors would like to give special thanks to N. Boston, K. Buzzard, and B. Conrad for providing so much valuable feedback on earlier versions of this

. 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. In this article, I discuss material which is related to the recent

In lectures at the Newton Institute in June of 1993, Andrew Wiles announced a proof of a large part of the Taniyama-Shimura Conjecture and, as a consequence, Fermat's Last Theorem. This report for non-experts discusses the mathematics involved in Wiles' lectures, including the necessary background and the mathematical history.

Abstract. It is conjectured that there exist only finitely many isomorphism classes of endomorphism algebras of abelian varieties of bounded dimension over a fixed number field. We explore this conjecture when particularized to quaternion endomorphism algebras of abelian surfaces of GL2-type over Q by giving a moduli interpretation which translates the question into the diophantine arithmetic of Shimura curves embedded in Hilbert surfaces. We address the resulting problems on these curves by local and global methods, including Chabauty techniques on explicit equations of Shimura curves. 1.

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This article explains Serre's conjectures relating mod p Galois representations of Gal( Q=Q) to modular forms mod p, with special emphasis on the aspects related to Wiles' recent breakthrough on the Shimura-Taniyama conjecture

In this paper, which follows closely the talk given at the conference, I will sketch an example of a non-trivial element of K2 of a certain threefold, whose existence is related to the vanishing of an incomplete L-function of a modular form at s = 1. To explain how this fits into a general picture, we begin with a simple account, for the non-specialist, of some of the conjectures (mostly due to Beilinson) which relate ranks of K-groups and orders of L-functions, supplemented by examples coming from modular forms. The picture presented is in some respects wildly distorted; among the important topics which are given little mention are: (i) the connection between special values of L-functions and higher regulators, which is at the heart of the Beilinson conjectures; (ii) the conjectures of Birch and Swinnerton-Dyer, and their generalisation by Beilinson and Bloch; (iii) the theory of (mixed) motives, which underlies the constructions of the last section. But I hope that it may be of some use as a gentle introduction to the subject, and to

In this paper we list the elliptic curves defined over Q ( √ 5) with good reduction away from 2. We use the results of the previous papers [2] and [3], referred to as I and II respectively. By Theorem 1.14 of I, such a curve must have a point of order 2 defined over Q ( √ 5) and by Theorem 2.3 of II, if t ∈ Q ( √ 5) is the corresponding value of the Hauptmodul on X0(2) then either t or t ′ = 4096/t satisfies one of the equations t = 64u/v, u + v = x 2 (1) or t = 64v/2 a u, 2 a u + v = x 2 where u, v are units, x ∈ Q ( √ 5) and a ≥ 0. We solve these equations and determine the corresponding j-invariants and obtain a global minimal equation in each isomorphism class by Tate’s algorithm [5]. There are 368 isomorphism classes of these curves.