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Abstract. We give a quantitative bound for the number of S-integral points on an elliptic curve over a number field K in terms of the number of primes dividing the denominator of the j-invariant, the degree [K: Q], and the number of primes in S. Let K be a number field of degree d and MK the set of places of K. Let E/K be an elliptic curve with quasi-minimal Weierstrass equation E: y 2 = x 3 + Ax + B. If ∆ = 4A 3 + 27B 2 is the discriminant of this equation, recall that quasi-minimal means that |N K/Q(∆) | is minimized subject to the condition that A, B ∈ OK. Let S ⊂ MK be a finite set of s places containing all the archimedean ones, and denote the ring of S-integers by OS. Let j be the j-invariant of E. In [Sil6], Silverman proved that if j is integral, then #{P ∈ E(K) : x(P) ∈ OS} can be bounded in terms of the field K, #S, and the rank of E(K). More generally, Silverman proved that if the j-invariant is non-integral for at most δ places of K, then that set can be bounded in terms of the previously mentioned constants and δ. This is a special case of a conjecture of Lang asserting the existence of such a bound which is independent of δ. However, Silverman did not explicitly compute the constants involved. In this paper, using more explicit methods, we compute the dependence of the bounds on the various constants. In particular, as a consequence of Proposition 11, we have the following Theorem. For elliptic curves E/K of sufficiently large height, the number of S-integral points is at most 2 · 10 11 dδ(j) 3d (32 · 10 9) rδ(j)+s. For elliptic curves E defined over Q of sufficiently large height, the number of S-integral points is at most 32 · 10 11 (32 · 10 9) rδ(j)+s.

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We construct examples of families of curves of genus 2 or 3 over Q whose Jacobians split completely and have various large rational torsion subgroups. For example, the rational points on a certain elliptic surface over P of positive rank parameterize a family of genus-2 curves over Q whose Jacobians each have 128 rational torsion points. Also, we find the genus-3 curve ) = 0, whose Jacobian has 864 rational torsion points.

Let M be a motive which is defined over a number field and admits an action of a finite dimensional semisimple Q-algebra A. We formulate and study a conjecture for the leading coefficient of the Taylor expansion at 0 of the A-equivariant L-function of M. This conjecture simultaneously generalizes and refines the Tamagawa number conjecture of Bloch, Kato, Fontaine, Perrin-Riou et al. and also the central conjectures of classical Galois module theory as developed by Fröhlich, Chinburg, M. Taylor et al. The precise formulation of our conjecture depends upon the choice of an order A in A for which there exists a ‘projective A-structure ’ on M. The existence of such a structure is guaranteed if A is a maximal order, and also occurs in many natural examples where A is non-maximal. In each such case the conjecture with respect to a non-maximal order refines the conjecture with respect to a maximal order. We develop a theory of determinant functors for all orders in A by making use of the category of virtual objects introduced by Deligne.

Abstract. It is well-known that a finite graph can be viewed, in many respects, as a discrete analogue of a Riemann surface. In this paper, we pursue this analogy further in the context of linear equivalence of divisors. In particular, we formulate and prove a graph-theoretic analogue of the classical Riemann-Roch theorem. We also prove several results, analogous to classical facts about Riemann surfaces, concerning the Abel-Jacobi map from a graph to its Jacobian. As an application of our results, we characterize the existence or non-existence of a winning strategy for a certain chip-firing game played on the vertices of a graph. 1.

Dedicated to the memory of S. Chowla.

Wiles ’ proof of Fermat’s Last Theorem puts to rest one of the most famous unsolved problems in mathematics, a question that has been a wellspring for much of modern algebraic number theory. While celebrating Wiles ’ achievement, one also feels a twinge of regret at Fermat’s demise. Is the Holy Grail of number theorists to become a mere footnote in the history books? Hoping to keep some of the spirit of Fermat alive, I would like to discuss the generalized Fermat equation x p + y q = z r, (1) where p, q and r are fixed exponents. As in the case of Fermat’s Last Theorem, one is interested in integer solutions (x, y, z), which are non-trivial in the sense that xyz ̸ = 0. One might expect the equation above to have no such solutions if the exponents p, q, and r are large enough. But observe that, if p = q is odd, and r = 2, then any solution to a p +b p = c (of which there is an abundant supply!) yields the solution (ac, bc, c p+1 2) to the equation xp + yp = z2. A similar construction works whenever the exponents p, q, and r are pairwise coprime. However, the solutions produced in this way are not very interesting: the integers x, y and z have a large common factor. ∗This is a transcription of the author’s Aisenstadt prize lecture given at the CRM in March 1997. It is a pleasure to thank Andrew Granville and Loïc Merel for stimulating collaborations related to the topics of this essay, as well as Dan Abramovich for many helpful conversations over the years. This research was supported by CICMA and by grants from the Sloan Foundation, NSERC and FCAR. 1 Accordingly, one calls a solution (x, y, z) to the generalized Fermat equation primitive if gcd(x, y, z) = 1. Main Question: What are the non-trivial primitive solutions to the generalized Fermat equation? In [DG], Andrew Granville and I made the following conjecture:

. 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...

The primary topic of this thesis is the construction of explicit projective equations for the modular curves X 0 (N ). The techniques may also be used to obtain equations for X + 0 (p) and, more generally, X 0 (N )=Wn . The thesis contains a number of tables of results. In particular, equations are given for all curves X 0 (N ) having genus 2 g 5. Equations are also given for all X + 0 (p) having genus 2 or 3, and for the genus 4 and 5 curves X + 0 (p) when p 251. The most successful tool used to obtain these equations is the canonical embedding, combined with the fact that the differentials on a modular curve correspond to the weight 2 cusp forms. A second method, designed specifically for hyperelliptic curves, is given. A method for obtaining equations using weight 1 theta series is also described. Heights of modular curves are studied and a discussion is given of the size of coefficients occurring in equations for X 0 (N ). Finally, the explicit equations are used to study the...