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132
Rational Points on Elliptic Curves
, 1992
"... Abstract. We give a quantitative bound for the number of Sintegral points on an elliptic curve over a number field K in terms of the number of primes dividing the denominator of the jinvariant, 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 ..."
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Cited by 71 (1 self)
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Abstract. We give a quantitative bound for the number of Sintegral points on an elliptic curve over a number field K in terms of the number of primes dividing the denominator of the jinvariant, 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 quasiminimal Weierstrass equation E: y 2 = x 3 + Ax + B. If ∆ = 4A 3 + 27B 2 is the discriminant of this equation, recall that quasiminimal 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 Sintegers by OS. Let j be the jinvariant 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 jinvariant is nonintegral 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 Sintegral 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 Sintegral points is at most 32 · 10 11 (32 · 10 9) rδ(j)+s.
Fourier Coefficients of Half Integral Weight Modular Forms Modulo l
 Annals of Mathematics
, 1998
"... Dedicated to the memory of S. Chowla. ..."
RiemannRoch and AbelJacobi theory on a finite graph
 Adv. Math
"... Abstract. It is wellknown 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 graphtheoretic analogue of the classic ..."
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Cited by 38 (4 self)
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Abstract. It is wellknown 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 graphtheoretic analogue of the classical RiemannRoch theorem. We also prove several results, analogous to classical facts about Riemann surfaces, concerning the AbelJacobi map from a graph to its Jacobian. As an application of our results, we characterize the existence or nonexistence of a winning strategy for a certain chipfiring game played on the vertices of a graph. 1.
Tamagawa Numbers for Motives with (NonCommutative) Coefficients
 DOCUMENTA MATH.
, 2001
"... Let M be a motive which is defined over a number field and admits an action of a finite dimensional semisimple Qalgebra A. We formulate and study a conjecture for the leading coefficient of the Taylor expansion at 0 of the Aequivariant Lfunction of M. This conjecture simultaneously generalizes a ..."
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Cited by 37 (11 self)
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Let M be a motive which is defined over a number field and admits an action of a finite dimensional semisimple Qalgebra A. We formulate and study a conjecture for the leading coefficient of the Taylor expansion at 0 of the Aequivariant Lfunction of M. This conjecture simultaneously generalizes and refines the Tamagawa number conjecture of Bloch, Kato, Fontaine, PerrinRiou 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 Astructure ’ 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 nonmaximal. In each such case the conjecture with respect to a nonmaximal 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.
Large Torsion Subgroups Of Split Jacobians Of Curves Of Genus Two Or Three
 FORUM MATH
, 1998
"... 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 genus2 curves over Q whose Jac ..."
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Cited by 33 (7 self)
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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 genus2 curves over Q whose Jacobians each have 128 rational torsion points. Also, we find the genus3 curve ) = 0, whose Jacobian has 864 rational torsion points.
Canonical periods and congruence formulae
 Duke Math. J
, 1999
"... The purpose of this article is to show how congruences between the Fourier coefficients of Hecke eigenforms give rise to corresponding congruences between the algebraic parts of the critical values of the associated Lfunctions. This study was initiated by B. Mazur in his fundamental work on the Eis ..."
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Cited by 24 (1 self)
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The purpose of this article is to show how congruences between the Fourier coefficients of Hecke eigenforms give rise to corresponding congruences between the algebraic parts of the critical values of the associated Lfunctions. This study was initiated by B. Mazur in his fundamental work on the Eisenstein ideal (see [Maz77] and [Maz79]) where it was made clear that congruences for analytic Lvalues were closely related to
Finiteness theorems in geometric classfield theory, Enseign
 Math
, 1981
"... Mit dem Zugriff auf den vorliegenden Inhalt gelten die Nutzungsbedingungen als akzeptiert. Die angebotenen Dokumente stehen für nichtkommerzielle Zwecke in Lehre, Forschung und für die private Nutzung frei zur Verfügung. Einzelne Dateien oder Ausdrucke aus diesem Angebot können zusammen mit diesen ..."
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Cited by 21 (3 self)
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Mit dem Zugriff auf den vorliegenden Inhalt gelten die Nutzungsbedingungen als akzeptiert. Die angebotenen Dokumente stehen für nichtkommerzielle Zwecke in Lehre, Forschung und für die private Nutzung frei zur Verfügung. Einzelne Dateien oder Ausdrucke aus diesem Angebot können zusammen mit diesen Nutzungsbedingungen und unter deren Einhaltung weitergegeben werden.
Faltings plus epsilon, Wiles plus epsilon, and the generalized Fermat equation
 C. R. Math. Rep. Acad. Sci. Canada
, 1997
"... 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 Grai ..."
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Cited by 17 (3 self)
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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 nontrivial 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 nontrivial primitive solutions to the generalized Fermat equation? In [DG], Andrew Granville and I made the following conjecture: