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Modularity of certain potentially BarsottiTate Galois representations
 J. Amer. Math. Soc
, 1999
"... Conjectures of Langlands, Fontaine and Mazur [22] predict that certain Galois representations ρ: Gal(Q/Q) → GL2(Qℓ) (where ℓ denotes a fixed prime) should arise from modular forms. This applies in particular to representations defined by the action of Gal(Q/Q) on the ℓadic Tate ..."
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Cited by 63 (6 self)
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Conjectures of Langlands, Fontaine and Mazur [22] predict that certain Galois representations ρ: Gal(Q/Q) → GL2(Qℓ) (where ℓ denotes a fixed prime) should arise from modular forms. This applies in particular to representations defined by the action of Gal(Q/Q) on the ℓadic Tate
The GL2 main conjecture for elliptic curves without complex multiplication
 Publ. I.H.E.S. 101 (2005
"... The main conjectures of Iwasawa theory provide the only general method known at present for studying the mysterious relationship between purely arithmetic problems and the special values of complex Lfunctions, typified by the conjecture of Birch and Swinnerton ..."
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Cited by 34 (11 self)
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The main conjectures of Iwasawa theory provide the only general method known at present for studying the mysterious relationship between purely arithmetic problems and the special values of complex Lfunctions, typified by the conjecture of Birch and Swinnerton
Rational Points on Modular Elliptic Curves
"... Based on an NSFCBMS lecture series given by the author at the University of Central Florida in Orlando from August 8 to 12, 2001, this monograph surveys some recent developments in the arithmetic of modular elliptic curves, with special emphasis on the Birch and SwinnertonDyer conjecture, the ..."
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Cited by 33 (9 self)
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Based on an NSFCBMS lecture series given by the author at the University of Central Florida in Orlando from August 8 to 12, 2001, this monograph surveys some recent developments in the arithmetic of modular elliptic curves, with special emphasis on the Birch and SwinnertonDyer conjecture, the construction of rational points on modular elliptic curves, and the crucial role played by modularity in shedding light on these questions.
Average Frobenius Distribution of Elliptic Curves
 Internat. Math. Res. Notices
, 1998
"... this paper average estimates related to the LangTrotter conjecture. The average distribution fits the one predicted by the conjecture, and the conjectural constant C E,r of Lang and Trotter is confirmed by our results, as seen in Section 2. Average estimates for the case r 0 were already obtained ..."
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Cited by 31 (4 self)
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this paper average estimates related to the LangTrotter conjecture. The average distribution fits the one predicted by the conjecture, and the conjectural constant C E,r of Lang and Trotter is confirmed by our results, as seen in Section 2. Average estimates for the case r 0 were already obtained by Fouvry and Murty [6], and we obtain a generalization of their results for any r Z. The techniques of Fouvry and Murty do not seem to extend to the general case r Z. Our proof then differs significantly from theirs. In the following, we fix r Z, and we denote by E(a, b) the elliptic curve Y b with a, b Z. Then . Following [11], we define 2 # t log t # . Theorem 1.2. Let r be an integer,A,B# 1. For every c>0, we have 1 , (2) where . (3) The constants in the Osymbol depend only on c and r. As the infinite product of (3) converges to a positive number, the constant C r is nonzero, even if some C E,r can be zero, as mentioned above. From the last theorem, we immediately obtain that the LangTrotter conjecture is true "on average." Corollary 1.3. Let #>0. If A,B>x , we have as x ##, . In analogy with the classical terminology, we can say that the average order of E(a,b) (x)isC r ( # x/ log x). Using the same techniques, we can also prove that the normal order of # E(a,b) (x)isC r ( # x/log x). Then,# C r ( # x/ log x) for "almost all" E(a, b) rather than on average (see Corollary 1.5). We are grateful to A. Granville for suggesting this application of our techniques
ABC Allows Us to Count Squarefrees
 Internat. Math. Res. Notices
, 1998
"... . We show several consequences of the abcconjecture for questions in analytic number theory which were of interest to Paul Erdos: For any given polynomial f(x) 2 Z[x], we deduce, from the abcconjecture, an asymptotic estimate for the frequency with which f(n) is squarefree, when n is an integer ..."
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Cited by 26 (0 self)
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. We show several consequences of the abcconjecture for questions in analytic number theory which were of interest to Paul Erdos: For any given polynomial f(x) 2 Z[x], we deduce, from the abcconjecture, an asymptotic estimate for the frequency with which f(n) is squarefree, when n is an integer (and also deduce such estimates for binary homogenous forms). Amongst several applications of this result, we deduce that there is a squarefree number in every interval of length O(x " ) around x, and give the asymptotic formula, predicted by Erdos, for the average moments for the gaps between squarefree numbers. 1. Introduction. For any given polynomial f(x) 2 Z[x], we investigate what proportion of the integers f(1); f(2); f(3); : : : are squarefree 1 ? The values, at integers, taken by certain polynomials, are always divisible by a square for not entirely obvious reasons (for example n(n \Gamma 1)(n \Gamma 2)(n \Gamma 3) is always divisible by 8). We take care of this as follo...
Equirépartition des petits points
, 1997
"... Soit E une courbe elliptique sur le corps C des nombres complexes. On note E[n] (resp E[n]) le sous–groupe des points de n–torsion (resp l’ensemble des points d’ordre exactement n). Une simple inspection permet de voir que les points de torsion sont denses dans E pour la topologie de C. Ils sont mêm ..."
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Cited by 26 (4 self)
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Soit E une courbe elliptique sur le corps C des nombres complexes. On note E[n] (resp E[n]) le sous–groupe des points de n–torsion (resp l’ensemble des points d’ordre exactement n). Une simple inspection permet de voir que les points de torsion sont denses dans E pour la topologie de C. Ils sont même équidistribués
Kolyvagin's descent and MordellWeil groups over ring class fields
, 2000
"... Introduction Let E=Q be a modular elliptic curve with the modular parametrization: : X 0 (N ) ! E; where X 0 (N ) is the complete curve over Q which classies pairs of elliptic curves related by a cyclic Nisogeny. The curve E is equipped with the collection of Heegner points dened over ring clas ..."
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Cited by 24 (11 self)
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Introduction Let E=Q be a modular elliptic curve with the modular parametrization: : X 0 (N ) ! E; where X 0 (N ) is the complete curve over Q which classies pairs of elliptic curves related by a cyclic Nisogeny. The curve E is equipped with the collection of Heegner points dened over ring class elds of suitable imaginary quadratic elds. More precisely, let K be an imaginary quadratic eld in which all rational primes dividing N are split and let O be the order of K of conductor c prime to N . There exists a proper Oideal N such that the natural projection of complex tori C=O ! C=N 1
Galois representations with conjectural connections to arithmetic cohomology
 Duke Math. J
"... In this paper we extend a conjecture of A. Ash and W. Sinnott relating niveau 1 Galois representations to the mod p cohomology of congruence subgroups of SLn(Z) to include Galois representations of higher niveau. We then present computational evidence for our conjecture in the case n = 3 in the form ..."
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Cited by 22 (10 self)
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In this paper we extend a conjecture of A. Ash and W. Sinnott relating niveau 1 Galois representations to the mod p cohomology of congruence subgroups of SLn(Z) to include Galois representations of higher niveau. We then present computational evidence for our conjecture in the case n = 3 in the form of threedimensional Galois representations which appear to correspond to cohomology eigenclasses as predicted by the conjecture. Our examples include Galois representations with nontrivial weight and level, as well as irreducible threedimensional representations that are in no obvious way related to lowerdimensional representations. In addition, we prove that certain symmetric square representations are actually attached to cohomology eigenclasses predicted by the conjecture. 1.
Arithmetic height functions over finitely generated fields
 Inventiones Mathematicae 140
, 2000
"... ABSTRACT. In this paper, we propose a new height function for a variety defined over a finitely generated field over Q. For this height function, we will prove Northcott’s theorem and Bogomolov’s conjecture, so that we can recover the original Raynaud’s theorem (ManinMumford’s conjecture). CONTENTS ..."
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Cited by 21 (8 self)
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ABSTRACT. In this paper, we propose a new height function for a variety defined over a finitely generated field over Q. For this height function, we will prove Northcott’s theorem and Bogomolov’s conjecture, so that we can recover the original Raynaud’s theorem (ManinMumford’s conjecture). CONTENTS