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79
On the Meromorphic Continuation of Degree Two LFunctions
 DOCUMENTA MATH.
, 2006
"... We prove that the Lfunction of any regular (distinct Hodge numbers), irreducible, rank two motive over the rational numbers has meromorphic continuation to the whole complex plane and satisfies the expected functional equation. ..."
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Cited by 28 (2 self)
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We prove that the Lfunction of any regular (distinct Hodge numbers), irreducible, rank two motive over the rational numbers has meromorphic continuation to the whole complex plane and satisfies the expected functional equation.
Compatibility of local and global Langlands correspondences
 J. Amer. Math. Soc
, 2007
"... Abstract. We prove the compatibility of local and global Langlands correspondences for GLn, which was proved up to semisimplification in [HT]. More precisely, for the ndimensional ladic representation Rl(Π) of the Galois group of a CMfield L attached to a conjugate selfdual regular algebraic cusp ..."
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Cited by 20 (7 self)
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Abstract. We prove the compatibility of local and global Langlands correspondences for GLn, which was proved up to semisimplification in [HT]. More precisely, for the ndimensional ladic representation Rl(Π) of the Galois group of a CMfield L attached to a conjugate selfdual regular algebraic cuspidal automorphic representation Π, which is square integrable at some finite place, we show that Frobenius semisimplification of the restriction of Rl(Π) to the decomposition group of a prime v of L not dividing l corresponds to Πv by the local Langlands correspondence.
Cyclicity of elliptic curves modulo p and elliptic curve analogues of Linnik’s problem
, 2001
"... 1 Let E be an elliptic curve defined over Q and of conductor N. For a prime p ∤ N, we denote by E the reduction of E modulo p. We obtain an asymptotic formula for the number of primes p ≤ x for which E(Fp) is cyclic, assuming a certain generalized Riemann hypothesis. The error terms that we get are ..."
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Cited by 14 (3 self)
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1 Let E be an elliptic curve defined over Q and of conductor N. For a prime p ∤ N, we denote by E the reduction of E modulo p. We obtain an asymptotic formula for the number of primes p ≤ x for which E(Fp) is cyclic, assuming a certain generalized Riemann hypothesis. The error terms that we get are substantial improvements of earlier work of J.P. Serre and M. Ram Murty. We also consider the problem of finding the size of the smallest prime p = pE for which the group E(Fp) is cyclic and we show that, under the generalized Riemann hypothesis, pE = O � (log N) 4+ε � if E is without complex multiplication, and pE = O � (log N) 2+ε � if E is with complex multiplication, for any 0 < ε < 1. 1
Do all elliptic curves of the same order have the same difficulty of discrete log
 Advances in Cryptology — ASIACRYPT 2005, Lecture Notes in Computer Science
"... Abstract. The aim of this paper is to justify the common cryptographic practice of selecting elliptic curves using their order as the primary criterion. We can formalize this issue by asking whether the discrete log problem (dlog) has the same difficulty for all curves over a given finite field with ..."
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Cited by 13 (4 self)
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Abstract. The aim of this paper is to justify the common cryptographic practice of selecting elliptic curves using their order as the primary criterion. We can formalize this issue by asking whether the discrete log problem (dlog) has the same difficulty for all curves over a given finite field with the same order. We prove that this is essentially true by showing polynomial time random reducibility of dlog among such curves, assuming the Generalized Riemann Hypothesis (GRH). We do so by constructing certain expander graphs, similar to Ramanujan graphs, with elliptic curves as nodes and low degree isogenies as edges. The result is obtained from the rapid mixing of random walks on this graph. Our proof works only for curves with (nearly) the same endomorphism rings. Without this technical restriction such a dlog equivalence might be false; however, in practice the restriction may be moot, because all known polynomial time techniques for constructing equal order curves produce only curves with nearly equal endomorphism rings.
Galois representations
 Proceedings of the International Congress of Mathematicians, Beijing, 2002, vol I. World Scientific
"... In the first part of this paper we try to explain to a general mathematical audience some of the remarkable web of conjectures linking representations of Galois groups with algebraic geometry, complex analysis and discrete subgroups of Lie groups. In the second part we briefly review some limited re ..."
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Cited by 9 (0 self)
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In the first part of this paper we try to explain to a general mathematical audience some of the remarkable web of conjectures linking representations of Galois groups with algebraic geometry, complex analysis and discrete subgroups of Lie groups. In the second part we briefly review some limited recent progress on these conjectures.
Construction and examples of higherdimensional modular CalabiYau manifolds
, 2006
"... We construct several examples of higherdimensional CalabiYau manifolds and prove their modularity. ..."
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Cited by 8 (2 self)
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We construct several examples of higherdimensional CalabiYau manifolds and prove their modularity.
On Fourier coefficients of Maass waveforms for PSL(2
 Z), Minnesota Supercomputer Institute Research Report
"... Dedicated to the memory ofD. H. Lehmer Abstract. In this paper, we use machine experiments to test the validity of the SatoTate conjecture for Maass waveforms on PSL(2, Z)\H. We also elaborate on Stark's iterative method for calculating the Fourier coefficients of such forms. 1. Introductory remark ..."
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Cited by 8 (2 self)
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Dedicated to the memory ofD. H. Lehmer Abstract. In this paper, we use machine experiments to test the validity of the SatoTate conjecture for Maass waveforms on PSL(2, Z)\H. We also elaborate on Stark's iterative method for calculating the Fourier coefficients of such forms. 1. Introductory remarks Around 20 years ago, D. H. Lehmer [19] empirically investigated the extent to which the numbers Çp = r(p)p~xx^2 obey SatoTate statistics as p> oo. Here, t(«) is the usual Ramanujan taufunction1 and the proposed statistics assert that m n r N[p<x:6peE] 2 f. 2 (1.1) lim ————^ = sin Odd x^oo n(x) n JE for ¡tp = 2cos(f9p) and any Jordan measurable E ç [0, n]. [n(x) is the usual counting function for the primes.] The corresponding assertion for é,p itself will then read: (1.2) lim *lPg*:p/n = 1 t ^^ x»oo n(x) In JE for E C [2, 2]. In this form, the proposed distribution coincides with the socalled Wigner semicircle law familiar from the study of spectra of random
Torsion points on modular curves
 Invent. Math
, 1999
"... Abstract. Let N ≥ 23 be a prime number. In this paper, we prove a conjecture of Coleman, Kaskel, and Ribet about the Qvalued points of the modular curve X0(N) which map to torsion points on J0(N) via the cuspidal embedding. We give some generalizations to other modular curves, and to noncuspidal em ..."
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Cited by 8 (2 self)
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Abstract. Let N ≥ 23 be a prime number. In this paper, we prove a conjecture of Coleman, Kaskel, and Ribet about the Qvalued points of the modular curve X0(N) which map to torsion points on J0(N) via the cuspidal embedding. We give some generalizations to other modular curves, and to noncuspidal embeddings of X0(N) into J0(N). 1.
A Uniform open image theorem for ℓadic representations
, 2008
"... Abstract. Let k be a field finitely generated over Q and let X be a smooth, separated and geometrically connected curve over k. Fix a prime ℓ. A representation ρ: π1(X) → GLm(Zℓ) is said to be geometrically strictly rationally perfect if any open subgroup of ρ(π1(Xk)) has finite abelianization. Typ ..."
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Cited by 8 (6 self)
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Abstract. Let k be a field finitely generated over Q and let X be a smooth, separated and geometrically connected curve over k. Fix a prime ℓ. A representation ρ: π1(X) → GLm(Zℓ) is said to be geometrically strictly rationally perfect if any open subgroup of ρ(π1(Xk)) has finite abelianization. Typical examples of such representations are those arising from the action of π1(X) on the generic ℓadic Tate module Tℓ(Aη) of an abelian scheme A over X or, more generally, from the action of π1(X) on the ℓadic etale cohomology groups Hi (Yη, Qℓ), i ≥ 0 of the geometric generic fiber of a smooth proper scheme Y over X. Let G denote the image of ρ. Any krational point x on X induces a splitting x: Γk: = π1(Spec(k)) → π1(X) of the canonical restriction epimorphism π1(X) → Γk so one can define the closed subgroup Gx: = ρ ◦ x(Γk) ⊂ G. The main result of this note is the following uniform open image theorem. Under the above assumptions, for any geometrically strictly rationally perfect representation ρ: π1(X) → GLm(Zℓ) the set Xρ of all x ∈ X(k) such that Gx is not open in G is finite and there exists an integer Bρ ≥ 1 such that [G: Gx] ≤ Bρ, x ∈ X(k) � Xρ.