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Elliptic Curves And Primality Proving
 Math. Comp
, 1993
"... The aim of this paper is to describe the theory and implementation of the Elliptic Curve Primality Proving algorithm. ..."
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Cited by 162 (22 self)
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The aim of this paper is to describe the theory and implementation of the Elliptic Curve Primality Proving algorithm.
Evidence for a Spectral Interpretation of the Zeros of LFunctions
, 1998
"... By looking at the average behavior (nlevel density) of the low lying zeros of certain families of Lfunctions, we find evidence, as predicted by function field analogs, in favor of a spectral interpretation of the nontrivial zeros in terms of the classical compact groups. This is further supported ..."
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Cited by 33 (7 self)
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By looking at the average behavior (nlevel density) of the low lying zeros of certain families of Lfunctions, we find evidence, as predicted by function field analogs, in favor of a spectral interpretation of the nontrivial zeros in terms of the classical compact groups. This is further supported by numerical experiments for which an efficient algorithm to compute Lfunctions was developed and implemented. iii Acknowledgements When Mike Rubinstein woke up one morning he was shocked to discover that he was writing the acknowledgements to his thesis. After two screenplays, a 40000 word manifesto, and many fruitless attempts at making sushi, something resembling a detailed academic work has emerged for which he has people to thank. Peter Sarnak from Chebyshev's Bias to USp(1). For being a terrific advisor and teacher. For choosing problems suited to my talents and involving me in this great project to understand the zeros of Lfunctions. Zeev Rudnick and Andrew Oldyzko for many disc...
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 31 (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.
Congruence subgroups and rational conformal field theory ,mathQA/9909080
"... We address here the question of whether the characters of an RCFT are modular functions for some level N, i.e. whether the representation of the modular group SL2(Z) coming from any RCFT ( is trivial) on some congruence subgroup. We prove that if the 1 1 matrix T, associated to ∈ SL2(Z), has odd ord ..."
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Cited by 10 (0 self)
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We address here the question of whether the characters of an RCFT are modular functions for some level N, i.e. whether the representation of the modular group SL2(Z) coming from any RCFT ( is trivial) on some congruence subgroup. We prove that if the 1 1 matrix T, associated to ∈ SL2(Z), has odd order, then this must be so. When 0 1 the order of T is even, we present a simple test which if satisfied — and we conjecture it always will be — implies that the characters for that RCFT will also be level N. We use this to explain three curious observations in RCFT made by various authors. This is the presubmission copy. We are interested in receiving any feedback. The published version of this paper will assume slightly more mathematical sophistication; both versions have equivalent content but this one is a little more pedagogical
On Atkin and Swinnerton–Dyer congruence relations (2), prepeint 2005
"... Abstract. In this paper we exhibit a noncongruence subgroup Γ whose space of weight 3 cusp forms S3(Γ) admits a basis satisfying the AtkinSwinnertonDyer congruence relations with two weight 3 newforms for certain congruence subgroups. This gives a modularity interpretation of the motive attached t ..."
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Cited by 8 (5 self)
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Abstract. In this paper we exhibit a noncongruence subgroup Γ whose space of weight 3 cusp forms S3(Γ) admits a basis satisfying the AtkinSwinnertonDyer congruence relations with two weight 3 newforms for certain congruence subgroups. This gives a modularity interpretation of the motive attached to S3(Γ) by A. Scholl and also verifies the AtkinSwinnertonDyer congruence conjecture for this space. 1.
An extension of Hecke’s converse theorem
 Internat. Math. Res. Notices
, 1995
"... Abstract. Associated to a newform f(z) is a Dirichlet series Lf(s) with functional equation and Euler product. Hecke showed that if the Dirichlet series F(s) has a functional equation of the appropriate form, then F(s) = Lf(s) for some holomorphic newform f(z) on Γ(1). Weil extended this result to ..."
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Cited by 5 (1 self)
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Abstract. Associated to a newform f(z) is a Dirichlet series Lf(s) with functional equation and Euler product. Hecke showed that if the Dirichlet series F(s) has a functional equation of the appropriate form, then F(s) = Lf(s) for some holomorphic newform f(z) on Γ(1). Weil extended this result to Γ0(N) under an assumption on the twists of F(s) by Dirichlet characters. We show that, at least for small N, the assumption on twists can be replaced by an assumption on the local factors of the Euler product of F(s).
Finite index subgroups of the modular group and their modular forms
, 2007
"... Abstract. Classically, congruence subgroups of the modular group, which can be described by congruence relations, play important roles in group theory and modular forms. In reality, the majority of finite index subgroups of the modular group are noncongruence. These groups as well as their modular f ..."
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Cited by 3 (1 self)
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Abstract. Classically, congruence subgroups of the modular group, which can be described by congruence relations, play important roles in group theory and modular forms. In reality, the majority of finite index subgroups of the modular group are noncongruence. These groups as well as their modular forms are central players of this survey article. Differences between congruence and noncongruence subgroups and modular forms will be discussed. We will mainly focus on three interesting aspects of modular forms for noncongruence subgroups: the unbounded denominator property, modularity of the Galois representation arising from noncongruence cuspforms, and Atkin and SwinnertonDyer congruences. 1.
Integrable Lagrangians and modular forms
, 707
"... We investigate nondegenerate Lagrangians of the form f(ux,uy,ut)dxdy dt such that the corresponding EulerLagrange equations (fux)x + (fuy)y + (fut)t = 0 are integrable by the method of hydrodynamic reductions. We demonstrate that the integrability conditions, which constitute an involutive overde ..."
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Cited by 2 (2 self)
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We investigate nondegenerate Lagrangians of the form f(ux,uy,ut)dxdy dt such that the corresponding EulerLagrange equations (fux)x + (fuy)y + (fut)t = 0 are integrable by the method of hydrodynamic reductions. We demonstrate that the integrability conditions, which constitute an involutive overdetermined system of fourth order PDEs for the Lagrangian density f, are invariant under a 20parameter group of Liepoint symmetries whose action on the moduli space of integrable Lagrangians has an open orbit. The density of the ‘masterLagrangian ’ corresponding to this orbit is shown to be a modular form in three variables defined on a complex hyperbolic ball. We demonstrate how the knowledge of the symmetry group allows one to linearise the integrability conditions.
QUADRATIC MINIMA AND MODULAR FORMS
, 1998
"... Abstract. We give upper bounds on the size of the gap between the constant term and the next nonzero Fourier coefficient of an entire modular form of given weight for Γ0(2). Numerical evidence indicates that a sharper bound holds for the weights h ≡ 2 ( mod 4). We derive upper bounds for the minimu ..."
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Cited by 1 (0 self)
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Abstract. We give upper bounds on the size of the gap between the constant term and the next nonzero Fourier coefficient of an entire modular form of given weight for Γ0(2). Numerical evidence indicates that a sharper bound holds for the weights h ≡ 2 ( mod 4). We derive upper bounds for the minimum positive integer represented by level two even positivedefinite quadratic forms. Our data suggest that, for certain meromorphic modular forms and p = 2, 3, the porder of the constant term is related to the basep expansion of the order of the pole at infinity.