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On modular signs
"... Abstract. We consider some questions related to the signs of Hecke eigenvalues or Fourier coefficients of classical modular forms. One problem is to determine to what extent those signs, for suitable sets of primes, determine uniquely the modular form, and we give both individual and statistical res ..."
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Abstract. We consider some questions related to the signs of Hecke eigenvalues or Fourier coefficients of classical modular forms. One problem is to determine to what extent those signs, for suitable sets of primes, determine uniquely the modular form, and we give both individual and statistical results. The second problem, which has been considered by a number of authors, is to determine the size, in terms of the conductor and weight, of the first signchange of Hecke eigenvalues. Here we improve the recent estimate of Iwaniec, Kohnen and Sengupta. 1.
Hyperelliptic curves, Lpolynomials, and random matrices, to appear
 in the lecture notes of AGCT11, Contemporary Mathematics
"... Abstract. We analyze the distribution of unitarized Lpolynomials ¯Lp(T) (as p varies) obtained from a hyperelliptic curve of genus g ≤ 3 defined over Q. In the generic case, we find experimental agreement with a predicted correspondence (based on the KatzSarnak random matrix model) between the dis ..."
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Abstract. We analyze the distribution of unitarized Lpolynomials ¯Lp(T) (as p varies) obtained from a hyperelliptic curve of genus g ≤ 3 defined over Q. In the generic case, we find experimental agreement with a predicted correspondence (based on the KatzSarnak random matrix model) between the distributions of ¯Lp(T) and of characteristic polynomials of random matrices in the compact Lie group USp(2g). We then formulate an analogue of the SatoTate conjecture for curves of genus 2, in which the generic distribution is augmented by 24 exceptional distributions, each corresponding to a compact subgroup of USp(4). In every case but two, we exhibit a curve closely matching the proposed distribution, and can find no curves unaccounted for by our classification. 1.
SOME ASPECTS AND APPLICATIONS OF THE RIEMANN HYPOTHESIS OVER FINITE FIELDS
"... Abstract. We give a survey of some aspects of the Riemann Hypothesis over finite fields, as it was proved by Deligne, and its applications to analytic number theory. In particular, we concentrate on the formalism leading to Deligne’s Equidistribution Theorem. 1. ..."
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Abstract. We give a survey of some aspects of the Riemann Hypothesis over finite fields, as it was proved by Deligne, and its applications to analytic number theory. In particular, we concentrate on the formalism leading to Deligne’s Equidistribution Theorem. 1.
THREEDIMENSIONAL DISCRETE SYSTEMS OF HIROTAKIMURA TYPE AND DEFORMED LIEPOISSON ALGEBRAS
"... Abstract. Recently Hirota and Kimura presented a new discretization of the Euler top with several remarkable properties. In particular this discretization shares with the original continuous system the feature that it is an algebraically completely integrable biHamiltonian system in three dimension ..."
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Abstract. Recently Hirota and Kimura presented a new discretization of the Euler top with several remarkable properties. In particular this discretization shares with the original continuous system the feature that it is an algebraically completely integrable biHamiltonian system in three dimensions. The HirotaKimura discretization scheme turns out to be equivalent to an approach to numerical integration of quadratic vector fields that was introduced by Kahan, who applied it to the twodimensional LotkaVolterra system. The Euler top is naturally written in terms of the so(3) LiePoisson algebra. Here we consider algebraically integrable systems that are associated with pairs of LiePoisson algebras in three dimensions, as presented by Gümral and Nutku, and construct birational maps that discretize them according to the scheme of Kahan and HirotaKimura. We show that the maps thus obtained are also biHamiltonian, with pairs of compatible Poisson brackets that are oneparameter deformations of the original LiePoisson algebras, and hence they are completely integrable. For comparison, we also present analogous discretizations for three biHamiltonian systems that have a transcendental invariant, and finally we analyze all of the maps obtained from the viewpoint of Halburd’s Diophantine integrability criterion.
University of Kent,
, 810
"... Abstract. Recently Hirota and Kimura presented a new discretization of the Euler top with several remarkable properties. In particular this discretization shares with the original continuous system the feature that it is an algebraically completely integrable biHamiltonian system in three dimension ..."
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Abstract. Recently Hirota and Kimura presented a new discretization of the Euler top with several remarkable properties. In particular this discretization shares with the original continuous system the feature that it is an algebraically completely integrable biHamiltonian system in three dimensions. The HirotaKimura discretization scheme turns out to be equivalent to an approach to numerical integration of quadratic vector fields that was introduced by Kahan, who applied it to the twodimensional LotkaVolterra system. The Euler top is naturally written in terms of the so(3) LiePoisson algebra. Here we consider algebraically integrable systems that are associated with pairs of LiePoisson algebras in three dimensions, as presented by Gümral and Nutku, and construct birational maps that discretize them according to the scheme of Kahan and HirotaKimura. We show that the maps thus obtained are also biHamiltonian, with pairs of compatible Poisson brackets that are oneparameter deformations of the original LiePoisson algebras, and hence they are completely integrable. For comparison, we also present analogous discretizations for three biHamiltonian systems that have a transcendental invariant, and finally we analyze all of the maps obtained from the viewpoint of Halburd’s Diophantine integrability criterion. 1.
Is it plausible?
, 2012
"... Rough notes in preparation for a lecture at the joint AMSMAA conference, Jan. 5, 2012 We mathematicians have handy ways of discovering what stands a chance of being true. And we have a range of different modes of evidence that help us form these expectations; such as: analogies with things that are ..."
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Rough notes in preparation for a lecture at the joint AMSMAA conference, Jan. 5, 2012 We mathematicians have handy ways of discovering what stands a chance of being true. And we have a range of different modes of evidence that help us form these expectations; such as: analogies with things that are indeed true, computations, special case justifications, etc. They abound, these methods—explicitly formulated, or not. They lead us, sometimes, to a mere hint of a possibility that a mathematical statement might be plausible. They lead us, other times, to substantially firm—even though not yet justified—belief. They may lead us astray. Our endgame, of course, is understanding, verification, clarification, and most certainly: proof; truth, in short. Consider the beginning game, though. With the word “plausible ” in my title, you can guess that I’m a fan of George Pólya’s classic Mathematics and Plausible Reasoning ([?]). George Pólya (18871985) I think that it is an important work for many reasons, but mainly because Pólya is pointing to an activity that surely takes up the majority of time, and energy, of anyone engaged in thinking 1 about mathematics, or in trying to work towards a new piece of mathematics. Usually under limited knowledge and much ignorance, often plagued by mistakes and misconceptions, we wrestle
cloth, US $75.00
, 2013
"... The story of equidistribution in number theory began about a hundred years ago with H. Weyl’s paper [18] concerning the distribution of sequences of real numbers modulo 1, and more generally that of points in euclidean space modulo a lattice. The theme of equidistribution has since become one of the ..."
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The story of equidistribution in number theory began about a hundred years ago with H. Weyl’s paper [18] concerning the distribution of sequences of real numbers modulo 1, and more generally that of points in euclidean space modulo a lattice. The theme of equidistribution has since become one of the most important unifying viewpoints in number theory. Equidistribution statements exist in many areas of number theory, which would otherwise seem to be very distant, and lead to sometimes surprising connections and applications (as can be seen, for instance, with the quantum unique ergodicity conjecture [15], expander graphs, especially Cayley graphs [13], ergodic theory of large groups [2], or the SatoTate conjecture [14], to give only examples taken from recent articles or reviews in the Bulletin of the AMS). Because equidistribution is a twin of the probabilistic idea of convergence in law, it also introduces strong links between arithmetic and fields, such as probability theory or ergodic theory. In addition to applications, one might add that equidistribution theorems are often by themselves extremely beautiful. In this review, we will work with the following definition, which is sufficient: given a compact topological space X and a Borel probability measure μ on X (i.e., a Borel measure such that μ(X) = 1), and given a sequence (Yn) of (nonempty) finite sets1 together with maps X θn: Yn − → X, one says that (Yn,θn) becomes equidistributed in X with respect to μ if, for any continuous function f: X − → C, wehave
MODULAR SIGNS, OR YET ANOTHER RECOGNITION PROBLEM FOR MODULAR FORMS
"... There are many results in the arithmetic of modular forms which are, more or less, concerned with various ways of characterizing a given (primitive) cusp form f from its siblings, starting from the fact that Fourier coefficients, hence the Lfunction, determine uniquely a cusp form f relative to a c ..."
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There are many results in the arithmetic of modular forms which are, more or less, concerned with various ways of characterizing a given (primitive) cusp form f from its siblings, starting from the fact that Fourier coefficients, hence the Lfunction, determine uniquely a cusp form f relative to a congruence subgroup Γ of SL(2, Z) 1, going through stronger forms of the multiplicity one theorem for automorphic representations, and then to various explicit forms of these statements, where only finitely many coefficients are required (say at primes p � X, for some explicit X depending on the parameters defining f), and to “statistic ” versions of the latter, where X can be reduced drastically, provided one accepts some possible exceptions. Some of these statements were strongly suggested by the analogy with the problem of the least quadraticnonresidue, which is a problem of great historic importance in analytic number theory. 2 Recently, Lau and Wu [LW2] have found, for real characters, a precise “threshold ” y(Q) for which the upper bound on the number of real characters of conductor q � Q with value +1 for primes p � y(Q) almost coincides with a lower bound for this number. Then they proved in [LW1] a corresponding upperbound for recognition problems for modular