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A Course in Arithmetic
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TLDR
In this article, the theorem on arithmetic progressions modular forms is proved for finite fields p-adic fields Hilbert symbol quadratic forms over Qp, and over Q integral quadratics forms with discriminant +-1.Abstract:
Part 1 Algebraic methods: finite fields p-adic fields Hilbert symbol quadratic forms over Qp, and over Q integral quadratic forms with discriminant +-1. Part 2 Analytic methods: the theorem on arithmetic progressions modular forms.read more
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Modular Hypergeometric Residue Sums of Elliptic Selberg Integrals
TL;DR: In this paper, it was shown that the residue expansion of an elliptic Selberg integral gives rise to an integral representation for a multiple modular hypergeometric series, and a conjectural evaluation formula for the integral then implies a closed summation formula.
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Nonlinear aspects of the renormalization group flows of Dyson's hierarchical model
TL;DR: In this paper, Dyson's Hierarchical Model (HM) is shown to be equivalent to both Wilson's approximate recursion formula and Polchinski's equation in the local potential approximation (despite the very small difference with the exponents of the latter).
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Length spectra andp-spectra of compact flat manifolds
TL;DR: In this article, the authors compare and contrast various length vs Laplace spectra of compact flat Riemannian manifolds, and give a pair of 4-dimensional manifolds that are isospectral on p-forms for p = 1, 3 and a length of a closed geodesic that occurs in one manifold but cannot occur in the other.
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How many primes decompose completely in an infinite unramified Galois extension of a global field
Limiting distributions and zeros of artin l-functions
TL;DR: In this paper, a limiting distribution for the set of conjugacy classes C1,..., Cr and the set PL/K;1,2,...,r is constructed, which can be expressed as an infinite product of J0(x) Bessel functions evaluated at zeros of corresponding Artin L-functions.