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A Course in Arithmetic
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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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Tame coverings of arithmetic schemes
TL;DR: In this article, the authors investigated tame fundamental groups of schemes of finite type over Spec(Z) and showed that the tame fundamental group π 1(X, X − X) is a quotient of the etale fundamental group of a regular algebraic scheme.
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Kronecker’s Limit Formula, Holomorphic Modular Functions, and q-Expansions on Certain Arithmetic Groups
TL;DR: For all such arithmetic groups of genus up to and including three, it is proved that the corresponding function field admits two generators whose q-expansions have integer coefficients, has lead coefficient equal to one, and has minimal order of pole at infinity.
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Global discrepancy and small points on elliptic curves
Matthew Baker,Clayton Petsche +1 more
TL;DR: In this article, the authors define the ''global discrepancy'' of a finite set Z of algebraic points on E which in a precise sense measures how far the set is from being adelically equidistributed.
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On bilinear forms represented by trees
TL;DR: In this article, the adjacency matrix of a weighted graph determines an integral bilinear form, and the trees with unimodular adjACency matrices are described with special emphasis on the definite and semidefinite cases, since they arise as configuration graphs of good divisors in compact complex surfaces.