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A Course in Arithmetic
01 Jan 1973-
TL;DR: 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.
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01 Jan 1986TL;DR: It is shown here how Elliptic Curves over Finite Fields, Local Fields, and Global Fields affect the geometry of the elliptic curves.
Abstract: Algebraic Varieties.- Algebraic Curves.- The Geometry of Elliptic Curves.- The Formal Group of Elliptic Curves.- Elliptic Curves over Finite Fields.- Elliptic Curves over C.- Elliptic Curves over Local Fields.- Elliptic Curves over Global Fields.- Integral Points on Elliptic Curves.-Computing the Mordell Weil Group.- Appendix A: Elliptic Curves in Characteristics.-Appendix B: Group Cohomology (H0 and H1).
4,680 citations
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01 Jan 1999
TL;DR: In this paper, Algebraic integral integers, Riemann-Roch theory, Abstract Class Field Theory, Local Class Field theory, Global Class Field and Zeta Functions are discussed.
Abstract: I: Algebraic Integers.- II: The Theory of Valuations.- III: Riemann-Roch Theory.- IV: Abstract Class Field Theory.- V: Local Class Field Theory.- VI: Global Class Field Theory.- VII: Zeta Functions and L-series.
2,824 citations
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TL;DR: In this article, the authors considered string propagation on the quotient of a flat torus by a discrete group and obtained an exactly soluble and more or less realistic method of string compactification.
1,870 citations
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TL;DR: By studying the partition function of N = 4 topologically twisted supersymmetric Yang-Mills on four-manifolds, this paper made an exact strong coupling test of the Montonen-Olive strong-weak duality conjecture.
1,381 citations
Cites background or methods from "A Course in Arithmetic"
...Moreover, it is known [49] that every self-dual lattice can be transformed to an odd, indefinite self-dual lattice by taking a direct sum, if necessary, with I or I....
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...24 How to compute such sums is explained in [49], as was pointed out to us by Dick Gross....
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...Then [49] every indefinite odd self-dual lattice(25) is a direct sum of the form...
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TL;DR: In this article, a new theory of closed orientable superstrings is constructed as a chiral combination of the closed D = 26 bosonic and D = 10 fermionic strings.
1,111 citations