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Abelian varieties
, 2008
"... These notes are an introduction to the theory of abelian varieties, including the arithmetic of abelian varieties and Faltings’s proof of certain finiteness theorems. The orginal version of the notes was distributed during the teaching of an advanced graduate course. Alas, the notes are still in ver ..."
Abstract

Cited by 160 (8 self)
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These notes are an introduction to the theory of abelian varieties, including the arithmetic of abelian varieties and Faltings’s proof of certain finiteness theorems. The orginal version of the notes was distributed during the teaching of an advanced graduate course. Alas, the notes are still in very rough form.
Algebraic Number Theory
 www.jmilne.org/math
, 2009
"... Version 3.06 May 28, 2014An algebraic number field is a finite extension of Q; an algebraic number is an element of an algebraic number field. Algebraic number theory studies the arithmetic of algebraic number fields — the ring of integers in the number field, the ideals and units in the ring of int ..."
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Cited by 7 (0 self)
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Version 3.06 May 28, 2014An algebraic number field is a finite extension of Q; an algebraic number is an element of an algebraic number field. Algebraic number theory studies the arithmetic of algebraic number fields — the ring of integers in the number field, the ideals and units in the ring of integers, the extent to which unique factorization holds, and so on. An abelian extension of a field is a Galois extension of the field with abelian Galois group. Class field theory describes the abelian extensions of a number field in terms of the arithmetic of the field. These notes are concerned with algebraic number theory, and the sequel with class field theory. BibTeX information
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"... I agree that this thesis shall be available in accordance with the regulations governing the University of Warwick theses. I agree that the summary of this thesis may be submitted for publication. I agree that the thesis may be photocopied (single copies for study purposes only). Theses with no rest ..."
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I agree that this thesis shall be available in accordance with the regulations governing the University of Warwick theses. I agree that the summary of this thesis may be submitted for publication. I agree that the thesis may be photocopied (single copies for study purposes only). Theses with no restriction on photocopying will also be made available to the British Library for microfilming. The British Library may supply copies to individuals or libraries. subject to a statement from them that the copy is supplied for nonpublishing purposes. All copies supplied by the British Library will carry the following statement: “Attention is drawn to the fact that the copyright of this thesis rests with its author. This copy of the thesis has been supplied on the condition that anyone who consults it is understood to recognise that its copyright rests with its author and that no quotation from the thesis and no information derived from it may be published without the author’s written consent.”
Contents
"... Abstract: The paper reviews existing results about the statistical distribution of zeros for three main types of zeta functions: numbertheoretical, geometrical, and dynamical. It provides necessary background and some details about the proofs of the main results. ..."
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Abstract: The paper reviews existing results about the statistical distribution of zeros for three main types of zeta functions: numbertheoretical, geometrical, and dynamical. It provides necessary background and some details about the proofs of the main results.
Prepared for submission to JHEP Uniqueness of TwoLoop Master Contours
"... Abstract: Generalizedunitarity calculations of twoloop amplitudes are performed by expanding the amplitude in a basis of master integrals and then determining the coefficients by taking a number of generalized cuts. In this paper, we present a complete classification of the solutions to the maxima ..."
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Abstract: Generalizedunitarity calculations of twoloop amplitudes are performed by expanding the amplitude in a basis of master integrals and then determining the coefficients by taking a number of generalized cuts. In this paper, we present a complete classification of the solutions to the maximal cut of integrals with the doublebox topology. The ideas presented here are expected to be relevant for all twoloop topologies as well. We find that these maximalcut solutions are naturally associated with Riemann surfaces whose topology is determined by the number of states at the vertices of the doublebox graph. In the case of four massless external momenta we find that, once the geometry of these Riemann surfaces is properly understood, there are uniquely defined master contours producing the coefficients of the doublebox integrals in the basis decomposition of the twoloop amplitude. This is in perfect analogy with the situation in oneloop generalized unitarity. In addition, we point out that the chiral integrals recently introduced by ArkaniHamed et al. can be used as master integrals for the doublebox contributions to the twoloop amplitudes in any gauge theory. The infrared finiteness of these integrals allow for their coefficients as well as their integrated expressions to be evaluated in strictly four dimensions, providing significant technical simplification. We evaluate these integrals at four points and obtain remarkably compact results.