The geometry of Gauss' composition law
Item
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Title
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The geometry of Gauss' composition law
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Identifier
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d_2009_2013:14d1637bfed8:10544
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identifier
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10767
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Creator
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Baishanski, Yelena,
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Contributor
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Lucien Szpiro
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Date
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2010
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Language
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English
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Publisher
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City University of New York.
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Subject
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Mathematics | Theoretical mathematics | arakelov theory | binary quadratic forms | function fields | Gauss composition | number fields
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Abstract
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We examine Gauss composition of quadratic forms from both an arithmetic and geometric perspective. Gauss' identification of a composition law for primitive integral binary quadratic forms of given discriminant D ---which provides the set FD of SL2( Z )-equivalence classes of such forms with a group structure---essentially amounts to the discovery of the class group of an order in a quadratic number field. We consider quadratic extensions of the field of rational functions k(u), where k is an algebraically closed field, and seek an analogue of Gauss composition in this context.;A quadratic extension of k(u) corresponds to the function field of a curve C with affine model t2 = D(u) for some polynomial D = D(u) in k[u], which is of odd degree if C has a smooth ramified point at infinity. Focusing on this case---the analogue of quadratic number fields with one complex place at infinity---we extend the notion of the degree of a Weil divisor on a curve to Cartier divisors on C, and find a bijection between the set of SL 2(k[u])-equivalence classes of primitive forms with coefficients in k[ u] of discriminant D, and the group Pic 0(C) of isomorphism classes of degree zero lines bundles on C..;In parallel fashion, we reinterpret the arithmetic case using Arakelov's invention of metrics associated to the infinite places of a number field. Given an invertible A-module L for an order A in a number field K, we have for each infinite place sigma of K a corresponding one-dimensional C -vector space Lsigma with a positive non-degenerate hermitian metric || ||sigma (which is then determined by the length ||x||2 of any element x of Lsigma). Using a notion of degree of an invertible metrized module---which mirrors the notion of degree used in the geometric case, yielding in both cases a "product formula" deg(f) = 0 for a principal divisor (f)---we find for D < 0 a bijection between classes of positive definite forms in FD and the compactified Picard group Pic0c (A) of isometry classes of degree zero invertible A-modules.
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Type
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dissertation
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Source
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2009_2013.csv
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degree
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Ph.D.
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Program
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Mathematics
