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Fourier coefficients of modular forms on G2
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In this article, a theory of Fourier coefficients for modular forms on the split exceptional group G2 over ℚ was developed, where the coefficients are derived from the Fourier coefficient theory of modular forms.Abstract:
We develop a theory of Fourier coefficients for modular forms on the split exceptional group G2 over ℚ.read more
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Lectures on on Black Holes, Topological Strings and Quantum Attractors (2.0)
TL;DR: The relation between the macroscopic entropy of four-dimensional BPS black holes and the microscopic counting of states, beyond the thermodynamical, large charge limit, was discussed in this paper.
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Lectures on black holes, topological strings and quantum attractors*
TL;DR: In this article, the relation between the macroscopic entropy of four-dimensional BPS black holes and the microscopic counting of states, beyond the thermodynamical, large charge limit, is discussed.
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BPS black holes, quantum attractor flows and automorphic forms
Abstract: We propose a program for counting microstates of four-dimensional BPS black holes in N{>=}2 supergravities with symmetric-space valued scalars by exploiting the symmetries of timelike reduction to three dimensions. Inspired by the equivalence between the four-dimensional attractor flow and geodesic flow on the three-dimensional scalar manifold, we radially quantize stationary, spherically symmetric BPS geometries. Connections between the topological string amplitude, attractor wave function, the Ooguri-Strominger-Vafa conjecture and the theory of automorphic forms suggest that black hole degeneracies are counted by Fourier coefficients of modular forms for the three-dimensional U-duality group, associated to special unipotent representations which appear in the supersymmetric Hilbert space of the quantum attractor flow.
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Higher composition laws III: The parametrization of quartic rings
TL;DR: In the first two articles of this series, we investigated various higher analogues of Gauss composition, and showed how several algebraic objects involving orders in quadratic and cubic fields could be explicitly parametrized.
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On the Davenport-Heilbronn theorems and second order terms
TL;DR: In this paper, the Davenport-Heilbronn theorems for the number of 3-torsion elements in the class groups of quadratic fields having bounded discriminant were proved.
References
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Book
A Course in Arithmetic
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.
Journal ArticleDOI
Groupes Réductifs
Armand Borel,Jacques Tits +1 more
TL;DR: In this paper, the authors present conditions générales d'utilisation (http://www.numdam.org/conditions), i.e., Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
Journal ArticleDOI
Sums involving the values at negative integers of L-functions of quadratic characters
TL;DR: For each integer r > 1, the authors defines an arithmetic function H(r, N) which for r = 1 is the class number of (not necessarily primitive) quadratic forms of discriminant N and for r > 2 the function is a modular form of weight r + 1/2 on Fo(4 ).
Book
Modular Forms and Functions
TL;DR: In this article, the authors present a general characterization of modular forms, including groups of matrices and bilinear mappings, groups of level 2 and sums of squares, hecke operators and congruence groups.