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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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Modeling Convolutions of $L$-Functions
Steven J. Miller,Ralph Morrison +1 more
Abstract: A number of mathematical methods have been shown to model the zeroes of $L$-functions with remarkable success, including the Ratios Conjecture and Random Matrix Theory In order to understand the structure of convolutions of families of $L$-functions, we investigate how well these methods model the zeros of such functions Our primary focus is the convolution of the $L$-function associated to Ramanujan's tau function with the family of quadratic Dirichlet $L$-functions, for which JB Conrey and NC Snaith computed the Ratios Conjecture's prediction Our main result is performing the number theory calculations and verifying these predictions for the one-level density for suitably restricted test functions up to square-root error term Unlike Random Matrix Theory, which only predicts the main term, the Ratios Conjecture detects the arithmetic of the family and makes detailed predictions about their dependence in the lower order terms Interestingly, while Random Matrix Theory is frequently used to model behavior of L-functions (or at least the main terms), there has been little if any work on the analogue of convolving families of L-functions by convolving random matrix ensembles We explore one possibility by considering Kronecker products; unfortunately, it appears that this is not the correct random matrix analogue to convolving families
New Results in Integer and Lattice Programming
TL;DR: The notion of a c-compact lattice basis B ∈ Rn×n that facilitates to represent the Voronoi relevant vectors with coefficients bounded by c is introduced, which allows to reduce the space requirement of Micciancio's & Voulgaris’ algorithm for the closest vector problem from exponential to polynomial, while the running time becomes exponential in c.
Dissertation
Galois groups and anabelian reconstruction
TL;DR: In this paper, the authors investigated the problem of recovering arithmetic structure on a field F from small quotients of its absolute Galois group, and obtained strong versions of the Birational Section Conjecture for curves over p -adic fields.
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Polynomial interpolation of modular forms for Hecke groups
TL;DR: For example, the authors showed that for any non-negative integer n = 3, there is a rational function with coefficients in the quadratic form f(n) = F(n(m)/a_{-1}(m)^n.
Journal ArticleDOI
A Bertrand Postulate for a Subclass of Primes
TL;DR: In this paper, the authors considered the subclass of primes with Legendre symbol (d/p ) = +1 and showed that for a large enough (x; 2x] contain a prime of this type, a small (x, 2x) prime can be found.