Venkatesh, Akshay 1981Overview
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Most widely held works by
Akshay Venkatesh
Limiting forms of the trace formula (Robert P. Langlands)
by Akshay Venkatesh
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2 editions published in 2002 in English and held by 2 WorldCat member libraries worldwide We carry out the first nontrivial cases of the limiting process proposed by Langlands in his manuscript Beyond Endoscopy, with technical variations that enable us to treat the limit unconditionally. This gives an elementary proof, on GL(2), of the classification of forms such that the symmetric square Lfunction has a pole (including, implicitly, the construction of these forms). The result of this may be seen as one of the simplest cases of the "pipedream" Langlands proposes. We also apply similar methods to derive a converse theorem, and to produce a result that generalizes Duke's estimate on the dimension of weight 1 forms to arbitrary number fieldsbut is sharper, even over Q , than Duke's original estimate
Algebraic modular forms on definite orthogonal groups
by Daniel Kim Murphy
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1 edition published in 2013 in English and held by 1 WorldCat member library worldwide The present thesis examines explicit computation with modular forms on definite orthogonal groups over the rational numbers. The space of functions on the genus of a definite quadratic form affords a natural representation of the Hecke algebra, which realizes the function space as a space of modular forms. The algorithm of the thesis computes the genus of a quadratic form as well as Hecke operators on the associated space of modular forms. A formula derived in the thesis yields Satake parameters from Hecke eigenvalues. Explicit examples of Satake parameters verify the correctness of the algorithm and the formula by their agreement with the predictions of the Arthur conjectures
Dichotomy between structure and randomness in combinatorial number theory
by Xuancheng Shao
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1 edition published in 2014 in English and held by 1 WorldCat member library worldwide This dissertation is about exploring the dichotomy between structure and randomness in analytic number theory and additive combinatorics. We explore various problems including bounds for character sums, Goldbachtype questions, and a Freimantype result
Some problems in multiplicative number theory
by Junsoo Ha
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1 edition published in 2014 in English and held by 1 WorldCat member library worldwide Let q be a power of a prime p. In the first part of this thesis, we establish the upper bound of the least prime primitive root mod q by p^3.1. We say a polynomial in F_q [T] is msmooth if all of its irreducible factors are of degree less than or equal to m. Let N(n, m) be the number of solutions to the polynomial equation X+Y=2Z where all variables are msmooth polynomials of degree n. In the second part of this thesis, we establish a lower bound on N(n, m) when (8+d) log_q n < = m < = n^1/2 for small d, and prove the analog of the xyz conjecture of Lagarias and Soundararajan in the polynomial rings over finite fields
Moments of automorphic Lfunctions and related problems
by Ian Petrow
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1 edition published in 2013 in English and held by 1 WorldCat member library worldwide We present in this dissertation several theorems on the subject of moments of automorphic Lfunctions. In chapter 1 we give an overview of this area of research and summarize our results. In chapter 2 we give asymptotic main term estimates for several different moments of central values of Lfunctions of a fixed GL_2 holomorphic cusp form f twisted by quadratic characters. When the sign of the functional equation of the twist L(s, f \otimes \chi_d) is 1, the central value vanishes and one instead studies the derivative L'(1/2, f \otimes \chi_d). We prove two theorems in the root number 1 case which are completely out of reach when the root number is +1. In chapter 3 we turn to an average of GL_2 objects. We study the family of cusp forms of level q^2 which are given by f \otimes \chi, where f is a modular form of prime level q and \chi is the quadratic character modulo q. We prove a precise asymptotic estimate uniform in shifts for the second moment with the purpose of understanding the offdiagonal main terms which arise in this family. In chapter 4 we prove an precise asymptotic estimate for averages of shifted convolution sums of Fourier coefficients of fulllevel GL_2 cusp forms over shifts. We find that there is a transition region which occurs when the square of the average over shifts is proportional to the length of the shifted sum. The asymptotic in this range depends very delicately on the constant of proportionality: its second derivative seems to be a continuous but nowhere differentiable function. We relate this phenomenon to periods of automorphic forms, multiple Dirichlet series, automorphic distributions, and moments of RankinSelberg Lfunctions
Distribution problems in number theory
by Robert Daniel Hough
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1 edition published in 2012 in English and held by 1 WorldCat member library worldwide Four separate problems are considered. The first chapter concerns the average equidistribution of Heegner points attached to elements of fixed order in the class group of imaginary quadratic fields. Some discussion of the CohenLenstra heuristics is also included. Chapter 2 treats the distribution of the logarithm of two families of Lfunctions at the central point. The most easily stated consequence is that the central values of Lfunctions attached to modular forms of large weight converge to 0 in distribution as the weight tends to infinity. Chapter 3 concerns omega results for large sums of Dirichlet characters. The length N of the sum is treated as a parameter that is allowed to vary with the modulus q of the characters. For N small compared to q the large values exhibited are related to the distribution of smooth numbers, while for larger N the results are analogous to omega bounds known for the corresponding Lfunctions. The final chapter gives a mixing time analysis for the random kcycle walk on the symmetric group. The main new analytic ingredient is an asymptotic formula for many character ratios evaluated at a kcycle
Part I, Symplectic ice, Part II, Global and local Kubota symbols
by Dmitriy Ivanov
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1 edition published in 2010 in English and held by 1 WorldCat member library worldwide My Ph. D. thesis consists of two parts. In the first part I construct an icemodel (a sixvertex model). I show the partition function of this ice model is equal to the product of an irreducible character of the symplectic group and a deformation of the Weyl denominator. A similar result was originally proved by Hamel and King, but the Boltzmann weights (for the vertices at the caps) that I use are different then the ones that are used by Hamel and King. Also, my proof of this result uses YangBaxter equation (while the proof of Hamel and King does not). This gives us a 6vertex models for characters of the symplectic group, but this result can also be interpreted as an example of an exactly solved model, that is, an ice model whose partition function can be computed explicitly (this is of interest to people who work in statistical mechanics). In the second part of my thesis I work with the global and local Kubota symbols. I introduce a local Kubota symbol, and show that it satisfies some properties that are satisfies by the global Kubota symbol; in particular, it satisfies the reciprocity law. Using the parallel between the properties of the global and local Kubota symbols, I give a new proof of Kubota's theorem (for the case when p does not divide n), and then generalize this theorem to the case when p divides n. In the last chapter of my thesis I work with the global Kubota symbol. I investigate the levels for the global Kubota symbols for particular fields, and I obtain new levels for these fields which improve the results of Kubota, and of Bass, Milnor and Serre
Controlling ramification in number fields
by Simon RubinsteinSalzedo
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1 edition published in 2012 in English and held by 1 WorldCat member library worldwide This thesis focuses on two aspects of limited ramification and is split up into two independent sections. The first section (which comprises the second and third chapters) is on the distribution of class groups of cyclic cubic fields. We propose an explanation for the discrepancy between the observed number of cyclic cubics whose 2class group is C_2 x C_2 and the number predicted by the CohenLenstra heuristics, in terms of an invariant living in a quotient of the Schur multiplier group. We also show that, in some cases, the definition of the invariant can be simplified greatly, and we compute 10^5 examples. The second section (which comprises the fourth and fifth chapters) discusses branched covers of algebraic curves, especially covers of elliptic curves with one branch point. We produce some techniques that allow us to write down explicit equations for such maps, and then we give examples of number fields which arise from such covers. Finally, we present some possibilities for future works that the author hopes to pursue
Conic degeneration and the determinant of the laplacian
by David Alexander Sher
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1 edition published in 2012 in English and held by 1 WorldCat member library worldwide We consider a family of smooth Riemannian manifolds which degenerate to a manifold with a conical singularity. Such families arise in various settings in spectral theory, including the study of the isospectral problem. We investigate the behavior of the determinant of the Laplacian under the degeneration. Our main result is an approximation formula for the determinant, including all terms which do not vanish in the limit. The key idea is a uniform parametrix construction for the heat kernel on the degenerating family of manifolds, which enables us to analyze the determinant via the heat trace. It becomes clear in the construction that we need to understand both the shorttime and longtime behavior of the heat kernel on an asymptotically conic manifold. Using techniques of Melrose and building on previous work of Guillarmou and Hassell, we give a complete description of the asymptotic structure of this heat kernel in all spatial and temporal regimes
Padic Hodge theory in rigid analytic families
by Rebecca Michal Bellovin
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1 edition published in 2013 in English and held by 1 WorldCat member library worldwide In this thesis, we study padic Hodge theory in rigid analytic families. Roughly speaking, padic Hodge theory is the study of padic representations of padic Galois groups. One introduces certain padic period rings B, such as B_{HT}, B_{dR}, B_{st}, and B_{cris}, and uses them to define functors D_B(.) from the category of padic Galois representations to various categories of linear algebra data. In the first half of this thesis, we study generalizations of these functors to families of padic Galois representations with rigid analytic coefficients. We prove that the functors D_{HT}(.) and D_{dR}(.) are coherent sheaves, and we prove that the Badmissible locus is a closed subspace of the base. In the second half of this thesis, we study the linear algebra data which arises from families of potentially semistable Galois representations valued in a connected reductive group G. We prove that for any G, the moduli space of linear algebra data is reduced and locally a complete intersection, and we deduce that potentially semistable deformation rings are generically smooth and equidimensional
The Fourteenth Takagi lectures 1516 November 2014, Tokyo
by Takagi lectures
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1 edition published in 2014 in English and held by 1 WorldCat member library worldwide
The behaviour of Lfunctions at the edge of the critical strip and applications
by Xiannan Li
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1 edition published in 2011 in English and held by 1 WorldCat member library worldwide A large number of problems in number theory can be reduced to statements about Lfunctions. In this thesis, we study Lfunctions at the edge of the critical strip, and relate these to a variety of objects of arithmetic interest
Gvalued flat deformations and local models
by Brandon William Allen Levin
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1 edition published in 2013 in English and held by 1 WorldCat member library worldwide We construct resolutions of Gvalued local Galois deformation rings by moduli spaces of Kisin modules with Gstructure when l = p. This generalizes Mark Kisin's work on potentially semistable deformation rings. In the case of flat deformations, we prove a structural result about these resolutions which relates the connected components of Gvalued flat deformation rings to the connected components of projective varieties in characteristic p, which are moduli spaces of linear algebra data. As a key step in the study of these resolutions, we prove a full faithfulness result in integral padic Hodge theory. We also generalize results of Pappas and Zhu on local models of Shimura varieties to groups arising from Weil restrictions
Microlocal analysis of lagrangian submanifolds of radial points
by Nick Haber
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1 edition published in 2013 in English and held by 1 WorldCat member library worldwide Microlocal analysis relies on correspondences between quantum physics and classical physics to give information about certain PDEs  for instance, linear variablecoefficient PDEs on manifolds. PDEs are interpreted as quantum systems. The corresponding classical systems tell us, for example, function spaces on which problems are solvable or almost solvable, existence and uniqueness results, and the structure of solution operators. Landmark papers of Hörmander and Duistermaat and Hörmander establish key results for the standard calculus of microlocal analysis, which gives a broad framework for dealing with variablecoefficient PDEs on manifolds. Their work is wellsuited for dealing with PDEs which, in a generalized sense, are hyperbolic, with corresponding classical dynamics looking like wave propagation of geometric optics. In this thesis, we aim to extend many of their results to situations in which the corresponding classical dynamics are less wellbehaved: those with a Lagrangian submanifold of radial points
Upper bounds and moments of Lfunctions
by Vorrapan Chandee
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1 edition published in 2010 in English and held by 1 WorldCat member library worldwide Lfunctions are some of the most studied objects in number theory. Although many crucial properties of Lfunctions remain mysterious, central conjectures such as the generalized Riemann hypothesis (GRH). This thesis concerns properties of Lfunctions. In particular, we focus on studying upper bounds and moments of $L$functions. Assuming GRH, we give effective explicit upper bounds for Lfunctions on the critical line and apply these bounds to determine what numbers are represented by a given ternary quadratic form. Moreover the best known version of the Lindelof hypothesis from the Riemann hypothesis (RH) is also derived. Another important way of understanding LH is through moments of Lfunctions. Information about moments sheds light on the distribution of values of \zeta(1/2 + it). We try to understand the joint distribution of quantities like \zeta(1/2 + it) and \zeta(1/2 + it + i). To study these we consider "shifted moments" of the zeta function and obtain good upper and lower estimates for such moments
Equivariant torsion and base change
by Michael Lipnowski
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1 edition published in 2013 in English and held by 1 WorldCat member library worldwide The CheegerMuller theorem provides a spectral means of obtaining information about torsion in the cohomology of compact manifolds. Using this remarkable theorem together with trace formula comparisons, we prove comparisons between torsion in the cohomology of two manifolds of arithmetic origin related by base change
Zerodistribution and size of the Riemann zetafunction on the critical line
by Maksym Radziwill
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1 edition published in 2013 in English and held by 1 WorldCat member library worldwide We investigate the analytic properties of the Riemann zetafunction. We focus on the average size of the Riemann zetafunction on the critical line and on the vertical distribution of its zeros on the critical line more
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