Kuznetsov's Trace Formula and the Hecke Eigenvalues of Maass Forms
| Author | : Andrew Knightly |
| Publisher | : |
| Total Pages | : 132 |
| Release | : 2013 |
| Genre | : Hecke operators |
| ISBN | : 9781470410063 |
Kuznetsovs Trace Formula And The Hecke Eigenvalues Of Maass Forms
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Kuznetsov's Trace Formula and the Hecke Eigenvalues of Maass Forms
| Author | : Andrew Knightly |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 144 |
| Release | : 2013-06-28 |
| Genre | : Mathematics |
| ISBN | : 0821887440 |
The authors give an adelic treatment of the Kuznetsov trace formula as a relative trace formula on $\operatorname{GL}(2)$ over $\mathbf{Q}$. The result is a variant which incorporates a Hecke eigenvalue in addition to two Fourier coefficients on the spectral side. The authors include a proof of a Weil bound for the generalized twisted Kloosterman sums which arise on the geometric side. As an application, they show that the Hecke eigenvalues of Maass forms at a fixed prime, when weighted as in the Kuznetsov formula, become equidistributed relative to the Sato-Tate measure in the limit as the level goes to infinity.
Trace Formulas and Their Applications on Hecke Eigenvalues
| Author | : Yingnan Wang |
| Publisher | : |
| Total Pages | : |
| Release | : 2017-01-26 |
| Genre | : |
| ISBN | : 9781361280928 |
This dissertation, "Trace Formulas and Their Applications on Hecke Eigenvalues" by Yingnan, Wang, 王英男, was obtained from The University of Hong Kong (Pokfulam, Hong Kong) and is being sold pursuant to Creative Commons: Attribution 3.0 Hong Kong License. The content of this dissertation has not been altered in any way. We have altered the formatting in order to facilitate the ease of printing and reading of the dissertation. All rights not granted by the above license are retained by the author. Abstract: The objective of the thesis is to investigate the trace formulas and their applications on Hecke eigenvalues, especially on the distribution of Hecke eigenvalues. This thesis is divided into two parts.. In the first part of the thesis, a review is firstly carried out for the equidistribution of Hecke eigenvalues as primes vary and for the expected size of the error term in this equidistribution problem. Then the Kuznetsov trace formula is applied to prove a result on the size of the error term in the asymptotic distribution formula of Hecke eigenvalues. These eigenvalues become equidistributed with respect to the p-adic Plancherel measures as Hecke eigenforms vary. Next, this problem is generalized to Satake parameters of GL2 representations with prescribed supercuspidal local representations. Such a generalization is novel to the case of classical automorphic forms. To achieve this result, a trace formula of Arthur-Selberg type with a couple of key refinements is used. In the second part of the thesis, a density theorem is proved which counts the number of exceptional nontrivial zeros of a family of symmetric power L-functions attached to primitive Maass forms in the critical strip. In addition, a large sieve inequality of Elliott-Montgomery-Vaughan type for primitive Maass forms is established. The density theorem and large sieve inequality have many applications. For instance, they are used to prove statistical results on Hecke eigenvalues of primitive Maass forms and the extreme values of the symmetric power L-functions attached to primitive Maass forms. DOI: 10.5353/th_b4832952 Subjects: Trace formulas Eigenvalues
Analytic Number Theory
| Author | : Carl Pomerance |
| Publisher | : Springer |
| Total Pages | : 378 |
| Release | : 2015-11-18 |
| Genre | : Mathematics |
| ISBN | : 3319222406 |
This volume contains a collection of research and survey papers written by some of the most eminent mathematicians in the international community and is dedicated to Helmut Maier, whose own research has been groundbreaking and deeply influential to the field. Specific emphasis is given to topics regarding exponential and trigonometric sums and their behavior in short intervals, anatomy of integers and cyclotomic polynomials, small gaps in sequences of sifted prime numbers, oscillation theorems for primes in arithmetic progressions, inequalities related to the distribution of primes in short intervals, the Möbius function, Euler’s totient function, the Riemann zeta function and the Riemann Hypothesis. Graduate students, research mathematicians, as well as computer scientists and engineers who are interested in pure and interdisciplinary research, will find this volume a useful resource. Contributors to this volume: Bill Allombert, Levent Alpoge, Nadine Amersi, Yuri Bilu, Régis de la Bretèche, Christian Elsholtz, John B. Friedlander, Kevin Ford, Daniel A. Goldston, Steven M. Gonek, Andrew Granville, Adam J. Harper, Glyn Harman, D. R. Heath-Brown, Aleksandar Ivić, Geoffrey Iyer, Jerzy Kaczorowski, Daniel M. Kane, Sergei Konyagin, Dimitris Koukoulopoulos, Michel L. Lapidus, Oleg Lazarev, Andrew H. Ledoan, Robert J. Lemke Oliver, Florian Luca, James Maynard, Steven J. Miller, Hugh L. Montgomery, Melvyn B. Nathanson, Ashkan Nikeghbali, Alberto Perelli, Amalia Pizarro-Madariaga, János Pintz, Paul Pollack, Carl Pomerance, Michael Th. Rassias, Maksym Radziwiłł, Joël Rivat, András Sárközy, Jeffrey Shallit, Terence Tao, Gérald Tenenbaum, László Tóth, Tamar Ziegler, Liyang Zhang.
An Introduction to Automorphic Representations
| Author | : Jayce R. Getz |
| Publisher | : Springer Nature |
| Total Pages | : 611 |
| Release | : |
| Genre | : |
| ISBN | : 3031411536 |
The Sine-Gordon Equation in the Semiclassical Limit: Dynamics of Fluxon Condensates
| Author | : Robert J. Buckingham |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 148 |
| Release | : 2013-08-23 |
| Genre | : Mathematics |
| ISBN | : 0821885456 |
The authors study the Cauchy problem for the sine-Gordon equation in the semiclassical limit with pure-impulse initial data of sufficient strength to generate both high-frequency rotational motion near the peak of the impulse profile and also high-frequency librational motion in the tails. They show that for small times independent of the semiclassical scaling parameter, both types of motion are accurately described by explicit formulae involving elliptic functions. These formulae demonstrate consistency with predictions of Whitham's formal modulation theory in both the hyperbolic (modulationally stable) and elliptic (modulationally unstable) cases.
On Some Aspects of Oscillation Theory and Geometry
| Author | : Bruno Bianchini |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 208 |
| Release | : 2013-08-23 |
| Genre | : Mathematics |
| ISBN | : 0821887998 |
The aim of this paper is to analyze some of the relationships between oscillation theory for linear ordinary differential equations on the real line (shortly, ODE) and the geometry of complete Riemannian manifolds. With this motivation the authors prove some new results in both directions, ranging from oscillation and nonoscillation conditions for ODE's that improve on classical criteria, to estimates in the spectral theory of some geometric differential operator on Riemannian manifolds with related topological and geometric applications. To keep their investigation basically self-contained, the authors also collect some, more or less known, material which often appears in the literature in various forms and for which they give, in some instances, new proofs according to their specific point of view.
On the Steady Motion of a Coupled System Solid-Liquid
| Author | : Josef Bemelmans |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 102 |
| Release | : 2013-10-23 |
| Genre | : Mathematics |
| ISBN | : 0821887734 |
We study the unconstrained (free) motion of an elastic solid B in a Navier-Stokes liquid L occupying the whole space outside B, under the assumption that a constant body force b is acting on B. More specifically, we are interested in the steady motion of the coupled system {B,L}, which means that there exists a frame with respect to which the relevant governing equations possess a time-independent solution. We prove the existence of such a frame, provided some smallness restrictions are imposed on the physical parameters, and the reference configuration of B satisfies suitable geometric properties.
Gromov, Cauchy and Causal Boundaries for Riemannian, Finslerian and Lorentzian Manifolds
| Author | : Jose Luis Flores |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 88 |
| Release | : 2013-10-23 |
| Genre | : Mathematics |
| ISBN | : 0821887750 |
Recently, the old notion of causal boundary for a spacetime V has been redefined consistently. The computation of this boundary ∂V on any standard conformally stationary spacetime V=R×M, suggests a natural compactification MB associated to any Riemannian metric on M or, more generally, to any Finslerian one. The corresponding boundary ∂BM is constructed in terms of Busemann-type functions. Roughly, ∂BM represents the set of all the directions in M including both, asymptotic and "finite" (or "incomplete") directions. This Busemann boundary ∂BM is related to two classical boundaries: the Cauchy boundary ∂CM and the Gromov boundary ∂GM. The authors' aims are: (1) to study the subtleties of both, the Cauchy boundary for any generalized (possibly non-symmetric) distance and the Gromov compactification for any (possibly incomplete) Finsler manifold, (2) to introduce the new Busemann compactification MB, relating it with the previous two completions, and (3) to give a full description of the causal boundary ∂V of any standard conformally stationary spacetime. J. L. Flores and J. Herrera, University of Malaga, Spain, and M. Sánchez, University of Granada, Spain. Publisher's note.
On the Regularity of the Composition of Diffeomorphisms
| Author | : H. Inci |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 72 |
| Release | : 2013-10-23 |
| Genre | : Mathematics |
| ISBN | : 0821887416 |
For M a closed manifold or the Euclidean space Rn we present a detailed proof of regularity properties of the composition of Hs-regular diffeomorphisms of M for s > 12dimM+1.
