Modular curves, Arakelov theory, algorithmic applications

This thesis is about arithmetic, analytic and algorithmic aspects of modular curves and modular forms. The arithmetic and analytic aspects are linked by the viewpoint that modular curves are examples of arithmetic surfaces. Therefore, Arakelov theory (intersection theory on arithmetic surfaces) occupies a prominent place in this thesis. Apart from this, a substantial part of it is devoted to studying modular curves over finite fields, and their Jacobian varieties, from an algorithmic viewpoint. The end product of this thesis is an algorithm for computing modular Galois representations. These are certain two-dimensional representations of the absolute Galois group of the rational numbers that are attached to Hecke eigenforms over finite fields. The running time of our algorithm is (under minor restrictions) polynomial in the length of the input. This main result is a generalisation of that of work of Jean-Marc Couveignes, Bas Edixhoven et al. Several intermediate results are developed in sufficient generality to make them of interest to the study of modular curves and modular forms in a wider sense.The research for this thesis was made possible by NWO (Netherlands Organisation for Scientific Research). Nederlandse Organisatie voor Wetenschappelijk Onderzoek Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO)UBL - phd migration 201

Distribution results for automorphic forms, their periods and masses

We explore a variety of topics in the analytic theory of automorphic forms. The main results of this thesis are about the arithmetic statistics of periods of automorphic forms and the distribution of masses of automorphic forms in the context of Quantum Chaos. We introduce a new technique for the study of the distribution of modular symbols. Answering an average version of a conjecture due to Mazur and Rubin for $\Gamma_0(N)$ and recovering results of Petridis and Risager using a different method, we show that modular symbols are asymptotically normally distributed, We apply our technique to obtain new results for congruence subgroups of Bianchi groups. Our novel insight is to use the behaviour of the smallest eigenvalue of the Laplace operator for twisted spaces. Our approach also recovers the first and the second moment of the distribution. In work joint with Asbjørn Nordentoft, we introduce an automorphic method for studying the residual distribution of modular symbols modulo primes. We obtain a refinement of a result of Lee and Sun, which solved an average version of another conjecture of Mazur and Rubin. In addition, we solve the full conjecture in some special cases. Furthermore, we generalise the results to quotients of general hyperbolic spaces. Lastly, we obtain a generalisation of the Quantum Unique Ergodicity for holomorphic cusp forms, as proved by Holowinsky and Soundararajan. We show that correlations of masses coming from off-diagonal terms dissipate as the weight tends to infinity. This corresponds to classifying the possible quantum limits along any sequence of Hecke eigenforms of increasing weight

Hybrid Symbolic-Numeric Computing in Linear and Polynomial Algebra

In this thesis, we introduce hybrid symbolic-numeric methods for solving problems in linear and polynomial algebra. We mainly address the approximate GCD problem for polynomials, and problems related to parametric and polynomial matrices. For symbolic methods, our main concern is their complexity and for the numerical methods we are more concerned about their stability. The thesis consists of 5 articles which are presented in the following order: Chapter 1, deals with the fundamental notions of conditioning and backward error. Although our results are not novel, this chapter is a novel explication of conditioning and backward error that underpins the rest of the thesis. In Chapter 2, we adapt Victor Y. Pan\u27s root-based algorithm for finding approximate GCD to the case where the polynomials are expressed in Bernstein bases. We use the numerically stable companion pencil of G. F. Jónsson to compute the roots, and the Hopcroft-Karp bipartite matching method to find the degree of the approximate GCD. We offer some refinements to improve the process. In Chapter 3, we give an algorithm with similar idea to Chapter 2, which finds an approximate GCD for a pair of approximate polynomials given in a Lagrange basis. More precisely, we suppose that these polynomials are given by their approximate values at distinct known points. We first find each of their roots by using a Lagrange basis companion matrix for each polynomial. We introduce new clustering algorithms and use them to cluster the roots of each polynomial to identify multiple roots, and then marry the two polynomials using a Maximum Weight Matching (MWM) algorithm, to find their GCD. In Chapter 4, we define ``generalized standard triples\u27\u27 X, zC1 - C0, Y of regular matrix polynomials P(z) in order to use the representation X(zC1 - C0)-1 Y=P-1(z). This representation can be used in constructing algebraic linearizations; for example, for H(z) = z A(z)B(z) + C from linearizations for A(z) and B(z). This can be done even if A(z) and B(z) are expressed in differing polynomial bases. Our main theorem is that X can be expressed using the coefficients of 1 in terms of the relevant polynomial basis. For convenience we tabulate generalized standard triples for orthogonal polynomial bases, the monomial basis, and Newton interpolational bases; for the Bernstein basis; for Lagrange interpolational bases; and for Hermite interpolational bases. We account for the possibility of common similarity transformations. We give explicit proofs for the less familiar bases. Chapter 5 is devoted to parametric linear systems (PLS) and related problems, from a symbolic computational point of view. PLS are linear systems of equations in which some symbolic parameters, that is, symbols that are not considered to be candidates for elimination or solution in the course of analyzing the problem, appear in the coefficients of the system. We assume that the symbolic parameters appear polynomially in the coefficients and that the only variables to be solved for are those of the linear system. It is well-known that it is possible to specify a covering set of regimes, each of which is a semi-algebraic condition on the parameters together with a solution description valid under that condition.We provide a method of solution that requires time polynomial in the matrix dimension and the degrees of the polynomials when there are up to three parameters. Our approach exploits the Hermite and Smith normal forms that may be computed when the system coefficient domain is mapped to the univariate polynomial domain over suitably constructed fields. Our approach effectively identifies intrinsic singularities and ramification points where the algebraic and geometric structure of the matrix changes. Specially parametric eigenvalue problems can be addressed as well. Although we do not directly address the problem of computing the Jordan form, our approach allows the construction of the algebraic and geometric eigenvalue multiplicities revealed by the Frobenius form, which is a key step in the construction of the Jordan form of a matrix

Mathematics & Statistics 2017 APR Self-Study & Documents

UNM Mathematics & Statistics APR self-study report, review team report, response report, and initial action plan for Spring 2017, fulfilling requirements of the Higher Learning Commission