Signs of Fourier coefficients of cusp forms and representations by quadratic polynomials

Dr Kane, Benjamin Robert   (Principal Investigator (PI))

Professor Lau Yuk Kam   (Co-Investigator)

36

2016-01-01

631972

Signs of Fourier coefficients of cusp forms and representations by quadratic polynomials

lattice theory, modular forms, quadratic forms, quadratic polynomials, spinor genera

Pure Mathematics

Physical Sciences (P)

17302515

General Research Fund (GRF)

2015

Completed

1 In a broad stroke, the classification of almost universal ternary quadratic polynomials which are sums of n-gonal numbers is the main goal. 2 A secondary goal is to apply the technique developed for the first objective for representations by a quadratic form related to elliptic curves. In particular, the goal is to prove than an algorithm for constructing supersingular elliptic curves given their endomorphism ring actually terminates successfully, by resolving a conjecture about quadratic forms. 3 The first step along the process of classifying almost universal quadratic polynomials is to establish a criterion which determines local obstructions to representability. In individual cases this is a straightforward calculation, but may become complicated in the generality needed for this project. 4 After establishing local representability, the next goal is to determine when the coefficients of certain unary theta functions cancel the main asymptotic growth of the number of representations expected by local conditions. This should involve an array of techniques and long calculations arising from lattice theory (and in particular cosets thereof). 5 Given the output from Objectives 3 and 4, establish a guess for the classification of almost universal quadratic polynomials 6 Prove a generalization of the infinitude of sign changes of Fourier coefficients of cusp forms which are orthogonal to unary theta functions. 7 Use Objective 6 to prove that the guess from Objective 5 is indeed correct, establishing the first goal, a classification.