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


Grant Data
Project Title
Signs of Fourier coefficients of cusp forms and representations by quadratic polynomials
Principal Investigator
Dr Kane, Benjamin Robert   (Principal investigator)
Co-Investigator(s)
Dr Lau Yuk Kam   (Co-Investigator)
Duration
36
Start Date
2016-01-01
Completion Date
2018-12-31
Amount
631972
Conference Title
Presentation Title
Keywords
modular forms, quadratic polynomials, quadratic forms, lattice theory, spinor genera
Discipline
Pure Mathematics
Panel
Physical Sciences (P)
Sponsor
RGC General Research Fund (GRF)
HKU Project Code
17302515
Grant Type
General Research Fund (GRF)
Funding Year
2015/2016
Status
On-going
Objectives
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.