Browsing by Subject "Number Theory"

Now showing 1 - 3 of 3
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    Hypergeometric Functions and Arithmetic Properties of Algebraic Varieties
    (2016-05) Goodson, Heidi
    In this thesis, we investigate the relationship between special functions and arithmetic properties of algebraic varieties. More specifically, we use Greene's finite field hypergeometric functions to give point count formulas for families of algebraic varieties over finite fields. We demonstrate that this is possible for a family of genus 3 curves and for families of higher dimensional varieties called Dwork hypersurfaces. We work out the calculations in great detail for Dwork K3 surfaces over fields whose order is congruent to 1 modulo 4. Furthermore, for K3 surfaces, we also give point count formulas in terms of finite field hypergeometric functions defined by McCarthy. This allows us to give formulas that hold for all primes. Inspired by a result of Manin for curves, we study the relationship between certain period integrals and the trace of Frobenius of these varieties. We show that these can be expressed in terms of ``matching" classical and finite field hypergeometric functions. Through congruences between classical and finite field hypergeometric functions that we prove, we show that the fundamental period is congruent to the trace of Frobenius for Dwork K3 surfaces and conjecture that this is true for higher dimensional Dwork hypersurfaces.
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    The Non-Split bessel Model on GSp(4) as an Iwahori-Hecke Algebra Module
    (2017-06) Grodzicki, William
    We realize the non-split Bessel model of Novodvorsky and Piatetski-Shapiro as a generalized Gelfand-Graev representation of GSp(4), as defined by Kawanaka. Our primary goal is to calculate the values of Iwahori-fixed vectors of unramified principal series representations in the Bessel model. On the path to achieving this goal, we will first use Mackey theory to realize the Bessel functional as an integral - as a result, we will reestablish the uniqueness and existence of a Bessel model for principal series representations, originally proved by Novodvorsky and Piatetski-Shapiro and by Bump, Friedberg, and Furusawa, respectively. Inspired by the work of Brubaker, Bump, and Friedberg, our method of calculation takes advantage of the connection between the Iwahori-fixed vectors in the Bessel model and a certain linear character of the Hecke algebra of GSp(4). We will also provide a detailed description of the conjectural program connecting characters of the Hecke algebra for a more general reductive group G with multiplicity-free models of principal series representations. In particular, we will focus on the role played by the Springer correspondence in this program. Additionally, using the formulas we develop for the Iwahori-fixed vectors, we provide an explicit alternator expression for the spherical vector in the Bessel model which matches previous results of Bump, Friedberg, and Furusawa.
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    A Uniform Approach towards the Local Gan-Gross-Prasad Conjecture
    (2024-06) Chen, Cheng
    The local Gan-Gross-Prasad conjecture is a generalization of the branching problem toclassical groups over local fields of characteristic zero. The conjecture speculates on the multiplicity, that is, the dimension of the Bessel models and Fourier-Jacobi models in an irreducible admissible representation. Equivalent conditions for the multiplicity equaling to one is given in [GP92, GP94, GGP12]. J.-L. Waldspurger did the pioneer’s work and proved the Bessel special orthogonal cases for tempered parameters over non-Archimedean local fields in [Wald10, Wald12a, Wald12b, Wald12c]. C. Mœglin and Waldspurger proved that case for generic parameters in [MW12]. The proof for the conjecture is almost completed but the proof for some cases used a different philosophy. This thesis aims to generalize Mœglin and Waldspurger’s approach to formulate a relatively uniform proof for all cases.