Real Quadratic Double Sums

q-HYPERGEOMETRIC DOUBLE SUMS AS MOCK THETA FUNCTIONS

Recently, Bringmann and Kane established two new Bailey pairs and used them to relate certain q-hypergeometric series to real quadratic fields. We show how these pairs give rise to new mock theta functions in the form of q-hypergeometric double sums. Additionally, we prove an identity between one of these sums and two classical mock theta functions introduced by Gordon and McIntosh.

Modular Invariants for Real Quadratic Fields and Kloosterman Sums

We investigate the asymptotic distribution of integrals of the j-function that are associated to ideal classes in a real quadratic field. To estimate the error term in our asymptotic formula, we prove a bound for sums of Kloosterman sums of half-integral weight which is uniform in every parameter. To establish this estimate we prove a variant of Kuznetsov’s formula where the spectral data is re...

Elliptic units for real quadratic fields

1. A review of the classical setting 2. Elliptic units for real quadratic fields 2.1. p-adic measures 2.2. Double integrals 2.3. Splitting a two-cocycle 2.4. The main conjecture 2.5. Modular symbols and Dedekind sums 2.6. Measures and the Bruhat-Tits tree 2.7. Indefinite integrals 2.8. The action of complex conjugation and of Up 3. Special values of zeta functions 3.1. The zeta function 3.2. Va...

On the Marcinkiewicz and (C, α)-means of the quadratical partial sums of double Walsh-Kaczmarz-Fourier series

The Real Holomorphy Ring and Sums of Powers in Rational and Elliptic Function Fields over The Field of Real Numbers

This thesis is devoted to a study of real holomorphy rings in R(x) and elliptic function fields R(x)( √ f(x)), f is a monic polynomial of degree 3 with no multiple roots in C over R and sums of powers in R(x). The objects of study are real valuation rings, real holomorphy rings and real places in the rational function fields R(x) and then investigate their extension to elliptic function fields ...