Half-integral weight p-adic coupling of weakly holomorphic and holomorphic modular forms [PDF]
In this paper, we consider p-adic limits of β −n g|U n2 for half-integral weight weakly holomorphic Hecke eigenforms g with eigenvalue λp = β + β under T p2 and prove that these equal classical Hecke eigenforms of the same weight. This result parallels the integral weight case, but requires a much more careful investigation due to a more complicated ...
Kathrin Bringmann+2 more
core +11 more sources
Mock theta functions and weakly holomorphic modular forms modulo 2 and 3 [PDF]
AbstractWe prove that the coefficients of the mock theta functions \begin{eqnarray*} f(q) = \sum_{n=1}^{\infty} \frac{ q^{n^2}}{(1+q)^2 (1+q^2)^2 \cdots (1+q^n)^2 } \end{eqnarray*} and \begin{eqnarray*} \omega(q)=1+\sum_{n=1}^\infty \frac{q^{2n^2+2n}}{(1+q)^2(1+q^3)^2\cdots (1+q^{2n+1})^2} \end{eqnarray*} possess no linear congruences modulo 3.
Scott Ahlgren, Byungchan Kim
core +9 more sources
p-Adic Properties of Coefficients of Weakly Holomorphic Modular Forms [PDF]
We examine the Fourier coefficients of modular forms in a canonical basis for the spaces of weakly holomorphic modular forms of weights 4, 6, 8, 10, and 14, and show that these coefficients are often highly divisible by the primes 2, 3, and 5.
Darrin Doud, Paul A. Jenkins
core +6 more sources
Algebraic de Rham theory for weakly holomorphic modular forms of level one [PDF]
We establish an Eichler-Shimura isomorphism for weakly modular forms of level one. We do this by relating weakly modular forms with rational Fourier coefficients to the algebraic de Rham cohomology of the modular curve with twisted coefficients. This leads to formulae for the periods and quasi-periods of modular forms.
Francis Brown, Richard Hain
semanticscholar +8 more sources
Zeros of weakly holomorphic modular forms of levels 2 and 3 [PDF]
Added a reference, corrected ...
Sharon Anne Garthwaite, Paul M. Jenkins
core +6 more sources
Zeros of weakly holomorphic modular forms of level 4 [PDF]
Let [Formula: see text] be the space of weakly holomorphic modular forms of weight k and level 4 that are holomorphic away from the cusp at ∞. We define a canonical basis for this space and show that for almost all of the basis elements, the majority of their zeros in a fundamental domain for Γ0(4) lie on the lower boundary of the fundamental domain ...
Andrew Haddock, Paul Jenkins
semanticscholar +8 more sources
Zagier duality for level $p$ weakly holomorphic modular forms [PDF]
We prove Zagier duality between the Fourier coefficients of canonical bases for spaces of weakly holomorphic modular forms of prime level $p$ with $11 \leq p \leq 37$ with poles only at the cusp at $\infty$, and special cases of duality for an infinite class of prime levels. We derive generating functions for the bases for genus 1 levels.
Paul Jenkins, Grant Molnar
semanticscholar +7 more sources
Classification of congruences for mock theta functions and weakly holomorphic modular forms [PDF]
Let $f(q)$ denote Ramanujan's mock theta function \[f(q) = \sum_{n=0}^{\infty} a(n) q^{n} := 1+\sum_{n=1}^{\infty} \frac{q^{n^{2}}}{(1+q)^{2}(1+q^{2})^{2}\cdots(1+q^{n})^{2}}.\] It is known that there are many linear congruences for the coefficients of $f(q)$ and other mock theta functions. We prove that if the linear congruence $a(mn+t) \equiv 0 \pmod{
Nickolas Andersen
semanticscholar +9 more sources
ZEROS OF CERTAIN WEAKLY HOLOMORPHIC MODULAR FORMS
Seiichi Hanamoto
semanticscholar +4 more sources
L-Series for Vector-Valued Weakly Holomorphic Modular Forms and Converse Theorems [PDF]
Abstract We introduce the L-series of weakly holomorphic modular forms using Laplace transforms and give their functional equations. We then determine converse theorems for vector-valued harmonic weak Maass forms, Jacobi forms, and elliptic modular forms of half-integral weight in Kohnen plus space.
Subong Lim, Wissam Raji
semanticscholar +4 more sources