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On higher congruences between cusp forms and Eisenstein series
higher congruences cusp forms Eisenstein series Number Theory
2012/6/30
In this paper we present several finite families of congruences between cusp forms and Eisenstein series of higher weights at powers of prime ideals. We formulate a conjecture which describes properti...
Linear Mahler Measures and Double L-values of Modular Forms
Linear Mahler Measures Double L-values of Modular Forms Number Theory
2012/6/27
We consider the Mahler measure of the polynomial 1+x_1+x_2+x_3+x_4, which is the first case not yet evaluated explicitly. A conjecture due to F. Rodriguez-Villegas represents this Mahler measure as a ...
On a completed generating function of locally harmonic Maass forms
locally harmonic Maass forms indefinite theta functions holomorphic projection mock modular forms modular forms
2012/6/21
While investigating the Doi-Naganuma lift, Zagier defined integral weight cusp forms $f_D$ which are naturally defined in terms of binary quadratic forms of discriminant $D$. It was later determined b...
Locally harmonic Maass forms and rational period functions
hyperbolic Poincare series harmonic weak Maass forms cusp forms lifting maps Shimura lift Shintani lift
2012/6/21
In this paper we define a new type of modular object and construct explicit examples of such functions. Our functions are closely related to cusp forms constructed by Zagier which played an important ...
Formulas for central critical values of twisted L-functions attached to paramodular forms
Formulas central critical values of twisted L-functions paramodular forms Number Theory
2012/6/7
In the 1980s B\"ocherer formulated a conjecture relating the central values of the imaginary quadratic twists of the spin L-function attached to a Siegel modular form $F$ to the Fourier coefficients o...
Residual automorphic forms and spherical unitary representations of exceptional groups
Arthur’s conjectures unitary dual residual Eisenstein series automorphic realizations
2012/5/25
Arthur has conjectured that the unitarity of a number of representations can be shown by finding appropriate automorphic realizations. This has been verified for classical groups by Moeglin and for th...
Eichler Cohomology of Generalized Modular Forms of Real Weights
Eichler Cohomology Generalized Modular Forms Number Theory
2012/5/9
In this paper, we prove the Eichler cohomology theorem of weakly parabolic generalized modular forms of real weights on subgroups of finite index in the full modular group. We explicitly establish the...
Computing power series expansions of modular forms
power series expansions of modular forms Number Theory
2012/5/9
We exhibit a method to numerically compute power series expansions of modular forms on a cocompact Fuchsian group, using the explicit computation a fundamental domain and linear algebra.
On the Petersson scalar product of arbitrary modular forms
Petersson scalar product arbitrary modular forms Number Theory
2012/4/18
We consider a natural extension of the Petersson scalar product to the entire space of modular forms of integral weight $k\ge 2$ for a finite index subgroup of the modular group. We show that Hecke op...
Ramanujan type congruences for modular forms of several variables
Congruences for modular forms Cusp forms Number Theory
2012/4/17
We give congruences between the Eisenstein series and a cusp form in the cases of Siegel modular forms and Hermitian modular forms. We should emphasize that there is a relation between the existence o...
Integrality Properties of the CM-values of Certain Weak Maass Forms
Integrality Properties CM-values Certain Weak Maass Forms Number Theory
2011/9/15
Abstract: In a recent paper, Bruinier and Ono prove that the coefficients of certain weight -1/2 harmonic Maass forms are traces of singular moduli for weak Maass forms. In particular, for the partiti...
Mock period functions, sesquiharmonic Maass forms, and non-critical values of L-functions
Mock period functions sesquiharmonic Maass forms non-critical values of L-functions
2011/8/24
Abstract: We introduce a new technique of completion for 1-cohomology which parallels the corresponding technique in the theory of mock modular forms. This technique is applied in the context of non-c...
Higher-order Fourier analysis of $\mathbb{F}_p^n$ and the complexity of systems of linear forms
Higher-order Fourier analysis linear forms
2010/11/24
Consider a subset $A$ of $\mathbb{F}_p^n$ and a decomposition of its indicator function as the sum of two bounded functions $1_A=f_1+f_2$. For every family of linear forms, we find the smallest degre...
In this paper, it is shown that if F(x , y) is an irreducible binary form with integral coefficients and degree $n \geq 3$, then provided that the absolute value of the discriminant of F is large enou...
Formulas for the Fourier Coefficients of Cusp Form for Some Quadratic Forms
representation of numbers quadratic forms generalized theta series Fourier coefficient of cusp forms
2010/2/26
In this paper, representations of positive integers by certain quadratic forms Qp defined for odd prime p are examined. The number of representations of positive integer n by the quadratic form Qp, is...