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A characterization of ordinary modular eigenforms with CM
modular forms with complex multiplication Galois represenations Number Theory
2012/6/12
We show that a $p$-ordinary modular eigenform $f$ of weight $k\geq 2$, with $p$-adic Galois representation $\rho_f$ and $\mod{p^m}$ reductions $\rho_{f,m}$, and with complex multiplication(CM) is char...
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...
In the present work we investigate the largest possible gaps between consecutive numbers which can be written as the difference of two primes. The best known upper bounds are the same as those concern...
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...
Quadratic congruences on average and rational points on cubic surfaces
Quadratic congruences rational points Manin’s conjecture cubic surfaces universal torsors
2012/5/24
We investigate the average number of solutions of certain quadratic congruences. As an application, we establish Manin's conjecture for a cubic surface whose singularity type is A_5+A_1.
Mersenne Primes in Real Quadratic Fields
Mersenne Primes Real Quadratic Fields Number Theory
2012/5/24
The concept of Mersenne primes is studied in real quadratic fields of class number 1. Computational results are given. The field $Q(\sqrt{2})$ is studied in detail with a focus on representing Mersenn...
We are interested in classifying those sets of primes P such that when we sieve out the integers up to x by the primes in P^c we are left with roughly the expected number of unsieved integers. In part...
On a senary cubic form
Manin-Peyre conjecture singular cubic fourfold multiple Dirichlet series universal torsor
2012/5/9
A strong form of the Manin-Peyre conjecture with a power saving error term is proved for a certain cubic fourfold.
A study on multiple zeta values from the viewpoint of zeta-functions of root systems
multiple zeta values zeta-functions of root systems Number Theory
2012/5/9
We study multiple zeta values (MZVs) from the viewpoint of zeta-functions associated with the root systems which we have studied in our previous papers. In fact, the $r$-ple zeta-functions of Euler-Za...
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...
Error term improvements for van der Corput transforms
Asymptotic analysis exponential sum trigonometric sum
2012/5/9
We improve the error term in the van der Corput transform for exponential sums
\sum_{a \le n \le b} g(n) exp(2\pi i f(n)).
For many functions g and f, we can extract the next term in the asymptoti...
Littlewood asked how small the ratio $||f||_4/||f||_2$ (where $||.||_\alpha$ denotes the $L^\alpha$ norm on the unit circle) can be for polynomials $f$ having all coefficients in $\{1,-1\}$, as the de...
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 Mordell-Tornheim sums and multiple zeta values
Tornheim series Witten zeta function Euler sums multiple harmonic series multiple zeta values
2012/5/9
We prove that any Mordell-Tornheim sum with positive integer arguments can be expressed as a rational linear combination of multiple zeta values of the same weight and depth. By a result of Tsumura, i...
Solving the Odd Perfect Number Problem: Some New Approaches
Odd perfect number Euler factor inequalities OPN components non-injective and non-surjective mapping
2012/6/29
A conjecture predicting an injective and surjective mapping $X = \displaystyle\frac{\sigma(p^k)}{p^k}, Y = \displaystyle\frac{\sigma(m^2)}{m^2}$ between OPNs $N = {p^k}{m^2}$ (with Euler factor $p^k$)...