On the special values of certain L-series related to half-integral weight modular forms

Let h be a cuspidal Hecke eigenform of half-integral weight, and En/2+1/2 be Cohen’s Eisenstein series of weight n/2+1/2. For a Dirichlet character χ we define a certain linear combination R(χ)(s, h,En/+1/2) of the Rankin-Selberg convolution products of h and En/2+1/2 twisted by Dirichlet characters related with χ. We then prove a certain algebraicity result for R(χ)(l, h,En/2+1/2) with l integers

Non-vanishing of LL-functions associated to cusp forms of half-integral weight

In this article, we prove non-vanishing results for $L$-functions associated to holomorphic cusp forms of half-integral weight on average (over an orthogonal basis of Hecke eigenforms). This extends a result of W. Kohnen to forms of half-integral weight.Comment: 8 pages, Accepted for publication in Oman conference proceedings (Springer

On certain finiteness questions in the arithmetic of modular forms

We investigate certain finiteness questions that arise naturally when studying approximations modulo prime powers of p-adic Galois representations coming from modular forms. We link these finiteness statements with a question by K. Buzzard concerning p-adic coefficient fields of Hecke eigenforms. Specifically, we conjecture that for fixed N, m, and prime p with p not dividing N, there is only a finite number of reductions modulo p^m of normalized eigenforms on \Gamma_1(N). We consider various variants of our basic finiteness conjecture, prove a weak version of it, and give some numerical evidence.Comment: 25 pages; v2: one of the conjectures from v1 now proved; v3: restructered parts of the article; v4: minor corrections and change

Performance Criteria for Relational Database Normalization

The fourth normal form where data redundancy is eliminated is a more efficient construct for storage and user access

Virtually abelian K\"ahler and projective groups

We characterise the virtually abelian groups which are fundamental groups of compact K\"ahler manifolds and of smooth projective varieties. We show that a virtually abelian group is K\"ahler if and only if it is projective. In particular, this allows to describe the K\"ahler condition for such groups in terms of integral symplectic representations

Moduli interpretation of Eisenstein series

Let L >= 3. Using the moduli interpretation, we define certain elliptic modular forms of level Gamma(L) over any field k where 6L is invertible and k contains the Lth roots of unity. These forms generate a graded algebra R_L, which, over C, is generated by the Eisenstein series of weight 1 on Gamma(L). The main result of this article is that, when k=C, the ring R_L contains all modular forms on Gamma(L) in weights >= 2. The proof combines algebraic and analytic techniques, including the action of Hecke operators and nonvanishing of L-functions. Our results give a systematic method to produce models for the modular curve X(L) defined over the Lth cyclotomic field, using only exact arithmetic in the L-torsion field of a single Q-rational elliptic curve E^0.Comment: 29 pages, amslatex. Version 6: corrected a sign misprint in equation (4.6) (thanks to N. Mascot for pointing it out). Final accepted versio

Endomorphisms of superelliptic jacobians

Let K be a field of characteristic zero, n>4 an integer, f(x) an irreducible polynomial over K of degree n, whose Galois group is doubly transitive simple non-abelian group. Let p be an odd prime, Z[\zeta_p] the ring of integers in the p-th cyclotomic field, C_{f,p}:y^p=f(x) the corresponding superelliptic curve and J(C_{f,p}) its jacobian. Assuming that either n=p+1 or p does not divide n(n-1), we prove that the ring of all endomorphisms of J(C_{f,p}) coincides with Z[\zeta_p].Comment: Several typos have been correcte

Interpolated sequences and critical LL-values of modular forms

Recently, Zagier expressed an interpolated version of the Ap\'ery numbers for $\zeta(3)$ in terms of a critical $L$-value of a modular form of weight 4. We extend this evaluation in two directions. We first prove that interpolations of Zagier's six sporadic sequences are essentially critical $L$-values of modular forms of weight 3. We then establish an infinite family of evaluations between interpolations of leading coefficients of Brown's cellular integrals and critical $L$-values of modular forms of odd weight.Comment: 23 pages, to appear in Proceedings for the KMPB conference: Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theor

Restricted sum formula of alternating Euler sums

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