d The action of the derivation e = q ~ on the q-expansions of modular forms in characteristic p is one of the fundamental tools in the Serre/Swinnerton-Dyer theory of mod p modular forms. In this note, we extend the basic results about this action, already known for P > 5 and level one, to arbitrary p and arbitrary prime-to-p level. !. Review of modular forms in characteristic p We fix an algeb...

Modular towers, a notion due to M. Fried, are towers of Hurwitz spaces, with levels corresponding to the characteristic quotients of the p-universal Frattini cover of a fixed finite group G (with p a prime divisor of |G|). The tower of modular curves X1(pn) (n>0) is the original example: the group G is then the dihedral group Dp. There are diophantine conjectures on modular towers, inspired by ...

In this paper we investigate the parallelization of two modular algorithms. In fact, we consider the modular computation of Gröbner bases (resp. standard bases) and the modular computation of the associated primes of a zero–dimensional ideal and describe their parallel implementation in Singular. Our modular algorithms to solve problems over Q mainly consist of three parts, solving the problem ...

‎in this paper, based on [a. razani, v. rako$check{c}$evi$acute{c}$ and z. goodarzi, nonself mappings in modular spaces and common fixed point theorems, cent. eur. j. math. 2 (2010) 357-366.] a fixed point theorem for non-self contraction mapping $t$ in the modular space $x_rho$ is presented. moreover, we study a new version of krasnoseleskii's fixed point theorem for $s+t$, where $t$ is a cont...

The main result of the paper is a description of the maximal ideals in the modular group algebras of the finitary symmetric and alternating groups (provided the characteristic p of the ground field is greater than 2). For the symmetric group there are exactly p − 1 such ideals and for the alternating group there are (p − 1)/2 of them. The description is obtained in terms of the annihilators of ...

Ho stein and Hulse recently introduced the notion of shifted convolution Dirichlet series for pairs of modular forms f1 and f2. The second two authors investigated certain special values of symmetrized sums of such functions, numbers which are generally expected to be mysterious transcendental numbers. They proved that the generating functions of these values in the h-aspect are linear combinat...

We use the relationship between Jacobi forms and vector-valued modular forms to study the Fourier expansions of Jacobi forms of indexes p, p2, and pq for distinct odd primes p, q. Specifically, we show that for such indexes, a Jacobi form is uniquely determined by one of the associated components of the vector-valued modular form. However, in the case of indexes of the form pq or p2, there are ...

For any natural number l and any prime p ≡ 1 (mod 4) not dividing l there is a Hermitian modular form of arbitrary genus n over L := Q[ √ −l] that is congruent to 1 modulo p which is a Hermitian theta series of an OL-lattice of rank p− 1 admitting a fixed point free automorphism of order p. It is shown that also for non-free lattices such theta series are modular forms.

Course description: This course will give an introduction to the theory of overconvergent modular symbols. This theory mirrors the theory of overconvergent modular forms in that both spaces encode the same systems of Hecke-eigenvalues. Moreover, the theory of overconvergent modular symbols has the great feature of being easily computable and is intimately connected to the theory of p-adic L-fun...

We bound the j-invariant of integral points on a modular curve in terms of the congruence group defining the curve. We apply this to prove that the modular curve Xsplit(p ) has no non-trivial rational point if p is a sufficiently large prime number. Assuming the GRH, one can replace p by p. AMS 2000 Mathematics Subject Classification 11G18 (primary), 11G05, 11G16 (secondary).