We propose to associate a modular form (an infinite number of) complex valued functions on the $p$-adic numbers $\mathbb{Q}_p$ for each prime $p$. elaborate correspondence and study its consequence in terms of Mellin transforms $L$-functions related forms. Further we discuss case products Dirichlet their duals, which are convolution $\vartheta$-series. The latter intriguingly similar non-holomo...

In this paper, we compute the number of fuzzy subgroups of some classes of non-abeilan groups. Explicit formulas are givenfor dihedral groups $D_{2n}$, quasi-dihedral groups $QD_{2^n}$, generalized quaternion groups $Q_{4n}$ and modular $p$-groups $M_{p^n}$.

Let p be prime, N be a positive integer prime to p, and k be an integer. Let Pk(t) be the characteristic series for Atkin’s U operator as an endomorphism of p-adic overconvergent modular forms of tame level N and weight k. Motivated by conjectures of Gouvêa and Mazur, we strengthen a congruence in [W] between coefficients of Pk and Pk′ for k ′ p-adically close to k. For p − 1 | 12, N = 1, k = 0...

Let V be a finite dimensional representation of a p-group, G, over a field, k, of characteristic p. We show that there exists a choice of basis and monomial order for which the ring of invariants, k[V ]G, has a finite SAGBI basis. We describe two algorithms for constructing a generating set for k[V ]G. We use these methods to analyse k[2V3]3 where U3 is the p-Sylow subgroup of GL3(Fp) and 2V3 i...

The specializations of the motivic elliptic polylogarithm on the universal elliptic curve to the modular curve are referred to as Eisenstein classes. In this paper, we prove that the syntomic realizations of the Eisenstein classes restricted to the ordinary locus of the modular curve may be expressed using p-adic Eisenstein-Kronecker series, which are p-adic modular forms defined using the two-...

This paper is about Hilbert modular forms on certain Hilbert modular varieties associated with a totally real field L. Let p be unramified in L. We reduce to the inert case and consider modular forms modulo p. We study the ideal of modular forms with q-expansion equal to zero modulo p, find canonical elements in it, and obtain as a corollary the congruences for the values of the zeta function o...

Let k be an algebraically closed field of characteristic p. Let X(p;N) be the curve parameterizing elliptic curves with full level N structure (where p N) and full level p Igusa structure. By modular curve, we mean a quotient of any X(p;N) by any subgroup of ((Z/peZ)× × SL2(Z/NZ)) /{±1}. We prove that in any sequence of distinct modular curves over k, the k-gonality tends to infinity. This exte...

This paper examines several modular inverse algorithms in GF (p) that have been proposed in literature. A survey of these algorithms attempts to study the evolution of modular inversion methods and trace key areas of improvement for hardware implementation efficiency.