For a set of integers A, we consider \(R(A)=\{a/b: a, b\in b\ne 0\}\). It is an open problem to study the denseness R(A) in p-adic numbers when A nonzero values attained by integral form. This has been answered for quadratic forms. Very recently, Antony and Barman have studied this diagonal binary cubic forms \(ax^3+by^3\), where b are integers. In article, We extend their results \(ax^n+by^n\)...

In this paper, we construct a splitting of the metaplectic cover of the reductive dual pairs of orthogonal and symplectic groups or the reductive dual pairs of unitary groups over a nonarchimedean local field with respect to a generalized lattice model of the Weil representation. We also prove a result concerning the splitting that we construct and the theta dichotomy for unitary group. The spl...

Ahlgren and Samart relate three cusp forms with complex multiplication to certain weakly holomorphic modular using p-adic bounds related their Fourier coefficients. In these examples, result strengthens a theorem of Guerzhoy, Kent, Ono which pairs CM via limits. use only the theory Hecke operators, whereas harmonic Maass forms. Here we extend Samart's work all cases where form space is one-dime...

We define exact functors from categories of Harish-Chandra modules for certain real classical groups to finite-dimensional modules over an associated graded affine Hecke algebra with parameters. We then study some of the basic properties of these functors. In particular, we show that they map irreducible spherical representations to irreducible spherical representations and, moreover, that they...

We prove modularity lifting theorems for l-adic Galois representations of any dimension satisfying a unitary type condition and a Fontaine-Laffaille type condition at l. This extends the results of Clozel, Harris and Taylor ([CHT08]), and the subsequent work by Taylor ([Tay08]). The proof uses the Taylor-Wiles method, as improved by Diamond, Fujiwara, Kisin and Taylor, applied to Hecke algebras...

In 1913 A. Fleck proved that if p is a prime, and n > 0 and r are integers then ∑ k≡r (mod p) (n k ) (−1) ≡ 0 ( mod pb(n−1)/(p−1)c ) . Only recently the significance of Fleck’s congruence was realized. It plays a fundamental role in Colmez’ and Wan’s investigation of the ψ-operator related to Fontaine’s theory and p-adic Langlands correspondence. In this talk we give a survey of the recent deve...

Aside from intrinsic interest there are three (related) reasons for studying the unitary representations of £-adic division algebras. (1) They provide heuristics for more difficult (e.g., noncompact) £-adic groups. (2) They would be basic building blocks in a theory of representations of reductive algebraic £-adic groups based on the philosophy of cusp forms. (3) There seem to be deep relations...