Ramanujan's function $\tau \left( n \right)$—A congruence property

RAMANUJAN'S FUNCTION r{n)—k CONGRUENCE PROPERTY

A Uniform Refinement Property for Congruence Lattices

The Congruence Lattice Problem asks whether every algebraic distributive lattice is isomorphic to the congruence lattice of a lattice. It was hoped that a positive solution would follow from E. T. Schmidt’s construction or from the approach of P. Pudlák, M. Tischendorf, and J. Tůma. In a previous paper, we constructed a distributive algebraic lattice A with א2 compact elements that cannot be ob...

On a congruence property of elliptic curves

Let d, α ∈ Z with d > 1. In this paper, we proved the following results about congruence properties of elliptic curves: (1) For elliptic curve E over the rational number field Q, if ♯Ẽp(Fp) ≡ α (mod d) hold for almost all primes p, then almost all supersingular primes p of E satisfy p ≡ α − 1 (mod d). In particular, α−1 is prime to d. Moreover, if φ(d) > 2, then E does not have complex multipli...

Elliptic Congruence Function Fields

Recently, the well-known Diie-Hellman key exchange protocol was extended to real quadratic congruence function elds in a non-group based setting. Here, the underlying key space was the set of reduced principal ideals. This set does not possess a group structure, but instead exhibits a so-called infrastructure. The techniques are the same as in the protocol based on real quadratic number elds. A...

$n$-factorization Property of Bilinear Mappings

In this paper, we define a new concept of factorization for a bounded bilinear mapping $f:Xtimes Yto Z$, depended on  a natural number $n$ and a cardinal number $kappa$; which is called $n$-factorization property of level $kappa$. Then we study the relation between $n$-factorization property of  level $kappa$ for $X^*$ with respect to $f$ and automatically boundedness and $w^*$-$w^*$-continuity...