Factor Congruence Lifting Property

Lifting the Determinantal Property

In this note we study standard and good determinantal schemes. We show that there exist arithmetically Cohen-Macaulay schemes that are not standard determinantal, and whose general hyperplane section is good determinantal. We prove that if a general hyperplane section of a scheme is standard (resp. good) determinantal, then the scheme is standard (resp. good) determinantal up to flat deformatio...

On the Unique-Lifting Property

We study the uniqueness of minimal liftings of cut generating functions obtained from maximal lattice-free polytopes. We prove a basic invariance property of unique minimal liftings for general maximal lattice-free polytopes. This generalizes a previous result by Basu, Cornuéjols and Köppe [3] for simplicial maximal lattice-free polytopes, thus completely settling this fundamental question abou...

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...