Let E/Q be an elliptic curve defined by the equation y 2 = x 3 + ax + b. For a prime p, p ∤ ∆ = −16(4a 3 + 27b 2) = 0, define N p = p + 1 − a p = |E(F p)|. As a precursor to their celebrated conjecture, Birch and Swinnerton-Dyer originally conjectured that for some constant c, p≤x,p∤∆ N p p ∼ c(log x) r , x → ∞. Let α p and β p be the eigenvalues of the Frobenius at p. Define ˜ c n =    α k ...

0. Introduction. In D], Dyer has introduced the concept of O algebra, this is a basic quasi-hereditary Koszul algebra with an anti-involution, such that its quadratic dual is also quasi-hereditary. Typical examples of such algebras are ones associated with blocks of category O of Bernstein-Gelfand-Gelfand BGG], see also Beilinson-Ginsburg-Soergel BGS]. The present paper grew out of a desire to ...

We now let # be the set of points P o such that M = M(P0); if sup M(Pt) < M Pit* (which happens in all cases known so far) we call this the second minimum, M2. R(0) is Euclidean if, and only if, M{P) < 1 for all rational triads [x,y,z]; and if 4> and \jj can be expressed as quadratic polynomials in 0, then the field is said to be cyclic. Heilbronn [1] has shown that Euclid's Algorithm holds in ...

We study the problem of sampling contingency tables (nonnegative integer matrices with specified row and column sums) uniformly at random. We give an algorithm which runs in polynomial time provided that the row sums ri and the column sums cj satisfy ri (n 3/ m log m), and cj (m 3/ n log n). This algorithm is based on a reduction to continuous sampling from a convex set. The same approach was t...