In the first part of the talk we describe an algorithm which computes a relative algebraic K-group as an abstract abelian group. We also show how this representation can be used to do computations in these groups. This is joint work with Steve Wilson. Our motivation for this project originates from the study of the Equivariant Tamagawa Number Conjecture which is formulated as an equality of an ...

For a (connected) smooth projective curve C over the rational numbers Q, it is known that the rational points CpQq depends on the genus g “ gpCq of C: (1) If g “ 0, then the local-global principle holds for C, i.e.: CpQq ‰ H if and only if CpQpq ‰ H for all primes p ď 8 (we understand Qp “ R when p “ 8). In other words, C is globally solvable if and only if it is locally solvable everywhere. An...

This paper provides evidence for the Birch and Swinnerton-Dyer conjecture for analytic rank 0 abelian varieties Af that are optimal quotients of J0(N) attached to newforms. We prove theorems about the ratio L(Af , 1)/ΩAf , develop tools for computing with Af , and gather data about certain arithmetic invariants of the nearly 20, 000 abelian varieties Af of level ≤ 2333. Over half of these Af ha...

We give upper and lower bounds for the Kazhdan-Lusztig polynomials of any Coxeter group W. If W is nite we prove that, for any k 0, the k-th coeecient of the Kazhdan-Lusztig polynomial of two elements u, v of W is bounded from above and below by a polynomial (which depends only on k) in l(v)?l(u). In particular, this implies the validity of Lascoux-Schutzenberger's conjecture for all suuciently...

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 ...