Let K be a number field, let OK be the ring of integers, let K be an algebraic closure of K and let OK be the ring of integers of K. Let M 0 K be the set of finite places and let M∞ K be the set of infinite places. Let Kv be the completion of K at v and let Ov be the ring of integers of Kv. Let ℘v, kv, qv be the maximal ideal of Ov, the residue field Ov/℘v and the size of the residue field |kv|...

In [MT1], B. Mazur and J. Tate present a “refined conjecture of Birch and Swinnerton-Dyer type” for a modular elliptic curve E. This conjecture relates congruences for certain integral homology cycles on E(C) (the modular symbols) to the arithmetic of E over Q. In this paper we formulate an analogous conjecture for E over suitable imaginary quadratic fields, in which the role of the modular sym...

We prove a conjecture of Thomas Lam that the face posets stratified spaces planar resistor networks are shellable. These called uncrossing partial orders. This shellability result combines with Lam's previous these same Eulerian to imply they CW posets, namely regular complexes. Certain subposets orders shown be isomorphic type A Bruhat order intervals; our shelling is coincide on intervals whi...

Let G be an algebraic group defined over a number field k. By choosing a lifting of G to a group scheme over 6' s c k, the ring of S-integers for some finite set of places S of k, we may define G(C,~), where (5~, c k~ is the ring of integers in the vadic completion of k for all non-archimedean places vr In this way, we can define the adelic points G(Ak). Since different choices of lifting will ...

1 Algebraic Groups 7 1.1 Group Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Restriction of Scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Algebraic Groups over a Nonalgebraically Closed Field K . . . . . . . . . . . 13 1.4 Structure of Algebraic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.5 Topologizing G(R...

Let Y C Pm be a subvariety of codimension d defined by an ideal / in charp > 0 with //'(Y, 0 (-1)) = 0. If t is an integer greater than log (d) and Hi( Y, f/l"+i) = 0 for n » 0 and i =1,2, then Pic(Y) is an extension of a finite p-primary group of exponent at most pt by Z[ 0 (1 )1 and Br'(Y)(p) is a group of exponent at most pl'. If Y is also smooth and defined over a finite field with dim Y < ...

Solving a conjecture of Hopkins and Mahowald, the second author [Ri] showed that Mitchell’s [Mi3] filtration {Fn,k}k=1 of ΩSU(n) splits stably, analogous to the Snaith [Sn2] splitting of BU . Crabb and Mitchell [C-M] then gave similar splittings of ΩU(n)/O(n) and ΩU(2n)/Sp(n). The first filtration Fn,1 is the inclusion CPn−1 ⊂ ΩSU(n), which was actually known to split off by the work of James [...

For each admissible monomial of Dyer-Lashof operations QI , we define a corresponding natural function Q̂I :TH̄∗(X) → H ∗(ΩnΣnX), called a Dyer-Lashof splitting. For every homogeneous class x in H∗(X), a Dyer-Lashof splitting Q̂I determines a canonical element y in H∗(ΩnΣnX) so that y is connected to x by the dual homomorphism to the operation QI . The sum of the images of all the admissible Dyer-...