Refined class number formulas and Kolyvagin systems

We use the theory of Kolyvagin systems to prove (most of) a refined class number formula conjectured by Darmon. We show that for every odd prime $p$, each side of Darmon's conjectured formula (indexed by positive integers $n$) is "almost" a $p$-adic Kolyvagin system as $n$ varies. Using the fact that the space of Kolyvagin systems is free of rank one over $\mathbf{Z}_p$, we show that Darmon's formula for arbitrary $n$ follows from the case $n=1$, which in turn follows from classical formulas

Finding large Selmer rank via an arithmetic theory of local constants

We obtain lower bounds for Selmer ranks of elliptic curves over dihedral extensions of number fields. Suppose $K/k$ is a quadratic extension of number fields, $E$ is an elliptic curve defined over $k$, and $p$ is an odd prime. Let $F$ denote the maximal abelian $p$-extension of $K$ that is unramified at all primes where $E$ has bad reduction and that is Galois over $k$ with dihedral Galois group (i.e., the generator $c$ of $Gal(K/k)$ acts on $Gal(F/K)$ by -1). We prove (under mild hypotheses on $p$) that if the rank of the pro-$p$ Selmer group $S_p(E/K)$ is odd, then the rank of $S_p(E/L)$ is at least $[L:K]$ for every finite extension $L$ of $K$ in $F$.Comment: Revised and improved. To appear in Annals of Mathematic

Disparity in Selmer ranks of quadratic twists of elliptic curves

We study the parity of 2-Selmer ranks in the family of quadratic twists of an arbitrary elliptic curve E over an arbitrary number field K. We prove that the fraction of twists (of a given elliptic curve over a fixed number field) having even 2-Selmer rank exists as a stable limit over the family of twists, and we compute this fraction as an explicit product of local factors. We give an example of an elliptic curve E such that as K varies, these fractions are dense in [0, 1]. More generally, our results also apply to p-Selmer ranks of twists of 2-dimensional self-dual F_p-representations of the absolute Galois group of K by characters of order p.Comment: This version corrects a typo in the published version. Just before the last displayed equation before Conjecture 7.12 (page 313 of the published version, page 23 of this manuscript), "...Sha(E/K) is finite" should be "...Sha(E^\chi/K) is finite". This typo does not affect anything else in the tex

A Proposal for a Comprehensive Restructuring of the Public Information System

After more than ten years of legislative, judicial and bureaucratic tinkering, the public information system created by the Freedom of Information Act (FOIA) is still far from satisfactory. The present public information system has not been successful because its drafters lacked imagination and failed to do the basic work necessary to create a sound foundation for such a comprehensive program. They failed to analyze the realistic goals of a public information system; they ignored the ultimate goals of improved government performance; they misrepresented the system\u27s costs, both in monetary expense to taxpayers and in diminished government performance. They considered neither alternative techniques nor the problem of designing the public information system as an integral part of the total governmental structure. Actual open government for the benefit of the general populace will be possible only if the basic weaknesses of the present system are explored in depth. This Article is an appeal to Congress to undertake the careful analysis necessary to construct a workable, useful public information system