Number Theory Seminar

I will explain a very explicit, conjectural relationship between first homology groups of modular curves modulo Eisenstein ideals and second K-groups of cyclotomic integer rings, in a form the mildly refines the published conjecture.  Taken up the modular and cyclotomic towers, this conjecture can be viewed as a refinement of the main conjecture of Iwasawa theory.  This Iwasawa-theoretic version of the original conjecture has been proven by Fukaya and Kato up to torsion when the relevant Kubota-Leopoldt p-adic L-function has no multiple zeros.  I will discuss joint work on a slight improvement of their result, along with work on a refinement of their method.  Finally, I will very briefly hint at work of Fukaya, Kato, and myself on higher-dimensional analogues of the conjecture.