These are notes for two talks given by Mark Hovey at the Summer School on the Interactions between Homotopy Theory and Algebra at the University of Chicago, July 26 to August 6, 2004. Because they are notes, they are a bit more chatty and a bit more likely to contain errors than a paper would be, so caveat lector. They are based on the papers [Hov02], [Gil04b], and [Gil04a], and concern the rel...

We introduce a natural way to define Selmer groups and $p$-adic $L$-functions for modular forms of weight 1. The corresponding Galois representation $\rho$ $\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$ is 2-dimensional Artin with odd determinant. Thus, the dimension $d^{+}$ (+1)-eigenspace complex conjugation Choose prime $p$ such that restriction local group $\mathr...

Abstract We are interested in characterising the commutative rings for which a 1-tilting cotorsion pair (