Rigid cohomology defined by Berthelot agrees with the formal
cohomology defined Monsky and Washnitzer for smooth affine varieites.
Motivated by this, mimicking the convergent theory of Ogus, we define a
site using weakly completed algebras. We show a certain sheaf cohomology of
this site agrees with Berthelot's rigid cohomology

In this talk we will focus on constructing "super-isolated abelian varieties". These are abelian varieties that have isogeny class which contains a single isomorphism class. Their motivation comes from security concerns in elliptic and hyperelliptic curve cryptography. Using a theorem of Honda and Tate, we transfer the problem of finding such varieties to a problem in algebraic number theory. Finding these varieties turns out to be related to finding primes of the form n2 + 1 and to solving Pell's equation.

Let E be a CM elliptic curve over the rationals with conductor N and p a prime coprime to 6N. If the p^{infty}-Selmer group of E has Z_{p}-corank one, we show that the analytic rank of E is also one (joint with Chris Skinner and Ye Tian). We plan to discuss the setup and strategy in the ordinary case.

The class group is a natural abelian group one can associate to a number field, and it is natural to ask how it varies in families. Cohen and Lenstra famously proposed a model for families of quadratic fields based on random matrices of large rank, and this was later generalized by Cohen-Martinet. However, their model was observed by Malle to have issues when the base field contains roots of unity. We study this in detail in the case of function fields using methods of Ellenberg-Venkatesh-Westerland, and based on this we propose a model in the number field setting. Our conjecture is based on keeping track not only of the underlying group structure, but also certain natural pairings one can define in the presence of roots of unity (joint with Lipnowski, Sawin).