Bryden Cais

Modular curves and Ramanujan's continued fraction

Abstract We use arithmetic models of modular curves to establish some properties of Ramanujans continued fraction. In particular, we give a new geometric proof that its singular values are algebraic units that generate specific abelian extensions of imaginary quadratic fields, and we use a mixture of geometric and analytic methods to construct and study an infinite family of two-variable polynomials over ℤ that are related to Ramanujans function in the same way that the classical modular polynomials are related to the classical j-function. We also prove that a singular value on the imaginary axis, necessarily real, lies in a radical tower in ℝ only if all odd prime factors of its degree over ℚ are Fermat primes; by computing some ray class groups, we give many examples where this necessary condition is not satisfied.

Dieudonné; crystals and Wach modules for p-divisible groups

Let k be a perfect field of characteristic p>2 and K an extension of F=FracW(k) contained in some F(μpr). Using crystalline Dieudonne theory, we provide a classification of p-divisible groups over R=OK[[t1,...,td]] in terms of finite height (φ,Γ)-modules over S:=W(k)[[u,t1,...,td]]. When d=0, such a classification is a consequence of (a special case of) the theory of Kisin–Ren; in this setting, our construction gives an independent proof of this result, and moreover allows us to recover the Dieudonne crystal of a p-divisible group from the Wach module associated to its Tate module by Berger–Breuil or by Kisin–Ren.

Random Dieudonné modules, random p-divisible groups, and random curves over finite fields

We describe a probability distribution on isomorphism classes of principally quasi-polarized

The geometry of Hida families I: \(\Lambda \)-adic de Rham cohomology

p

CANONICAL INTEGRAL STRUCTURES ON THE DE RHAM COHOMOLOGY OF CURVES

-divisible groups over a finite field

Canonical extensions of Néron models of Jacobians

k

On f-crystalline representations

of characteristic

A characterization of strictly APF extensions

p

Canonical Cohen Rings for Norm Fields

which can reasonably be thought of as a ‘uniform distribution’, and we compute the distribution of various statistics (

Breuil-Kisin modules via crystalline cohomology

p