We analyze stochastic volatility effects in the context of the bond market. The short rate model is of Vasicek type and the focus of our analysis is the effect of multiple scale variations in the volatility of this model. Using a singular perturbation approach we can identify a parsimonious representation of multiscale stochastic volatility effects. The results are illustrated with numerical simulations. We also present a framework for model calibration and look at applications to bond option pricing.

Evaluating certain Siegel modular functions at CM points on the moduli space of principally polarized abelian surfaces give algebraic
numbers which we call class invariants. The construction of class
invariants is motivated by explicit class field theory, specifically,
the construction of units with possible applications to Stark conjectures.
Class invariants can also be viewed as invariants of the binary sextic
defining a genus 2 curve whose Jacobian corresponds to the CM point
on the moduli space. The explicit construction of genus two curves
with CM is motivated by cryptographic applications.

When evaluating certain Siegel modular functions at CM points,
the coefficients of the minimal polynomials have striking factorizations.
In joint work with Eyal Goren, we studied primes that appear in the
factorization of the denominators, and proved a bound on such primes
closely related to the discriminant of the CM field. In more recent work,
we study the primes appearing simultaneously in the numerators of
CM values of certain Siegel modular functions in dimension 2.
This work generalizes the work of Gross and Zagier for the modular
j-function and is related to a conjecture of Bruinier and Yang on
intersection numbers.