1. CJM 2016 (vol 68 pp. 1227)
||Eigenvarieties for Cuspforms over PEL Type Shimura Varieties with Dense Ordinary locus|
Let $p \gt 2$ be a prime and let $X$ be a compactified PEL Shimura
variety of type (A) or (C) such that $p$ is an unramified prime
for the PEL datum and such that the ordinary locus is dense in
the reduction of $X$. Using the geometric approach of Andreatta,
Iovita, Pilloni, and Stevens we define the notion of families
of overconvergent locally analytic $p$-adic modular forms of
Iwahoric level for $X$. We show that the system of eigenvalues
of any finite slope cuspidal eigenform of Iwahoric level can
be deformed to a family of systems of eigenvalues living over
an open subset of the weight space. To prove these results, we
actually construct eigenvarieties of the expected dimension that
parameterize finite slope systems of eigenvalues appearing in
the space of families of cuspidal forms.
Keywords:$p$-adic modular forms, eigenvarieties, PEL-type Shimura varieties
2. CJM 2007 (vol 59 pp. 673)
||Hecke $L$-Functions and the Distribution of Totally Positive Integers |
Let $K$ be a totally real number field of degree $n$. We show that
the number of totally positive integers
(or more generally the number of totally positive elements of a given fractional ideal)
of given trace is evenly distributed around its expected value, which is
obtained from geometric considerations.
This result depends on unfolding an integral over
a compact torus.
Keywords:Eisenstein series, toroidal integral, Fourier series, Hecke $L$-function, totally positive integer, trace
Categories:11M41, 11F30, , 11F55, 11H06, 11R47
3. CJM 2003 (vol 55 pp. 933)
||Renormalized Periods on $\GL(3)$ |
A theory of renormalization of divergent integrals over torus
periods on $\GL(3)$ is given, based on a relative truncation. It
is shown that the renormalized periods of Eisenstein series have
unexpected functional equations.