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Search: MSC category 11F55 ( Other groups and their modular and automorphic forms (several variables) )

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1. CJM Online first

Brasca, Riccardo
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
Categories:11F55, 11F33

2. CJM 2007 (vol 59 pp. 673)

Ash, Avner; Friedberg, Solomon
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)

Beineke, Jennifer; Bump, Daniel
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.

Categories:11F12, 11F55

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