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Jan 28

Monica Nevins
Interpreting the local character expansion of p-adic SL(2)

The Harish-Chandra—Howe local character expansion expresses the character of an admissible representation of a p-adic group G as a linear combination with complex coefficients of the (finitely many!) Fourier transforms of nilpotent orbital integrals \(\widehat{\mu}_{\mathcal{O}}\) --- near the identity. Approaching from another direction: we can restrict the representation to any compact open subgroup K of G, obtaining its branching rules, which also describe the representation near the identity, in a different sense. We show that for G=SL(2,k), k a nonarchimedean local field, where the branching rules to maximal compact open subgroups K are known, each of these terms \(\widehat{\mu}_{\mathcal{O}}\) can be interpreted as the character \(\tau_{\mathcal{O}}\) of a representation of K, up to an error term arising from the zero orbit. Moreover, the irreducible components of \(\tau_{\mathcal{O}}\) are explicitly constructed from the K -orbits in \(\mathcal{O}\). This work in progress offers a conjectural alternative interpretation of branching rules of admissible representations.

Feb 4

Ke Chen
On CM points away from the Torelli locus

Coleman conjectured in 1980's that when g is an integer sufficiently large, the open Torelli locus T_g in the Siegel modular variety A_g should contain at most finitely many CM points, namely Jacobians of general curves of high genus should not admit complex multiplication. We show that certain CM points do not lie in T_g if they parametrize abelian varieties isogeneous to products of simple CM abelian varieties of low dimension. The proof relies on known results on Faltings height and Sato-Tate equidistributions. This is a joint work with Kang Zuo and Xin Lv.