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Session S09 - Number Theory in the Americas

Wednesday, July 14, 14:30 ~ 15:00 UTC-3

On a construction of Poincar\'e series for $SU(2,1)$

Roberto Miatello

U. N. Cordova, Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

Poincar\'e series are holomorphic automorphic forms on the upper half-plane, defined by Poincar\'e, and studied by Hecke and Petersson who showed in particular that the inner product of a cusp form $f$ and a Poincar\'e series $P_m$ is a nonzero multiple of the $m^{th}$ Fourier coefficient of $f$. As a consequence, the Poincar\'e series span the full space of cusp forms of weight $k$, for each weight $ k> 2$. In the case of a larger group like the group $G=SU(2,1)$, this construction cannot be directly generalized since there exist many non-generic cusp forms, i.e., those having all its Fourier coefficients equal to zero. Jointly with Roelof Bruggeman we have defined a holomorphic family of Poincar\'e series attached to a Stone-von Neumann representation of the maximal unipotent subgroup $N$ of $G$. After carrying out a meromorphic continuation, we study the singularities and the special values, showing in particular that, as in the classical case, the special values of the family span the spaces holomorphic (and antiholomorphic) cusp forms of the corresponding symmetric space $G/K$ (the complex hyperbolic space $CH^2$).

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