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Higher Green functions and their CM values

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Stephan Ehlen
Universität zu Köln
Wed, 2018-12-05 14:30 - 15:30
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

In my talk I will present results from joint work in progress with Jan Bruinier and Tonghai Yang.
We construct higher Green functions on orthogonal Shimura varieties of signature $(n,2)$ using a regularized theta lift.
We also consider their special values at certain CM points and prove algebraicity results for these special values.
For n=2 and n=1, our results concern the automorphic Green function on modular curves which plays an
important role in the arithmetic geometry. Gross and Zagier made a conjecture about the CM values
of certain linear combinations of Hecke translates of this Green function.
Special cases of the conjecture are known by work of Mellit and Viazovska.
We establish new cases and generalize and obtain generalization of the conjecture for signature $(n,2)$.

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