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Theta block conjecture for Siegel paramodular forms

Posted in
Haowu Wang
Mit, 2019-11-06 14:30 - 15:30
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

A Siegel paramodular form of level N is a Siegel modular form of degree two
with respect to the paramodular group of level N.  The Fourier-Jacobi
coefficients of such modular form are holomorphic Jacobi forms in the sense
of Eichler-Zagier. Conversely,  there are two ways to construct paramodular
forms from Jacobi forms, namely additive Jacobi lifts (a generalization of
Maass lifts) and Borcherds products. In 2013, Gritsenko, Poor and Yuen

proposed the theta-block conjecture which characterizes paramodular forms
which are simultaneously Borcherds products and additive Jacobi lifts.

Theta-block is a great way to construct holomorphic Jacobi forms of small
weight proposed by Gritsenko-Skoruppa-Zagier. They are defined as the
product of odd Jacobi theta-series divided by Dedekind eta-function.  In
this talk, I will prove the theta-block conjecture for two infinite series
of  theta blocks of minimal weight 2 related to root systems A_4 and

A_1+B_3, here the infinite series mean that the level of paramodular forms
goes to infinite. In some sense, the additive lift is like an infinite sum
and the Borcherds product has an infinite product expansion. Thus our
results give generalized Euler type identities of the form "infinite sum =
infinite product".

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