On a conjecture of Gross and Zagier on algebraicity

Tonghai Yang (University of Wisconsin)

17-Dec-2020, 22:00-23:00 (3 years ago)

Abstract: The automorphic Green function $G_s(z_1, z_2)$ for $SL_2(\Bbb Z)$, also called the resolvent kernel function for $\Gamma$, plays an important role in both analytic and algebra number theory, e.g. in the Gross-Zagier formula and Gross-Kohnen-Zagier formula. It is transcendental in nature, even its CM values are transcendental. It is quite interesting to have the following conjectural algebraicity property. For a weakly holomorphic modular form $f(\tau)=\sum\limits_{m} c_f(m) q^m$ of weight $-2j$ ($j \ge 0$), consider the linear combination \vskip -1pt $$ G_{1+j, f}(z_1, z_2) = \sum_{m >0} c_f(-m) m^j G_{1+j}^m(z_1, z_2) $$ \vskip -1pt \noindent where $G_s^m(z_1, z_2)$ is the Hecke correspondence of $G_s(z_1, z_2)$ under the Hecke operator $T_m$ on the first (or second) variable. Gross-Zagier conjectured in 1980s that for any two CM points $z_i$ of discriminants $d_i$ $$ (d_1 d_2)^{j/2} G_{j+1, f} (z_1, z_2) = \frac{w_{d_1}w_{d_2}}{4}\cdot \log|\alpha| $$ for some algebraic number $\alpha$, where $w_i$ is the number of units in $O_{d_i}$. In this talk, I will describe some progress on this conjecture. If time permits, I will also explain how one method to attack this conjecture also produces an analogue of the Gross-Kohnen-Zagier theorem in Kuga varieties.

In the RTG talk, I will explain regularized theta lifting (Borcherds product) and their CM value formula.

number theory

Audience: researchers in the topic


Columbia CUNY NYU number theory seminar

Organizers: Dorian Goldfeld*, Eric Urban, Fedor Bogomolov, Yuri Tschinkel, Alexander Gamburd, Victor Kolyvagin, Gautam Chinta
*contact for this listing

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