BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Tonghai Yang (University of Wisconsin) DTSTART:20201217T220000Z DTEND:20201217T230000Z DTSTAMP:20260423T024020Z UID:JNTS/11 DESCRIPTION:Title: On a conjecture of Gross and Zagier on algebraicity\nby Tonghai Yang (Un iversity of Wisconsin) as part of Columbia CUNY NYU number theory seminar\ n\n\nAbstract\nThe 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 t he Gross-Zagier formula and Gross-Kohnen-Zagier formula. It is transcenden tal in nature\, even its CM values are transcendental. It is quite inter esting to have the following conjectural algebraicity property. \nFor a we akly holomorphic modular form $f(\\tau)=\\sum\\limits_{m} c_f(m) q^m$ of w eight $-2j$ ($j \\ge 0$)\, consider the linear combination \n\\vskip -1pt\ n$$\nG_{1+j\, f}(z_1\, z_2) = \\sum_{m >0} c_f(-m) m^j G_{1+j}^m(z_1\, z_2 )\n$$\n\\vskip -1pt\n\\noindent\nwhere $G_s^m(z_1\, z_2)$ is the Hecke cor respondence of $G_s(z_1\, z_2)$ under the Hecke operator $T_m$ on the firs t (or second) variable. Gross-Zagier conjectured in 1980s that for any two CM points $z_i$ of discriminants $d_i$\n$$\n(d_1 d_2)^{j/2} G_{j+1\, f } (z_1\, z_2) = \\frac{w_{d_1}w_{d_2}}{4}\\cdot \\log|\\alpha|\n$$\nfor so me 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. I f time permits\, I will also explain how one method to attack this conject ure also produces an analogue of the Gross-Kohnen-Zagier theorem in Kuga v arieties. \n\nIn the RTG talk\, I will explain regularized theta lifting ( Borcherds product) and their CM value formula.\n LOCATION:https://researchseminars.org/talk/JNTS/11/ END:VEVENT END:VCALENDAR