Algebraic cycles and values of Green's functions -- Products of Elliptic Curves

Abstract

Gross and Zagier defined certain `higher Green's functions' on products of modular curves and conjectured that the value of these functions at complex multiplication points should be logarithms of algebraic numbers. This is now a theorem of Li and Bruinier-Li-Yang. We relate this conjecture to the existence of motivic cycles in the universal family of products of elliptic curves along the lines of Mellit and Zhang. Using this we are able to prove Zagier's conjecture in some cases when the two CM points have the same discriminant. This is originally a theorem of Viazovska. Li, Bruinier-Li-Yang, Bruinier-Ehlen-Yang, Viazovska and others relate this conjecture to Borcherds' lifts of weakly holomorphic modular forms. Their works, coupled with ours, suggest that there should be a link between motivic cycles in the universal family on the one hand and Borcherds lifts on the other. We explain why this is the case. This suggests a motivic interpretation of weakly holomorphic modular forms. In the special case we look at we show that indeed this is true.