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

TL;DR

This paper explores the relationship between algebraic cycles on products of elliptic curves and special values of Green's functions at CM points, proving the existence of infinitely many such cycles and relating them to Borcherds' lifts.

Contribution

It constructs infinitely many motivic cycles in the universal family of elliptic curve products and links these cycles to Borcherds' lifts of modular forms.

Findings

Proved the existence of infinitely many motivic cycles.

Established a connection between motivic cycles and Borcherds' lifts.

Worked out an example demonstrating algebraicity of certain values.

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. We relate this question to the existence of motivic cycles in the universal family of products of elliptic curves along the lines of Mellit and Zhang. We then construct infinitely many such cycles. In the appendix we work out an example of algebraicity. The work of Li, Bruinier-Ehlen-Yang, Viazovska and others relate this conjecture to Borcherds' lifts of weakly holomorphic modular forms. This suggests that there should be a link between motivic cycles in the universal family on the one hand and Borcherds' lifts on the other. We formulate a precise conjecture relating the two objects.