Self-intersection of Manin-Drinfeld Cycles and Taylor expansion of L-functions

Abstract

A rising philosophy in the theory of automorphic representations in number theory is that higher central derivatives of L-functions of automorphic forms should correspond to the intersection numbers of special cycles on moduli spaces. A classic early result along this philosophy was achieved by Gross and Zagier, who proved that the derivative of the L-function of an elliptic curve is equal, up to a constant, to the Néron-Tate height pairing of a special point called a Heegner point on the elliptic curve.

A more recent result was proven in the function field case by Yun and Zhang which showed that higher derivatives of the base change L-function of an unramified automorphic representation over PGL₂ over a function field are equal, up to a constant, to the self-intersection number, inside the moduli stack of PGL₂-shtukas, of the moduli stack of shtukas for an anisotropic torus.

We prove in the function field case that the higher derivatives of the square of the L-function of unramified automorphic representations over PGL₂ are equal, up to a constant, to the self-intersection number, inside the moduli stack of PGL₂-shtukas, of the moduli stack of shtukas for the split torus.