DOCUMENTA MATHEMATICA, Vol. 5 (2000), 365-444

Hans-Georg Rück and Ulrich Tipp

Heegner Points and $L$-Series of Automorphic Cusp Forms of Drinfeld Type

In their famous article [Gr-Za], Gross and Zagier proved a formula relating heights of Heegner points on modular curves and derivatives of $L$-series of cusp forms.

We prove the function field analogue of this formula. The classical modular curves parametrizing isogenies of elliptic curves are now replaced by Drinfeld modular curves dealing with isogenies of Drinfeld modules. Cusp forms on the classical upper half plane are replaced by harmonic functions on the edges of a Bruhat-Tits tree.

As a corollary we prove the conjecture of Birch and Swinnerton-Dyer for certain elliptic curves over functions fields whose analytic rank is equal to $1$.

2000 Mathematics Subject Classification: 11G40 11G50 11F67 11G09

Keywords and Phrases: Heegner points, Drinfeld modular curves, $L$-series, automorphic cusp forms, Gross-Zagier formula

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