Shtukas and the Taylor expansion of L-functions (II)

Abstract

For arithmetic applications, we extend and refine our previously published results to allow ramifications in a minimal way. Starting with a possibly ramified quadratic extension F'/F of function fields over a finite field in odd characteristic, and a finite set of places Σ of F that are unramified in F', we define a collection of Heegner-Drinfeld cycles on the moduli stack of PGL 2 -Shtukas with r-modifications and Iwahori level structures at places of Σ. For a cuspidal automorphic representation π of PGL 2 (AF) with square-free level Σ, and r∈Z≥0 whose parity matches the root number of πF', we prove a series of identities between (1) the product of the central derivatives of the normalized L-functions L (a) (π, 1/2)L (r-a) (π⊗η,1/2), where η is the quadratic idèle class character attached to F'/F, and 0≤a≤r; (2) the self intersection number of a linear combination of Heegner-Drinfeld cycles. In particular, we can now obtain global L-functions with odd vanishing orders. These identities are function-field analogues of the formulae of Waldspurger and Gross-Zagier for higher derivatives of L-functions.