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

Abstract

For arithmetic applications, we extend and refine our results in [10] 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 (A F) with square-free level Σ, and r ∈ Z ≥0 whose parity matches the root number of π F ′ , 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 formulas of Waldspurger and Gross-Zagier for higher derivatives of L-functions.

A Gross-Kohnen-Zagier type formula for moduli of shtukas with Iwahori level structures

“…To define the moduli of shtukas with level structures, Yun-Zhang introduced fractional twists and Atkin-Lehner involution for vector bundles with Iwahori level structures ( [7], Definition 3.2). Again, I only review what I need here, for the more general cases, you can just go to read their paper.…”

“…Yun and Zhang proved the following geometric properties of the Hecke stacks ( [7], Proposition 3.4).…”

“…Following the method of Yun-Zhang ( [7], Corollary 4.11), one can prove the following: Lemma 2.14. For any two ξ, ξ ∈ S(k), one has:…”

“…In their secomd volumn [7], Yun-Zhang generalized their previous work [6] to the moduli stack of shtukas with Iwahori level structure and Heegner-Drinfeld cycles coming from a double cover with ramification points away from the level. A natural question is whether one can do a similar thing for Heegner-Drinfeld cycles coming from two different double covers, i.e.…”

“…Yun-Zhang proved the following "coarse spectral decomposition" of the total cohomology of Sht r G (Σ; Σ ∞ ) ( [7], Theorem 3.41) (By total cohomology I mean not the cohomology of the generic fiber): Let V(ξ) = H r c (Sht r G (Σ; Σ ∞ ) × S ∞ ξ, Q l ), then one has the decompostion:…”