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Shtukas and the Taylor Expansion of L-Functions Annals of Mathematics 186 (2017), 767{911 https://doi.org/10.4007/annals.2017.186.3.2 Shtukas and the Taylor expansion of L-functions By Zhiwei Yun and Wei Zhang Abstract We define the Heegner{Drinfeld cycle on the moduli stack of Drinfeld Shtukas of rank two with r-modifications for an even integer r. We prove an identity between (1) the r-th central derivative of the quadratic base change L-function associated to an everywhere unramified cuspidal automorphic representation π of PGL2, and (2) the self-intersection number of the π- isotypic component of the Heegner{Drinfeld cycle. This identity can be viewed as a function-field analog of the Waldspurger and Gross{Zagier formula for higher derivatives of L-functions. Contents 1. Introduction 768 Part 1. The analytic side 781 2. The relative trace formula 781 3. Geometric interpretation of orbital integrals 788 4. Analytic spectral decomposition 797 Part 2. The geometric side 807 5. Moduli spaces of Shtukas 807 6. Intersection number as a trace 821 7. Cohomological spectral decomposition 845 Part 3. The comparison 869 8. Comparison for most Hecke functions 869 9. Proof of the main theorems 874 Keywords: L-functions, Drinfeld Shtukas, Gross{Zagier formula, Waldspurger formula AMS Classification: Primary: 11F67; Secondary: 14G35, 11F70, 14H60. Research of Z.Yun partially supported by the Packard Foundation and the NSF grant DMS-1302071/DMS-1736600. Research of W. Zhang partially supported by the NSF grants DMS-1301848 and DMS-1601144, and a Sloan research fellowship. c 2017 Department of Mathematics, Princeton University. 767 768 ZHIWEI YUN and WEI ZHANG Appendix A. Results from intersection theory 880 Appendix B. Super-positivity of L-values 901 Index 907 References 909 1. Introduction In this paper we prove a formula for the arbitrary order central derivative of a certain class of L-functions over a function field F = k(X) for a curve X over a finite field k of characteristic p > 2. The L-function under consid- eration is associated to a cuspidal automorphic representation of PGL2;F , or rather, its base change to a quadratic field extension of F . The r-th central derivative of our L-function is expressed in terms of the intersection number r of the \Heegner{Drinfeld cycle" on a moduli stack denoted by ShtG in the r introduction, where G = PGL2. The moduli stack ShtG is closely related to the moduli stack of Drinfeld Shtukas of rank two with r-modifications. One important feature of this stack is that it admits a natural fibration over the r-fold self-product Xr of the curve X over Spec k r r ShtG / X : The very existence of such moduli stacks presents a striking difference between a function field and a number field. In the number field case, the analogous spaces only exist (at least for the time being) when r ≤ 1. When r = 0, the 0 moduli stack ShtG is the constant groupoid over k (1.1) BunG(k) ' G(F )n(G(A)=K); where A is the ring of ad`elesand K is a maximal compact open subgroup of G(A). The double coset in the right-hand side of (1.1) remains meaningful for a number field F (except that one cannot demand the archimedean component of K to be open). When r = 1 the analogous space in the case F = Q is the moduli stack of elliptic curves, which lives over Spec Z. From such perspectives, our formula can be viewed as a simultaneous generalization (for function fields) of the Waldspurger formula [26] (in the case of r = 0) and the Gross{Zagier formula [11] (in the case of r = 1). Another noteworthy feature of our work is that we need not restrict our- selves to the leading coefficient in the Taylor expansion of the L-functions: our formula is about the r-th Taylor coefficient of the L-function regardless whether r is the central vanishing order or not. This leads us to speculate that, contrary to the usual belief, central derivatives of arbitrary order of mo- tivic L-functions (for instance, those associated to elliptic curves) should bear some geometric meaning in the number field case. However, due to the lack TAYLOR EXPANSION 769 r of the analog of ShtG for arbitrary r in the number field case, we could not formulate a precise conjecture. Finally we note that, in the current paper, we restrict ourselves to every- where unramified cuspidal automorphic representations. One consequence is that we only need to consider the even r case. Ramifications, particularly the odd r case, will be considered in subsequent work. Now we give more details of our main theorems. 1.1. Some notation. Throughout the paper, let k = Fq be a finite field of characteristic p > 2. Let X be a geometrically connected smooth proper curve over k. Let ν : X0 ! X be a finite ´etalecover of degree 2 such that X0 is also geometrically connected. Let σ 2 Gal(X0=X) be the nontrivial involution. Let F = k(X) and F 0 = k(X0) be their function fields. Let g and g0 be the genera of X and X0, then g0 = 2g − 1. We denote the set of closed points (places) of X by jXj. For x 2 jXj, let Ox be the completed local ring of X at x and let Fx be its fraction field. Let Q0 Q A = x2jXj Fx be the ring of ad`eles,and let O = x2jXj Ox be the ring of 0 integers inside A. Similar notation applies to X . Let × × × ηF 0=F : F nA =O / {±1g be the character corresponding to the ´etaledouble cover X0 via class field theory. Q Let G = PGL2. Let K = x2jXj Kx where Kx = G(Ox). The (spherical) 1 Hecke algebra H is the Q-algebra of bi-K-invariant functions Cc (G(A)==K; Q) with the product given by convolution. 1 1.2. L-functions. Let A = Cc (G(F )nG(A)=K; Q) be the space of ev- erywhere unramified Q-valued automorphic functions for G. Then A is an H -module. By an everywhere unramified cuspidal automorphic representa- tion π of G(AF ) we mean an H -submodule Aπ ⊂ A that is irreducible over Q. For every such π, EndH (Aπ) is a number field Eπ, which we call the co- efficient field of π. Then by the commutativity of H , Aπ is a one-dimensional Eπ-vector space. If we extend scalars to C, Aπ splits into one-dimensional HC-modules Aπ ⊗Eπ,ι C, one for each embedding ι : Eπ ,! C, and each Aπ⊗Eπ,ιC ⊂ AC is the unramified vectors of an everywhere unramified cuspidal automorphic representation in the usual sense. The standard (complete) L-function L(π; s) is a polynomial of degree −s−1=2 0 4(g − 1) in q with coefficients in the ring of integers OEπ . Let πF be the 0 base change to F , and let L(πF 0 ; s) be the standard L-function of πF 0 . This L-function is a product of two L-functions associated to cuspidal automorphic representations of G over F : L(πF 0 ; s) = L(π; s)L(π ⊗ ηF 0=F ; s): 770 ZHIWEI YUN and WEI ZHANG −s−1=2 Therefore, L(πF 0 ; s) is a polynomial of degree 8(g − 1) in q with coeffi- cients in Eπ. It satisfies a functional equation L(πF 0 ; s) = (πF 0 ; s)L(πF 0 ; 1 − s); where the epsilon factor takes a simple form −8(g−1)(s−1=2) (πF 0 ; s) = q : Let L(π; Ad; s) be the adjoint L-function of π. Denote −1=2 L(πF 0 ; s) (1.2) L (π 0 ; s) = (π 0 ; s) ; F F L(π; Ad; 1) where the the square root is understood as −1=2 4(g−1)(s−1=2) (πF 0 ; s) := q : Then we have a functional equation: L (πF 0 ; s) = L (πF 0 ; 1 − s): Note that the constant factor L(π; Ad; 1) in L (πF 0 ; s) does not affect the func- tional equation, and it shows up only through the calculation of the Petersson inner product of a spherical vector in π; see the proof of Theorem 4.5. Consider the Taylor expansion at the central point s = 1=2: r X (r) (s − 1=2) L (π 0 ; s) = L (π 0 ; 1=2) ; F F r! r≥0 i.e., r Ç å (r) d −1=2 L(πF 0 ; s) 0 0 L (πF ; 1=2) = r (πF ; s) : ds s=0 L(π; Ad; 1) If r is odd, by the functional equation we have (r) L (πF 0 ; 1=2) = 0: −s−1=2 s−1=2 Since L(π; Ad; 1) 2 Eπ, we have L (πF 0 ; s) 2 Eπ[q ; q ]. It follows that (r) r L (πF 0 ; 1=2) 2 Eπ · (log q) : The main result of this paper is to relate each even degree Taylor coefficient to the self-intersection numbers of a certain algebraic cycle on the moduli stack of Shtukas. We give two formulations of our main results, one using certain subquotient of the rational Chow group, and the other using `-adic cohomology. TAYLOR EXPANSION 771 1.3. The Heegner{Drinfeld cycles. From now on, we let r be an even in- r teger. In Section 5.2, we will introduce moduli stack ShtG of Drinfeld Shtukas r with r-modifications for the group G = PGL2.
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