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Shtukas and the Taylor expansion of $$L$$-functions. (English) Zbl 1385.11032
Let $$X'\rightarrow X$$ be an étale cover of degree 2 where $$X$$ and $$X'$$ are geometrically connected smooth complete curves over a finite field of characteristic $$> 2$$ and $$F\subset F'$$ function fields. Let $$G = \text{PGL}_2$$. Consider an everywhere unramified cuspidal automorphic representation $$\pi$$ of $$G(F_{\mathbb A})$$ and its base change $$\pi_{F'}$$ to $$F'$$. The main result of the article is a geometric expression for the central derivatives of the $$L$$-function of $$\pi_{F'}$$. The result is obtained by the comparison of two relative trace formulas. One of these is an adaptation of Jacquet’s relative trace formula, the other one is of geometric nature.
The authors define for each even number $$r$$ a cycle (called Heegner-Drinfeld cycle) on a moduli stack of shtukas for $$G$$ with $$r$$-modifications. The intersection number of this cycle with its transform by a Hecke function is defined. It is studied in two ways. Using intersection theory it is written as a trace. The decomposition of the cohomology of the stack under the Hecke action gives a decomposition of the Heegner-Drinfeld cycle and the spectral decomposition of the intersection number.
In order to compare the two relative trace formulas, the orbital integrals of the analytic formula are interpreted as traces of Frobenius. The orbital sides being identified (up to a factor $${(\text{log} q)}^r$$), the spectral sides are also equal. That gives the expression of the $$r$$-th derivative of the $$L$$-function as a self-intersection number of a component of the Heegner-Drinfeld cycle.

##### MSC:
 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols 14G35 Modular and Shimura varieties 11F70 Representation-theoretic methods; automorphic representations over local and global fields 14H60 Vector bundles on curves and their moduli
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