## Weil representation and arithmetic fundamental lemma.(English)Zbl 1486.11059

The theorem of Gross and Zagier relates the Néron-Tate heights of Heegner points on modular curves to the central derivative of certain $$L$$-functions. The arithmetic Gan-Gross-Prasad conjecture is a generalization of this theorem to higher-dimensional Shimura varieties. The author has proposed an approach to establish the conjecture using an “arithmetic” relative trace formula. The paper is a major breakthrough in this approach. The paper proves the fundamental lemma for the relative trace formula, with the assumption that the local field is $${\mathbb Q}_p$$ where $$p$$ is a large enough prime. The argument should work for all local fields with large residue characteristic, in the general case there are just some claims that are expected to be true but need to be checked.
A relative trace formula is a distribution which can be decomposed into a sum of orbital integrals. The fundamental lemma are identities for the orbital integrals over a local field, when the orbital integrals are associated to a base function (identity element of Hecke algebra). Once such identities are established, it is strong evidence that the trace formula approach will work, and some partial results can follow with relatively small amount of extra work.
The reviewer does not claim to understand the details of the proof of the fundamental lemma, however the brilliance of the idea behind it is clear. Previously Jacquet introduced an idea to use Fourier transform in the proof of fundamental lemma. The current paper further developed the idea, and uses Weil representation (which encodes Fourier transform as Weyl group action) in the proof. This is combined with a local-global argument where the local orbital integral identity is derived from a global one, and an induction argument. The paper also used the same argument to reprove the fundamental lemma for Jacquet-Rallis relative trace formula that is used for Gan-Gross-Prasad conjecture for unitary groups. That fundamental lemma was originally proved using arithmetic geometric method by Z. Yun [Duke Math. J. 156, No. 2, 167–227 (2011; Zbl 1211.14039)] for positive characteristic case and extended to any local field with large residue characteristic using a standard argument in logic. In comparison, the proof presented in current paper can be considered an elementary proof.

### MSC:

 11F27 Theta series; Weil representation; theta correspondences 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 14C25 Algebraic cycles 14G35 Modular and Shimura varieties

Zbl 1211.14039
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### References:

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