Braverman and Kazhdan have conjectured the existence of summation formulae that are essentially equivalent to the analytic continuation and functional equation of Langlands L-functions in great generality. Motivated by their conjectures and related conjectures of L. Lafforgue, Ngo, and Sakellaridis, Baiying Liu and I have proven a summation formula analogous to the Poisson summation formula for the subscheme cut out of three quadratic spaces $(V_i,Q_i)$ of even dimension by the equation $Q_1(v_1)=Q_2(v_2)=Q_3(v_3)$. I will sketch the proof of this formula in the first portion of the talk. In the second portion, time permitting, I will discuss how these summation formulae lead to functional equations for period integrals for automorphic representations of $GL_{n_1} \times GL_{n_2} \times \GL_{n_3}$ where the $n_i$ are arbitrary, and speculate on the relationship between these period integrals and Langlands L functions.

# Joint IAS/Princeton University Number Theory Seminar

## Summation formulae and speculations on period integrals attached to triples of automorphic representations

### Featuring

Jayce Getz

### Speaker Affiliation

Duke University; Member, School of Mathematics

### Affiliation

Mathematics

### Event Series

### Video

https://video.ias.edu/puias/2018/0327-JayceGetzDate & Time

March 27, 2018 | 4:45 – 5:45pm

### Location

Simonyi Hall 101