# Relative Trace Formulas

Date & Time

Organizers:
Nicolas Templier, Cornell University
Sug Woo Shin, UC Berkeley
Werner Mueller, Universität Bonn

Past years:

The symposium focused on the current status of knowledge of relative trace formulas and brought together experts on different specialties.

Discussion topics included:

• Period formulas and special values of L-functions. (notably recent progress on Gan-Gross-Prasad, and Ichino-Ikeda conjectures, and Yun-Zhang’s higher Gross-Zagier formula over function fields)
• Relative and arithmetic fundamental lemmas and connections with geometric representation theory.
• Recent results on non-vanishing of L-values and the subconvexity problem. (notably results of Blomer, Munshi, and others)
• Automorphic spectrum: bounds towards Ramanujan, relative analogues of Sato-Tate equidistribution, and local and global harmonic analysis.
• Symposium Report

Hervé Jacquet proposed a series of questions that arise in the study of relative trace formulas. He explained various approaches toward relative fundamental lemmas, notably the use of harmonic analysis in the study of Kloosterman integrals. He also detailed the treatment of hyperbolic classes in the relative trace formula of the general linear group over quadratic extensions.

Raphael Beuzart-Plessis explained results on the Ichino–Ikeda conjecture for unitary groups. First results were obtain by W. Zhang following Jacquet’s strategy of comparison of relative trace formulas. In the new work, the spectral transfer is fully established by developing a local relative trace formula, and new period formulas are established with improved assumptions.

Wee Teck Gan spoke on how to globalize distinguished supercuspidal representations over function fields. Similar results over number fields have been known by using Poincaré series, but at the expense of losing control at a place. Over function field case it is essential to keep control at all places. Gan and Lomeli made it possible by allowing small ramification at finitely many auxiliary places. Their result leads to complete the Langlands–Shahidi method over function fields at (finitely many) missing places.

Dihua Jiang gave a talk on the non-vanishing of the central value of certain $$L$$-functions and automorphic Bessel descent. For a cuspidal automorphic representation $$\tau$$ of $$\GL_n$$ it is a classical open problem to determine whether $$L(1/2,\tau\times\chi)$$ is nonzero for some characters $$\chi$$. This problem has a lot of applications to number theory and arithmetic geometry. The talk started with the proof of such a non-vanishing result when $$\tau$$ comes from a unitary group by base change or Langlands functorial transfer under the assumption that $$n\le 4$$. If the problem is put in a bigger context based on endoscopic classification, automorphic Bessel descents, and general theory of the tensor product $$L$$-functions, he has shown that such a non-vanishing problem is a consequence of refined structures of global Arthur packets, and the Glocal Gan-Gross-Prasad conjecture. He reported about what he and Lei Zhang have done on this aspect.

Solomon Friedberg explained in his talk how to obtain endoscopic lifts of automorphic representations via the converse theorem. For classical groups, Arthur has established the existence of endoscopic liftings using the trace formula. Using the converse theorem, S. Friedberg and his co-authors can also deal with the group $$\GSpin$$ which is not covered by Arthur. For the groups $$G$$ considered in this talk, the dual group $$\widehat G$$ admits an embedding into $$\widehat\GL_N=\GL_N(\C)$$. The main result states that every irreducible unitary cuspidal automorphic representation of $$G(\A)$$ has a functorial lift (with respect to the embedding) to $$\GL_N(\A)$$.

Valentin Blomer reviewed the relationships between relative trace formulas and analytic number theory. The case of GL(2), that is the Kuznetsov trace formula, has been much studied. Results include Linnik’s problem on sums of Kloosterman sums, bounds for the non-tempered spectrum, Selberg 3/16 bound, subconvexity of L-functions, prime geodesic theorem, and bounds for the largest prime factor in the sequence $$n^2+1$$. In higher rank, the geometric side is much more complicated, and results are more recent. It includes subconvex bounds for $$GL(3)$$ $$L$$-functions in the eigenvalue and level aspects, and simultaneous non-vanishing for $$L$$-functions attached to forms on $$GSp(4)$$.

Ritabrata Munshi gave an introduction to the delta symbol method. It is a variant of Kloosterman’s refinement of the circle method, where one exploits the Petersson trace formula, with optimal choice of level and weights. He outlines the history of subconvex bounds for $$L$$-functions of degree 1,2,3, starting with Weyl’s bound for Riemann zeta function, and Burgess bound for Dirichlet $$L$$ functions, which is still a numerical record. The delta symbol method has recently produced new subconvex estimates that are strong enough to imply results on the QUE problem, and also can be used to produce new proofs of previous subconvexity estimates.

Erez Lapid discussed in his talk the derivation of an upper-bound on the support of matrix coefficients of supercuspidal representations of the general linear group $$G=\GL_N(F)$$ for a non-archimedean local field $$F$$. Let $$(\pi,V)$$ be a supercuspidal representation of $$G$$ and let $$(\pi^\vee,V^\vee)$$ be its contragredient. Let $$v\in V$$, $$v^\vee\in V^\vee$$ be fixed under the principal congruence subgroup of level $$n$$. Then an explicit bound for the support of the corresponding matrix coefficient is given as a function of only $$N,n$$. This result is also a consequence of the classification of supercuspidal representations by Bushnell–Kutzko. The proof given by Lapid is independent of the classification. It is based on two ingredients. The first ingredient, which is special to the general linear group, is basic properties of local Rankin-Selberg integrals for $$G\times G$$. The second ingredient is Howe’s result on the integrality of the formal degree with respect to a suitable Haar measure.

Yifeng Liu presented his work on arithmetic Fourier–Jacobi (FJ) periods. Given two automorphic representations $$\pi_1,\pi_2$$ of $$U(N)$$ with a certain conjugate symplectic Hecke character $$\mu$$, the global Gan–Gross–Prasad conjecture asserts that the FJ period of $$\pi_1,\pi_2$$ is nonzero if and only if $L(1/2,\pi_1\times \pi_2\times \mu)\neq 0$. Liu gave an intricate construction of the arithmetic FJ map via certain unitary Shimura varieties and formulates the conjecture that the arithmetic FJ map is nonzero for $$\pi_1,\pi_2$$ if and only if $$L'(1/2,\pi_1\times \pi_2\times \mu)\neq 0$$. When this happens, he predicts the existence of a certain algebraic cycle explaining it.

Nicolas Templier announced results toward the Ramanujan conjecture over function fields, in work with Will Sawin. The result is the temperedness at unramified places of automorphic representations, subject to a local assumption at one place, stronger than supercuspidality. Such an assumption is necessary, as was first shown by Saito-Kurokawa and Howe-Piatetskii-Shapiro in the 1970’s. The method relies on combining the $$\ell$$-adic geometry of $$\Bun_G$$ (Deligne’s purity theorem, compactification of the diagonal, Beilinson-Drinfeld Grassmannian), and trace formulas (fundamental lemma, cyclic base change, character identities). It is different from that of L.Lafforgue’s proof of the Ramanujan bound for the group GL(n) using shtukas.

Aaron Pollack discussed in his talk the Fourier expansion of modular forms on exceptional groups. For classical modular forms for the group $$\SL_2(\Z)$$, the Fourier coefficients contain a wealth of information. For Siegel modular forms there is also a satisfactory theory of Fourier coefficients. On the other hand, there is a less refined notion of Fourier coefficients for a general automorphic form on a general reductive group and it is difficult to extract arithmetic information from these coefficients. For arithmetic applications it is therefore desirable to have a refined theory of Fourier coefficients analogous to that for the holomorphic forms in the classical case. In the talk it is shown how one can develop such a theory for exceptional groups.

Birgit Speh gave a talk on work with T. Kobayashi on bilinear invariant forms on tensor products of irreducible unitary representations of rank one orthogonal groups. This is the GGP conjecture at the archimedean cases for pairs of orthogonal groups, however for non-tempered representations, where the final form of the conjecture remains uncertain. In many cases, symmetry breaking operators are constructed, and the multiplicity of Hom spaces between representations are determined entirely.

Wei Zhang formulated a general conjecture relating the relative trace formula (RTF) and the geometry of Shimura varieties and Shtukas, over $$r$$-copies of “curves” (thus $$r\le 1$$ in the number field case). Given an RTF there are geometric objects $$\mathrm{Sh}_r(H)$$ in $$\mathrm{Sh}_r(G)$$. Assuming $$\dim\mathrm{Sh}_r(H)=\frac12 \dim\mathrm{Sh}_r(G)$$, his “template conjecture” precisely describes the intersection numbers between $$\mathrm{Sh}_r(H)$$ and its Hecke translates in terms of the $$r$$-th derivative of certain $$L$$-functions. He mentioned a few examples involving general linear groups or unitary/orthogonal groups based on his work with Z. Yun and Rapoport–Smithling.

Atsushi Ichino and Kartik Prasanna reported on their joint work in two talks titled Langlands functoriality and algebraic cycles. They consider two groups $$G_1$$ and $$G_2$$ arising from two quaternion algebras over a totally real field, giving rise to two Shimura varieties $$X_1$$ and $$X_2$$. Their main theorem roughly says that if $$X_1$$ and $$X_2$$ have equal dimension $$d$$ then for each Hecke eigensystem $$\pi$$, there exists a motivated cycle giving rise to an isomorphism between the $$\pi$$-isotypic part of the $$d$$-th cohomology (either Betti or étale) of $$X_1$$ and $$X_2$$. To prove, they embed $$X_1\times X_2$$ in a Shimura variety $$X$$ associated with $$U(2,2)$$, and then find a piece of $$H^2(X)$$ corresponding to a nontempered automorphic representation with certain nonvanishing period. The nonvanishing is obtained by the theta correspondence for a suitable see-saw diagram. The speakers discussed possible generalizations to higher-rank unitary groups, where the nonvanishing of periods is a key difficulty.

Michael Harris explained his joint work with H. Grobner and J. Lin. Motivated by Deligne’s conjecture on critical values of motivic $$L$$-functions, they prove the conjecture for “Rankin-Selberg” motives coming from automorphic representations of unitary groups, conditional on the Ichino–Ikeda–Harris conjecture and a nonvanishing conjecture for central $$L$$-values. The proof blends various tools from geometry and automorphic forms; for instance period integrals are interpreted as cup products in coherent cohomology.

A. Raghuram explained generalizations of Shimura’s classical results on special values of $$L$$-functions of modular forms. The general guiding principles are Deligne’s conjecture on special $$L$$-values and the Langlands–Shahidi method. The latter has precise analogues in terms of cohomology of locally symmetric spaces, so that the cohomology can be used to establish algebraicity results for special $$L$$-values. He exhibited examples of Rankin–Selberg, orthogonal, and Asai $$L$$-functions based on joint works.

Werner Müller explained in his talk recent results concerning the study of torsion in the cohomology of arithmetic groups. This is related to $$p$$-adic questions in the Langlands program. Torsion cohomology classes which are eigenclasses of Hecke operators are expected to correspond to Galois representations over finite fields. This makes it desirable to study the size of the torsion subgroup in the cohomology of arithmetic groups. An analytic tool to study this problem is the analytic torsion, which was first used in this context by Bergeron–Venkatesh. The analytic torsion is a spectral invariant of the underlying locally symmetric space, which is defined in terms of the spectrum of the Casimir operator acting on differential forms with values in a flat vector bundle. The talk gave an overview of recent developments and results for cocompact arithmetic groups. Future research is concerned with the extension of these results to the non-cocompact case.

Sug Woo Shin started by introducing Sarnak–Xue’s conjecture on the limit multiplicity of nontempered representations for arithmetic quotients of real semisimple groups. His theorem with Marshall bounds the dimension of cohomology of locally symmetric spaces for unitary groups of signature $$(N-1,1)$$. In every degree the theorem has a power saving over Sarnak–Xue’s conjecture. The proof involves analysis of cohomological $$A$$-packets, endoscopic classification for unitary groups, Savin’s result on limit multiplicity, among others.

Jasmin Matz discussed the analytic torsion for arithmetic quotients of the symmetric space $$\SL(n,\R)/\SO(n)$$. This is related to the problem of extending the results on the growth of torsion in the cohomology of cocompact arithmetic groups, described in the talk of W. Müller, to the finite volume case. Since the corresponding locally symmetric spaces are in general non-compact, this requires a regularization of the analytic torsion. The method is based on the Arthur trace formula. An important part is the study of the unipotent orbital integrals on the geometric side of the trace formula. At the moment, this can only be carried out for $$\GL(n)$$.

Yiannis Sakellaridis explained a formalism of transfer operators for comparison of relative trace formulas. The goal is to demonstrate, by example, the existence of transfer operators, that generalize the ones from endoscopy in the case of the stabilization of Arthur trace formula. Notably this includes the definition of certain integral transforms between suitable Schwartz spaces.

Bao-Chau Ngô outlined a program toward generalization of the Godement-Jacquet integral to the general case of functoriality. Some of the ideas had been laid down in an article of Braverman–Kazhdan, which contained several questions and conjectures. Recent advances include the construction of basic functions that correspond geometrically to unramified $$L$$-factors, the systematic use of monoid, definitive results in the analogous case of finite groups of Lie type with a modification of Deligne–Lusztig geometric construction of character sheaves.

• Agenda & Slides

#### Monday

 10:00 - 10:45 AM Herve Jacquet | Questions 11:00 - 11:45 AM Raphael Beuzart-Plessis | Comparison of Local Relative Characters and the Ichino-Ikeda Conjecture for Unitary Groups (PDF) 11:55 AM - 12:40 PM Wee-Teck Gan | Globalisation of Distinguished Supercuspidal Representations (PDF) 5:00 - 5:45 PM Dihua Jiang | Automorphic Bessel Descent and L(1/2,\pi \times \chi) \neq 0 5:55 - 6:40 PM Solomon Friedberg | Lifting via the Converse Theorem: New Results (PDF)

#### Tuesday

 10:00 - 10:45 AM Valentin Blomer | RTF in Analytic Number Theory 11:00 - 11:45 AM Ritabrata Munshi | On the Delta Method 11:55 AM - 12:40 PM Erez Lapid | On the Support of Matrix Coefficients of Supercuspidal Representations of the General Linear Group 5:00 - 5:45 PM Yifeng Liu | GGP in the Anti-Hermitian Case 5:55 - 6:40 PM Nicolas Templier | On the Ramanujan Conjecture Over Function Fields (PDF)

#### Wednesday

 5:00 - 5:45 PM Aaron Pollack | Fourier Expansion of Modular Forms on Exceptional Groups (PDF) 5:55 - 6:40 PM Birgit Speh | Some Results for Rank 1 Orthogonal Groups (PDF)

#### Thursday

 10:00 - 10:45 AM Wei Zhang | Sh(G,mu) and RTF 11:00 - 11:45 AM Atsushi Ichino | Langlands Functoriality and Algebraic Cycles 11:55 AM - 12:40 PM Michael Harris | L-functions and Periods of Automorphic Motives (PDF) 5:00 - 5:45 PM Kartik Prasana | Motivated Cycles for the Jacquet-Langlands Correspondence 5:55 - 6:40 PM Anantharam Raghuram | Special Values of Automorphic L-functions (PDF)

#### Friday

 10:00 - 10:45 AM Werner Müller | On the Growth of Torsion in the Cohomology of Arithmetic Groups (PDF) 11:00 - 11:45 AM Sug-Woo Shin | Cohomology Growth of U(N-1,1) and Endoscopy 11:55 AM - 12:40 PM Jasmin Matz | Torsion Growth in the Cohomology of Arithmetic Groups 5:00 - 5:45 PM Yiannis Sakellaridis | RTF and Haenkel Transforms 5:55 - 6:40 PM Bao-Chau Ngo | Toward a Generalization of the Godement-Jacquet Integral
• Participants
 Raphael Beuzart-Plessis Marseille Valentin Blomer Georg-August-Universität Göttingen Solomon Friedberg Boston College Wee-Teck Gan National University of Singapore Michael Harris Columbia University Atsushi Ichino Kyoto University Herve Jacquet Columbia University Dihua Jiang University of Minnesota Erez Lapid Weizmann Institute Yifeng Liu Northwestern University Jasmin Matz Hebrew University Werner Müller Universität Bonn Ritabrata Munshi Indian Statistical Institute, Kolkata Ngô Bảo Châu University of Chicago Aaron Pollack Duke University Kartik Prasana University of Michigan Anantharam Raghuram Institute for Advanced Study Yiannis Sakellaridis Rutgers University Sug Woo Shin UC Berkeley Birgit Speh Cornell University Nicolas Templier Cornell University Wei Zhang Massachusetts Institute of Technology