Analytic theory of GL(3) automorphic forms and applications

Transcription

1 Analytic theory of GL(3) automorphic forms and applications The American Institute of Mathematics The following compilation of participant contributions is only intended as a lead-in to the AIM workshop Analytic theory of GL(3) automorphic forms and applications. This material is not for public distribution. Corrections and new material are welcomed and can be sent to workshops@aimath.org Version: Mon Nov 10 14:22:

2 2 Table of Contents A. Participant Contributions Blomer, Valentin 2. Brumley, Farrell 3. Bump, Dan 4. DeCelles, Amy 5. Ernvall-Hytonen, Anne-Maria 6. Garrett, Paul 7. Harcos, Gergely 8. Khan, Rizwanur 9. Knightly, Andrew 10. Kontorovich, Alex 11. Lau, Yuk-Kam 12. Li, Xiaoqing 13. Luo, Wenzhi 14. Miller, Stephen 15. Rhoades, Robert 16. Royer, Emmanuel 17. Rudnick, Zeev 18. Shahidi, Freydoon 19. Templier, Nicolas 20. Trotabas, Denis 21. Ye, Yangbo 22. Young, Matthew

3 3 A.1 Blomer, Valentin Chapter A: Participant Contributions I have worked on the theory of automorphic forms, their Fourier coefficients, and L- functions, mostly on GL(2). I am very interested to see which classical and not so classical tools can be developed and are appropriate to solve the corresponding problems (subconvexity, non-vanishing, shifted convolution problem, arithmetic of Fourier coefficients etc.) for higher rank. It may be noted that even in Eisenstein case ζ 3, these problems often either trivial or unsolved. With G. Harcos and Ph. Michel I proved a subconvexity estimate for the Dedekind zeta function of a cubic field (a decomposable GL(3) L-function) in terms of the discriminant. Probably the most accessible genuine GL(3) candidate to start with is the symmetric square lift of a GL(2) cusp form. Recently I proved that a positive proportion of L(1/2, sym 2 f), f S k (N, chi) for N a large prime and chi primitive quadratic does not vanish. This seems to be first non-vanishing result in the level aspect for a GL(3) L-function. I would hope that the workshop will help to enriching our toolbox to understand GL(3) automorphic forms better. A.2 Brumley, Farrell Here are some suggestions for open problems: 1) Can one prove a version of the converse theorem on GL(3) which uses GL(1) twists unramified outside a finite number of places? (A folklore conjecture is that no twists are required to deduce modularity. According to this conjecture, such a strengthening is true; it s just a matter of what we can prove.) If I am not mistaken, what we know on GL(3) is that one can deduce modularity by GL(2) twists unramified outside a finite set of places, or by GL(1) twists for which we may impose the much weaker condition that they be unramified at a fixed finite set of places. I don t know of any applications to such a strengthening on GL(3) (a similar result has been known for a long time for GL(2)). Perhaps no applications exist, and the problem has only aesthetic interest. 2) This is a question of Sarnak : determine the asymptotic behavior of the number of cusp forms on GL(n) over a fixed number field of analytic conductor less than X. Any proof giving a sharp bound would illuminate the importance of ordering by analytic conductor (the definition of which was given by Iwaniec-Sarnak). For example, when the conductor of the central character is close to that of the cusp form pi the size of the analytic conductor drops. The potential subtleties in this range make the problem interesting. One can make a first start on GL(3) by trying to obtain uniformity in the conductor in the error term in Weyl s law. A.3 Bump, Dan I am interested in spherical Whittaker functions on GL(3). These are analogs of the K- Bessel function. They naturally arise in considering automorphic forms. They are solutions to an overdetermined system of differential equations, which have six solutions; the most important one is of rapid decay as the parameters y1 and y2 in the matrix diag(y1y2, y2, 1) go to infinity. But there are six solutions (analogs of the I-Bessel function) that are well behaved near y1 = y2 = 0, and for these I found series expansions. With Jonathan Huntley I found

4 4 asymptotic expansions for the Whittaker functions in general. A particularly interesting phenomenon is that when the parameters are specialized to the parameters of the cubic theta function the asymptotics simplify. This might be an analog of the Airy Bessel function, or of the fact that K 1/2 becomes an exponential. There are several interesting questions that one can ask. To mention one, there is a result of Polya which says that if y is real then K ν (y) = 0 implies that ν is purely imaginary. This is a sort of Riemann hypothesis. There is a corresponding statement that one can make for Whittaker functions (with little evidence but some a related p-adic assertion is true). A.4 DeCelles, Amy I am a graduate student studying under Paul Garrett at the University of Minnesota. I have been studying the spectral thoery of GL 3 automorphic forms and its applications to number theory. In particular, the method for breaking convexity employed by Anton Good s work in the early 1980 s and resurrected by Diaconu and Goldfeld in , has been extended recently by Diaconu-Garrett in such a way as to suggest a framework for obtaining spectral identities on GL n, n > 2, which yield a promising way of proving asymptotics for higher moments and perhaps breaking convexity for L-functions and periods. I would like to gain a better understanding of automorphic Schwartz spaces and the corresponding spaces of tempered distributions, in order to justify regularization arguments that suggest that certain results that hold for cusp forms have similar applications to Eisenstein series. Obtaining such results for Eisenstein series would be productive because the associated L-functions are attached to smaller groups, thus allowing a simplifying reduction. A.5 Ernvall-Hytonen, Anne-Maria I m especially interested in a truncated Voronoi-type formula for GL(3). With this I mean the kind of formula containing an integral with Bessel function (or something else equally oscillating). Compare to Jutila s formula for GL(2): a(n)n (κ 1)/2 e k (nh)f(n) = 2πk 1 ( 1) κ/2 ( b ( ) 4π nx a(n)e k n h) x (κ 1)/2 J κ 1 f(x)dx. k a n b This kind of formula often enables one to derive good upper bound estimates for sums, sometimes even sharp approximations. Also, one might hope for an approximate functional equation. A.6 Garrett, Paul My particular interests in this workshop are in understanding the applications to number theory, especially L-functions, of spectral methods and harmonic analysis on GL(3) and on larger reductive groups generally. For example, there are non-trivial spectral identities involving second integral moments of L-functions L(s, f F ) with f on GL(n) and F on GL(n 1) which may touch subconvexity in the t-aspect. The GL(3) case of this is already complicated, and the path from the identity to number-theoretic consequences is not obvious. Such spectral identities suggest various reasonable notions of natural families over which to average. Viewing these spectral identities as deformations of simpler identities, p-adic versions are also possible, suggesting results toward subconvexity in other aspects. n=1 a

5 I would like to understand how to leverage spectral ideas with more traditional mollification/amplification methods. A.7 Harcos, Gergely In collaboration with Valentin Blomer (and with an original kick from Akshay Venkatesh at a previous AIM workshop!) I developed a spectral method to study shifted convolution sums in the Fourier coefficients of classical cusp forms. The method has applications to subconvexity, and it has the advantage of generalizing to more complicated additive convolution sums and higher rank groups. On the technical side, however, things become complicated because one needs to know spectral theory quite explicitly over the underlying group. In particular, uniform upper bounds for the number of cusp forms w.r.t. congruence subgroups are needed, and such bounds are hard (impossible) to find in the literature. Another technical input is bounds for Sobolev norms for automorphic vectors, again with uniformity w.r.t. the various parameters of the representation. It would be nice to see developments along the above lines. A.8 Khan, Rizwanur I am interested in the analytic aspects of L-functions. I would like to see some of the results and methods of the theory of GL(1) and GL(2) L-functions carry over to higher rank. However the analysis and combinatorics seem to be more complicated in this case. I hope matters can get clarified in the upcoming workshop. A.9 Knightly, Andrew I am interested in the relative trace formula and applications to analytic number theory. Kuznetsov s formula is one example, and it would be fun to write down a representationtheoretic version for GL(3). Other new explicit relative trace formulas have appeared recently for GL(2). For example, by integrating a kernel function against a character over M N where M is the diagonal subgroup and N is unipotent, one gets a formula for averages of modular L-values in the critical strip. The resulting asymptotics give non-vanishing results and Lindelof-on-average. The method should ceratinly generalize to GL(3) and beyond, though explicit calculation quickly becomes too complicated. Other interesting possibilities include using the relative trace formula to study symmetric square L-functions or Rankin- Selberg L-funcions. A.10 Kontorovich, Alex I m interested in developing various aspects of GL(3) automorphic forms, such as Poincare series, Kloosterman sums, and a workable Kuznetsov formula. We ve already seen the success of the Voronoi formula on GL(3), together with Kuznetsov on GL(2), in Xiaoqing Li s work on subconvexity for self-dual GL(3) forms and GL(3)xGL(2); hopefully other applications can be obtained. Also in recent joint work with Jeff Hoffstein, we encountered an application for subconvexity of a double Dirichlet series formed out of quadratic twists of GL(3) forms. Even Lindelof-on-average is not known in this case. 5

6 6 A.11 Lau, Yuk-Kam Let F be a self-dual Hecke Maass form for GL(3), which is a lift of f on GL(2). The Fourier coefficients of F can be expressed in terms of the coefficients of f. Unravel the relation, if any, between the voronoi formulas for GL(3) and for GL(2). A.12 Li, Xiaoqing I am interests in subconvexity bounds of GL(3) L-functions in various aspects and their applications, especially the quantum unique ergodicity conjecture. I am also interested in the remainder term of Weyl s law on GL(n). Are there any lower order terms? What should be the best bound for the remainder? Are there any conjectures on it? A.13 Luo, Wenzhi I hope the workshop will compare the different existing techniques, identify the true obstacles, and explore new ideas or directions to the subconvexity problem. A.14 Miller, Stephen I am interested in analytic questions about automorphic cusp forms, in particular on the quotient SL(3, Z)\SL(3, R) which this workshop focuses on. Here are some topics in particular: * Weyl Law, and remainder terms for spectral asymptotics * Analytic continuation of Langlands L-functions * Voronoi-style summation formulas, and applications to subconvexity * Existence of zeroes on the critical line for GL(3) L-functions * Estimates for periods, such as uniform bounds in x for a(1, n)e(nx), over n < T. A.15 Rhoades, Robert I am interested in exploring new techniques geared toward proving non-vanishing of L-values. In particular, I am interested in exploring the possibility of a geometric version of the mollifier method. More generally, I am interested in geometric approaches for computing moments of L-values. A.16 Royer, Emmanuel I used to be interested in the values of the symmetric square L-functions at the edge of the critical strip both in anaytical (also in collaboration with Jie WU and Yuk-kam LAU) and in combinatorial aspect (in collaboration with Laurent HABSIEGER). In particular, I want to explore the possible relations between values at the edge of the critical strip and values in the center of eventually different L-functions. In collaboration with Guillaume RICOTTA, I also studied the one-level density of symmetric power L-functions. A.17 Rudnick, Zeev My expectations from the workshop are 1) It will stimulate people to think about GL(3) in greater detail.

7 2) There are several open problems in the analytic theory of GL(2)-automorphic forms which should benefit from a greater understanding of corresponding issues on GL(3). 3) There are some striking recent results in the analytic theory of automorphic forms, such as the work by Xiaqing Li and by Roman Holowinsky and Soundararajan which need to be digested by the experts and the workshop will be a good opportunity to do so. 4) Raise some new problems on GL(n), beyond the ones currently on people s to-do list. For instance there are some interesting questions in the geometry of numbers which could benefit from progress on GL(n) (I have in mind a recent paper of myself with Risager). A.18 Shahidi, Freydoon Understanding the cuspidal spectrum of GL(3), and for that matter GL(n), is of crucial importance. When an automorphic representaion is self-dual or self-dual up to a twist, the theory of endoscopy allows us to get them through Langlands functoriality from either quasisplit classical or general spin groups. In the case of GL(3), the transfer is that of Gelbart- Jacquet which covers basically most of our understanding of the spectrum for GL(3). The representations in the image of these transfers for GL(n) are far from the whole spectrum. For example, the image of the transfer from GL(2)xGL(3) to GL(6) has little intersection with the image of those transfers mentioned above. Whether analytic techniques to be studied in the workshop benefits this understanding or benefits from it remains to be seen and an appropriate topics to discuss in this workshop. A.19 Templier, Nicolas 1. An open problem I am currently working on, and which is related to the workshop focus is the following one. Let a be a non-cube integer. Consider R X := { ν q, ν3 a (mod q), 1 q X} (1) inside R/Z. One expects equidistribution of R X as X +. Hooley (early seventies) established the equidistribution of R X under a condition assumption (he labelled (R*)). The assumption (R*) is a sharp bound for incomplete Kloosterman sums on a short interval which goes beyond Weil s bound. Later works, notably the one by Duke-Friedlander-Iwaniec on bilinear Kloosterman fractions tend to show that assumption (R*) lies beyond current techniques. Heath-Brown (2002) established the equidistribution unconditionnaly when the modulus q is restricted to be highly composite in the definition of R X. I do believe that the equidistribution problem (but not assumption (R*)) has a solution in the framework of GL(3)-automorphic forms, especially making use of Poincaré series. The first step is to translate the work of Hooley and Heath-Brown from its classical setting to the automorphic setting. This is under progress. 2. On GL 3 -automorphic forms and their typical problems, there are many questions I would like to understand more during this workshop. Which criterions are equivalent to GL 3 - subconvexity? What are the arithmetics applications of GL 3 -non-vanishing? Is there any connection with Landau-Siegel zeros? Is functoriality a cornerstone toward the resolution of problems on GL 3 or is it of independant interest? For instance does it plays a role in the combinatorics of the Hecke algebra? In which extent some arguments are specific to GL 2 : because of special tricks or because some dimensions match? 7

8 8 A.20 Trotabas, Denis I studied non-vanishing questions about L-functions of Hilbert modular forms following Selberg s methods of mollified moments. The main tool to do this is an extension of Petersson formula. I would like to study this problem for GL(3) L-functions. The main questions I have are: is there a form of Petersson/Kuznetsov formula useful to attack these problems? Is there another harmonic/ergodic interpretation of mollification (ie a more geometric one, allowing one to study higher rank cases, where harmonic techniques alone are presently unefficient)? A.21 Ye, Yangbo I am interested in spectral methods for GL(3), including classical and adelic relative trace formulas and Voronoi formulas. Applications in mind are subconvexity bounds for automorphic L-functions and problems on primes on orbits. A.22 Young, Matthew I am generally interested in families of L-functions, and in particular the structure of moments of special values. One reason I am interested in GL(3) is that, at the least, it seems necessary to understand GL(3) (and higher) in order to solve specific problems on GL(1) and GL(2) (e.g. moments of the Riemann zeta function). At this workshop I hope to better understand the power and limitations of the current technology available for GL(3) (especially the summation formulas).