Applications of automorphic distributions to analytic number theory

1. Applications of automorphic distributions to analytic number theory  Stephen D. Miller Rutgers University and Hebrew University http://www.math.rutgers.edu/~sdmiller

5. Outline of the talk • Definition of automorphic distributions and connection to representation theory • Applications to • Constructing L-functions • Summation Formulas • Cancellation in sums with additive twists • Implication to moments • Existence of infinitely many zeroes on the critical line

6. Automorphic Distributions • Suppose G = real points of a split reductive group defined over Q. • ½ G = arithmetically defined subgroup • e.g.  = SL(n,Z) ½ SL(n,R) • or  = GL(n,Z) ½ GL(n,R) (if center taken into account appropriately) • An automorphic representation is an embedding of a unitary irreducible representation j : (,V) ! L2(nG) • Under this G-invariant embedding j, the smooth vectors V1 are sent to C1(nG). • Consider the “evaluation at the identity” map • : v  j(v)(e) • which is a continuous linear functional on V1 (with its natural Frechet topology). • Upshot:  2 ((V’)-1) - a -invariant distribution vector for the dual representation. • Because (,V) and (’,V’) play symmetric roles, we may switch them and henceforth assume 2(V-1).

7. Some advantages • The study of automorphic distributions is equivalent to the study of automorphic forms. • It appears many analytic phenomena are easier to see than in classical approaches: • For example, • However, this technique is not well suited to studying forms varying over a spectrum, just an individual form. Whittaker expansion (messy) Summation Formulas Hard Automorphic form Hard Easy? Easy? L-functions Easy?

8. Embeddings • A given representation (,V) may have several different models of representations • Different models may reveal different information. • Main example: all representations of G=GL(n,R) embed into principal series representations (,,V,): • V = { f : G!Cj f(gb) = f(g) -1(b) } , [(h)f](g) = f(h-1g) • Here b 2 B = lower triangular Borel subgroup, (b) = ,(b) =  |bj|(n+1)/2 - j - j sgn(bj)j , and bj are the diagonal elements of the matrix b. • (Casselman-Wallach Theorem) Embedding extends equivariantly to distribution vectors: V-1 embeds into V,-1 = {2 C-1(G) j(gb) = (g)-1(b)} as a closed subspace.

9. Another model for Principal Series • Principal series are modeled on sections of line bundles over the flag varieties G/B. • G/B has a dense, open “big Bruhat cell” N = {unit upper triangular matrices}. • Functions in V,1 are of course determined by their restriction to this dense cell; distributions, however, are not. • However, automorphic distributions have a large invariance group, so in fact are determined by their restriction to N. • Upshot: instead of studying automorphic forms on a large dimensional space G, we may study distributions on a space N which has < half the dimension. View 2 C-1(NÅnN). • Another positive: no special functions are needed. • A negative: requires dealing with distributions instead of functions, and hence some analytic overhead.

10. The line model for GL(2,R) For simplicity, set  = (,-),  = (0,0), and  = SL(2,Z) • Here N is one dimensional, isomorphic to R. • NÅ'Z • So 2 C-1(ZnR) is a distribution on the circle, hence has a Fourier expansion (x) = n2Z cn e(nx) with e(x) = e2 i x and some coefficients cn. • The G-action in the line model is • Therefore:

11. Forming distributions from holomorphic forms In general start with a q-expansion Restrict to x-axis: Here cn = an n(k-1)/2, where k is the weight. The distribution  inherits automorphy from F : If then

12. For Maass forms • Start with classical Fourier expansion • Get boundary distribution where again cn = an n- • Note of course that when  = (1-k)/2 the two cases overlap. This corresponds to the fact that the discrete series for weight k forms embeds into V for this parameter. • Upshot: uniformly, in both cases get distributions • satisfying

13. What can you do with Boundary value distributions? • Applications include: • Constructing L-functions • Summation Formulas • Cancellation in sums with additive twists • Implication to moments • Existence of infinitely many zeroes on the critical line • All of these give new proofs for GL(2), where these problems have been well-studied. • New summation formulas, and results on analytic continuation of L-functions have been proven using this method on GL(n).

14. Analytic Continuation of L-functions • GL(2) example: one has (say, for GL(2,Z) automorphic forms) • Formally, we would like to integrate (x) against the measure |x|s-1dx. However, there are potential singularities at x = 0 and 1. A priori, distributions can only be integrated against smooth functions of compact support. • If (x) is cuspidal then c0 = 0 and the Fourier series oscillates a lot near x = 1. More importantly, (x) has bounded antiderivatives of arbitrarily high order. This allows one to make sense of the integral when Re s is large or small. • Since x = 1 and x = 0 are related by x  1/x, the same is true near zero. • Thus the Mellin transform M(s) = sR(x)|x|s-1dx is holomorphically defined as a pairing of distributions. It satisfies the identity M(s) = M(1-s+2). • One computes straightforwardly, term by term, that which is the functional equation for the standard L-function. • The “archimedean integral” here is sR e(x)|x|s-1 sgn(x) dx, and (apparently) the only one that occurs in general.

15. A picture of Maass form antiderivative For the first Maass form for GL(2,Z) We of course cannot plot the distribution. Oscillation near zero

16. Zoom near origin Oscillation near zero

17. Weight one antiderivative

18. L-functions on other groups • Given a collection of automorphic distributions and an ambient group which acts with an open orbit on the product of their (generalized) flag varieties, one can also define a holomorphic pairing. • This condition is related to the uniqueness principal in Reznikov’s talk earlier today. • Main difference: we insert distribution vectors into the multilinear functionals (and justify). • These pairings can be used to obtain the analytic continuation of L-functions which have not been obtained by the Langlands-Shahidi or Rankin-Selberg methods. • Main example: Theorem (Miller-Schmid, 2005). Let F be a cusp form on GL(n) over Q, and S any finite set of places containing the ramified nonarchimedean places. Then Langlands partial L-function LS(s,Ext2F) is fully holomorphic, i.e. holomorphic on all of C, except perhaps for simple poles at s = 0 or 1 which occur for well-understood reasons. • In particular, if F is a cuspidal Hecke eigenform on GL(n,Z)n GL(n,R), the completed global L-function (s,Ext2 F) is fully holomorphic. • The main new contribution is the archimedean theory, which seems difficult to obtain using the Rankin-Selberg method. Similarly, the Langlands-Shahidi method gives the correct functional equation, but has difficulty eliminating the possibility of poles. • Pairings (formally, at least) also can be set up for nonarchimedean places also. Thus, this method represents a new, third method for obtaining the analytic properties of L-functions. It requires other models of unitary irreducible representations, such as the Kirillov model. • Two main reasons this works: • Ability to apply pairing theorem (which holds in great generality) • Ability to compute the pairings (so far in all cases reduces to one-dimensional integrals, but the reason for this is not understood).

19. Outline of the talk • Definition of automorphic distributions and connection to representation theory • Applications to • Constructing L-functions • Summation Formulas • Cancellation in sums with additive twists • Implication to moments • Existence of infinitely many zeroes on the critical line

20. Summation Formulas • Recall the Voronoi summation formula for GL(2): if • f(x) is a Schwartz function which vanishes to infinite order at the origin • an are the coefficients of a modular or Maass form for SL(2,Z) • a, c relatively prime integers, then where • This formula has many analytic uses for dualizing sums of coefficients (e.g. subconvexity, together with trace formulas). • It can be derived from the standard L-function (if a=0), and from its twists (general a,c). The usual proofs involve special functions, but the final answer does not. Is that avoidable?

21. The distributional vantage point • The Voronoi summation formula is simply the statement that the distribution (x) is automorphic…integrated against test functions. • Namely, • Integrate against g^(x), and get • This is equivalent to the Voronoi formula. • To justify the proof, use the oscillation of (x) near rationals (as in the analytic continuation of L(s,)).

22. Generalizations • One can make a slicker proof using the Kirillov model, in which (x) = n0 ann(x). • In this model (x) has group translates • When a,c(x) is integrated against a test function f(x), one gets exactly the LHS of the Voronoi formula. • The righthand side is (almost tautologically) equivalent to the automorphy of (x) under SL(2,Z) under the G-action in the Kirillov model. • However, the analytic justification of this argument – and especially its generalizations – gets somewhat technical.

23. A Voronoi-style formula for GL(3) • Theorem (Miller-Schmid, 2002) Under the same hypothesis, but instead with am,n the Fourier coefficients of a cusp form on GL(3,Z)nGL(3,R) for any q > 0 and • The proof uses automorphic distributions on N(Z)n N(R), where N is the 3-dimensional Heisenberg group. • The summation formula reflects identities which are satisfied by the various Fourier components. • The theorem can be applied to GL(2) via the symmetric square lift GL(2)! GL(3), giving nonlinear summation formulas (i.e. involving an2). This formula is used by Sarnak-Watson in their sharp bounds for L4-norms of eigenfunctions on SL(2,Z)nH.

24. Outline of the talk • Definition of automorphic distributions and connection to representation theory • Applications to • Constructing L-functions • Summation Formulas • Cancellation in sums with additive twists • Implication to moments • Existence of infinitely many zeroes on the critical line

25. Cancellation in sums with additive twists • Let an be the coefficients of a cusp form L-function on GL(d): S(T,x) = n6T an e(n x) , e(t) := e 2  i t • Since the an have unit size on average, we have the following two trivial bounds: • S(T,x) = O(T) • sR/Z |S(T,x)|2 dx = n6T |an|2 ~ cT • Folklore Cancellation Conjecture: S(T,x) = O(T1/2+), where the implied constant depends  but is uniform in x and T. • In light of the L2-norm statement, this is the best possible exponent.

26. Rationals vs. Irrationals • Fix x 2Q. S(T,x) can be smaller = Ox(T1/2-) (Landau). • For example, the sum S(T,0) = n6T an is typically quite small, because for example: • L(s) = n>1 an n-s is entire • Smoothed sums behave even better: decays rapidly in T (faster than any polynomial), for  say a Schwartz function on (0,1). [shift contour  to -1] • Similar behavior at other rationals (related to L-functions twisted by Dirichlet characters). • However, uniform bounds over rationals x are still not easy.

27. Brief history of results for irrationals • First considered by Hardy and Littlewood for classical arithmetic functions which are connected to degree 2 L-functions of automorphic forms on GL(2). • Typically for noncusp forms. • E.g., for an = r2(n) from before or d(n) = divisor function. • Later results by Walfisz, Erdos, etc. are sharp, but mainly apply to Eisenstein series. • No clean, uniform statement is possible in the Eisenstein case because of large main terms, which, however, are totally understood.

28. Bounds on S(T,x) for general cusp forms (on GL(d)) • Recall that we expect S(T,x) = n6T an e(nx) to be O(T1/2+) when an are the coefficients of an entire L-function. • according to the Langlands/Selberg/Piatetski-Shapiro philosophy, these are always L-functions of cusp forms on GL(2,AQ). • Main known result: S(T,x) = O(T1/2+).for cusp forms on GL(2) (degree 2 L-functions) • For holomorphic cusp forms, this is classical and straightforward to prove • But for Maass forms this is much more subtle. • Importance: used in Hardy-Littlewood’s seminal method to prove (s) has infinitely many zeroes on its critical line (we will see this again later).

29. Higher Rank? • Only general result is the trivial bound S(T,x) = O(T). • Theorem (Miller, 2004) For cusp forms on GL(3,Z)nGL(3,R) and an equal to the standard L-function coefficients, S(T,x) = O(T3/4+). • This is halfway between the trivial O(T) and optimal O(T1/2+) bounds. • We will see that the full conjecture implies the correct order of magnitude for the second moment of L(s)=n¸ 1an n-s, which beyond GL(2) is thought to be a problem as difficult as the Lindelof conjecture.

30. Outline of the talk • Definition of automorphic distributions and connection to representation theory • Applications to • Constructing L-functions • Summation Formulas • Cancellation in sums with additive twists • Implication to moments • Existence of infinitely many zeroes on the critical line

31. Distributions and integrals of L-functions on critical line • Recall the Mellin transform of the distribution t(x) = n 0 an|n|-e(nx) is • Let  be an even, smooth function of compact support on R*. By Parseval for any  (integrand is entire, so the contour may be shifted). • If (x) is an approximate identity (near x = 1), M(1/2+it) approximates the (normalized) characteristic function of the interval t 2 [-1/,1/]. • One can therefore learn the size of smoothed integrals of Mt(1/2+it) through properties of the distribution t(x) near x = 1. • When t vanishes to infinite order near x = 1, these smoothed integrals are very small. • This is related to cancellation in S(T,x) for particular values of x (in this case rational, but in general irrational). • Similarly, the multiplicative convolution tF has Mellin transform Mt(s)*M(s). Its L2-norm approximates the second moment of L(1/2+it), and is determined by the L2-norm of tF. The latter is controlled by the size of smooth variants of S(T,x) = n·T an e(nx). • Conclusion: cancellation in additive sums is related to moments.

32. Lindelöf conjecture and moment estimates • Lindelöf conjecture: L(1/2+it) = O((1+|t|)) for any  > 0. • Fundamental unsolved conjecture in analytic number theory. • Implied by GRH. • Equivalent to moment bounds: s-TT |L(½+it)|2k dt = O(T1+) for each fixed k ¸ 1. • The 2k-th moment for a cusp form on GL(d) is thought to be exactly as difficult to the 2nd moment on GL(dk). • The cancellation conjecture – or more precisely a variant for non-cusp forms – implies the Lindelöf conjecture (next slide), and is thus a very hard problem for d > 2.

33. Bounds on S(T,x) imply bounds on moments • Folklore theorem (known as early as the 60’s by Chandrasekharan, Narasimhan, Selberg): • If S(T,x) = O(T+) for some ½ · < 1, then s-TT |L(½ + it)|2 dt = O(T1 +  + (2-1) d), • Where d = the degree of the L-function • E.g. L-function comes from GL(d,AQ). • Thus  = 1/2 is very hard to achieve because it gives the optimal bound O(T1+) . • GL(3) result of O(T3/4+) unfortunately does not give new moment information. • Voronoi-style summation formulas with Schmid give an implication between: • squareroot cancellation in sums of d-1-hyperkloosterman sums weighted by an, and • Optimal cancellation S(T,x) = O(T1/2+) – and therefore Lindelöf also.

34. Outline of the talk • Definition of automorphic distributions and connection to representation theory • Applications to • Constructing L-functions • Summation Formulas • Cancellation in sums with additive twists • Implication to moments • Existence of infinitely many zeroes on the critical line

35. Connection to zeroes on the critical line • Suppose (for fictitious expositional simplicity)  = 0 for a cusp form on SL(2,Z). It is not difficult to handle arbitrary . • Let H(t) = M(1/2+it). Then H(t) = H(-t) is real. • Let 1/T be an approximate identity such that M(1/2+it) ¸ 0. • If L(s) has only a finite number of zeroes on the critical line, then the following integral must also be of order T: • But it cannot if (x) vanishes to infinite order at x=1 ( is concentrated near a point where  behaves as if it is zero). • In that case this integral decays as O(T-N) for any N > 0! • The above was for a cusp form on SL(2,Z). For congruence groups, the point x=1 changes to pq, q = level. The bound S(T,x) = O(T1/2+) shows that the last integral is still o(T) with room to spare. • New phenomena: numerically that integral decays only like T1/2 for q=11.

36. Higher rank? • Like the moment problem, nothing is known about infinitude of zeroes on the critical line for degree d > 2 L-functions. • In fact, aside from zeroes at s = 1/2 coming from algebraic geometry, it is not known there are any zeroes on the critical line for d > 2. • Possible approach: if a certain Fourier component of the automorphic distribution of a cusp form  on GL(4,Z)nGL(4,R) vanishes to infinite order at 1, then L(1/2+it,) = 0 for infinitely many t 2R.