Applications of automorphic distributions to analytic number theory

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

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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)?.

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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?
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Embeddings

• A given representation (?,V) may have several
  different models of representations
• Different models may reveal different
  information.
• Main example all representations of GGL(n,R)
  embed into principal series representations
  (??,?,V?,?)
• V f G! C j f(gb) f(g) ?-1(b) ,
  ?(h)f(g) f(h-1g)
• Here b 2 B lower triangular Borel subgroup,
• ?(b) ??,?(b) ? bj(n1)/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.

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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 lt 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.

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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) ?n2 Z cn e(nx)
• with e(x) e2? i x and some coefficients cn.
• The G-action in the line model is
• Therefore

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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
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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

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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).

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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 xs-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)xs-1dx
  is holomorphically defined as a pairing of
  distributions. It satisfies the identity M?(s)
  M?(1-s2?).
• 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)xs-1
  sgn(x)? dx, and (apparently) the only one that
  occurs in general.

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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
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Zoom near origin
Oscillation near zero
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Weight one antiderivative
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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 Reznikovs 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).

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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

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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 a0), and from its twists (general a,c). The
  usual proofs involve special functions, but the
  final answer does not. Is that avoidable?

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The distributional vantage point

• The Voronoi summation formula is simply the
  statement that the distribution ?(x) is
  automorphicintegrated 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,?)).

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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.

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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 gt 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.

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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

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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 an2 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.

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Rationals vs. Irrationals

• Fix x 2 Q. 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) ?ngt1 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.

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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.

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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-Shapi
  ro 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-Littlewoods seminal
  method to prove ?(s) has infinitely many zeroes
  on its critical line (we will see this again
  later).

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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.

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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

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Distributions and integrals of L-functions on
critical line

• Recall the Mellin transform of the distribution
  t(x) ?n? 0 ann-?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?(1/2it) approximates the (normalized)
  characteristic function of the interval t 2
  -1/?,1/?.
• One can therefore learn the size of smoothed
  integrals of Mt(1/2it) 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(1/2it), and
  is determined by the L2-norm of tF?. The latter
  is controlled by the size of smooth variants of
  S(T,x) ?nT an e(nx).
• Conclusion cancellation in additive sums is
  related to moments.

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Lindelöf conjecture and moment estimates

• Lindelöf conjecture L(1/2it) O?((1t)?) for
  any ? gt 0.
• Fundamental unsolved conjecture in analytic
  number theory.
• Implied by GRH.
• Equivalent to moment bounds s-TT L(½it)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 gt 2.

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Bounds on S(T,x) imply bounds on moments

• Folklore theorem (known as early as the 60s by
  Chandrasekharan, Narasimhan, Selberg)
• If S(T,x) O?(T??) for some ½ ? lt 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.

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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

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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?(1/2it). Then H(t) H(-t) is
  real.
• Let ?1/T be an approximate identity such that
  M?(1/2it) 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 x1 (? 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 gt 0!
• The above was for a cusp form on SL(2,Z). For
  congruence groups, the point x1 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 q11.

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Higher rank?

• Like the moment problem, nothing is known about
  infinitude of zeroes on the critical line for
  degree d gt 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 gt 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(1/2it,?) 0 for infinitely many t 2
  R.