#### Transcript Cancellation in sums of L

```Applications of automorphic
distributions to analytic
number theory
Stephen D. Miller
Rutgers University and Hebrew University
http://www.math.rutgers.edu/~sdmiller
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

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).
• 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,
Whittaker expansion
(messy)
Summation
Formulas
Automorphic form
L-functions
• However, this technique is not well suited to studying
forms varying over a spectrum, just an individual form.
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! 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|(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.
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
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:
Forming distributions from holomorphic forms
Restrict to x-axis:
Here cn = an n(k-1)/2, where k is the weight.
The distribution  inherits automorphy from F :
If
then
For Maass forms
•
•
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
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).
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.
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
Zoom near origin
Oscillation
near zero
Weight one antiderivative
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 RankinSelberg 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).
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

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?
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,)).
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.
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 SarnakWatson in their sharp bounds for L4-norms of eigenfunctions
on SL(2,Z)nH.
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

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.
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) = 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.
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.
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).
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.
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

Distributions and integrals of L-functions on
critical line
• Recall the Mellin transform of the distribution (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 M(1/2+it)
through properties of the distribution (x) near x = 1.
– When  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 F has Mellin transform
M(s)*M(s). Its L2-norm approximates the second moment of L(1/2+it),
and is determined by the L2-norm of F. 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.
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.
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.
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

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.
•
•
•
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 2 R.
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