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 Definition of automorphic distributions and connection to representation
theory
 Applications to
 Constructing Lfunctions
 Summation Formulas
 Cancellation in sums with additive twists
 Implication to moments
 Existence of infinitely many zeroes on the critical line

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 Suppose G = real points of a split reductive group defined over Q.
 G ½ G = arithmetically defined subgroup
 e.g. G = SL(n,Z) ½ SL(n,R)
 or G = 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 : (p,V) ! L^{2}(GnG)
 Under this Ginvariant embedding j, the smooth vectors V^{1} are
sent to C^{1}(GnG).
 Consider the “evaluation at the identity” map
 t: v a
j(v)(e)
 which is a continuous linear functional on V^{1} (with its
natural Frechet topology).
 Upshot: t 2 ((V’)^{}^{1})^{G }  a Ginvariant
distribution vector for the dual representation.
 Because (p,V) and (p’,V’) play symmetric roles, we may switch them
and henceforth assume t
2 (V^{}^{1})^{G}.

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 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:
 However, this technique is not well suited to studying forms varying
over a spectrum, just an individual form.

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 A given representation (p,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 (p_{l}_{,}_{d},V_{l}_{,}_{d}):
 V = { f : G! C j f(gb) = f(g) c^{1}(b) } ,
[p(h)f](g) = f(h^{1}g)
 Here b 2 B = lower triangular Borel subgroup,
 c(b) = c_{l}_{,}_{d}(b) = Õ b_{j}^{(n+1)/2
 j  }^{l}^{j} sgn(b_{j})^{d}^{j },

and b_{j}
are the diagonal elements of the matrix b.
 (CasselmanWallach Theorem) Embedding extends equivariantly to
distribution vectors:
V^{}^{1}
embeds into V_{l}_{,}_{d}^{}^{1}
= {s 2 C^{}^{1}(G)
j s(gb) = s(g)c^{1}(b)}
as a closed subspace.

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 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_{l}_{,}_{d}^{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 t 2 C^{}^{1}(NÅ GnN).
 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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 Here N is one dimensional, isomorphic to R.
 NÅ G ' Z
 So t 2 C^{}^{1}(ZnR)
is a distribution on the circle, hence has a Fourier expansion
 t(x) = å_{n}_{2}_{ }_{Z} c_{n}
e(nx)
 with e(x) = e^{2}^{p}^{ i x }and some
coefficients c_{n}.
 The Gaction in the line model is
 Therefore:

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 In general start with a qexpansion

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 Start with classical Fourier expansion
 Get boundary distribution

where again c_{n} = a_{n} n^{}^{n}
 Note of course that when n = (1k)/2 the two cases overlap. This corresponds to the fact that the
discrete series for weight k forms embeds into V_{n} for this parameter.
 Upshot: uniformly, in both cases get distributions
 satisfying

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 Applications include:
 Constructing Lfunctions
 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
wellstudied.
 New summation formulas, and results on analytic continuation of
Lfunctions have been proven using this method on GL(n).

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 GL(2) example: one has (say, for GL(2,Z) automorphic forms)
 Formally, we would like to integrate t(x) against the measure x^{s1}dx. However, there are potential
singularities at x = 0 and 1. A
priori, distributions can only be integrated against smooth functions of
compact support.
 If t(x) is cuspidal then c_{0
}= 0 and the Fourier series oscillates a lot near x = 1. More importantly, t(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 a 1/x, the same is true near zero.
 Thus the Mellin transform Mt(s)
= s_{R} t(x)x^{s1}dx
is holomorphically defined as a pairing of distributions. It satisfies the identity Mt(s) = Mt(1s+2n).
 One computes straightforwardly, term by term, that
 which is the functional
equation for the standard Lfunction.
 The “archimedean integral” here is s_{R} e(x)x^{s1}
sgn(x)^{d} dx, and
(apparently) the only one that occurs in general.

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 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
Lfunctions which have not been obtained by the LanglandsShahidi or
RankinSelberg methods.
 Main example:
 Theorem (MillerSchmid, 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
Lfunction L_{S}(s,Ext^{2}F) is fully holomorphic, i.e.
holomorphic on all of C, except perhaps for simple poles at s = 0 or 1
which occur for wellunderstood reasons.
 In particular, if F is a cuspidal Hecke eigenform on GL(n,Z)n GL(n,R),
the completed global Lfunction L(s,Ext^{2} F) is fully holomorphic.
 The main new contribution is the archimedean theory, which seems
difficult to obtain using the RankinSelberg method. Similarly, the LanglandsShahidi
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
Lfunctions. 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
onedimensional integrals, but the reason for this is not understood).

19

 Definition of automorphic distributions and connection to representation
theory
 Applications to
 Constructing Lfunctions
 Summation Formulas
 Cancellation in sums with additive twists
 Implication to moments
 Existence of infinitely many zeroes on the critical line

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 Recall the Voronoi summation formula for GL(2): if
 f(x) is a Schwartz function which vanishes to infinite order at the
origin
 a_{n} 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 Lfunction (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?

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 The Voronoi summation formula is simply the statement that the
distribution t(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 t(x) near rationals (as in the analytic continuation of L(s,t)).

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 One can make a slicker proof using the Kirillov model, in which t(x) = å_{n}_{¹}_{0} a_{n}d_{n}(x).
 In this model t(x) has group
translates
 When t_{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 t(x) under SL(2,Z)
under the Gaction in the Kirillov model.
 However, the analytic justification of this argument – and especially
its generalizations – gets somewhat technical.

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 Theorem (MillerSchmid, 2002)
Under the same hypothesis, but instead with a_{m,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
3dimensional 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 a_{n}^{2}). This formula is used by SarnakWatson
in their sharp bounds for L^{4}norms of eigenfunctions on SL(2,Z)nH.

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 Definition of automorphic distributions and connection to representation
theory
 Applications to
 Constructing Lfunctions
 Summation Formulas
 Cancellation in sums with additive twists
 Implication to moments
 Existence of infinitely many zeroes on the critical line

25

 Let a_{n} be the coefficients of a cusp form Lfunction on
GL(d):
 S(T,x) = å_{n}_{6}_{T} a_{n} e(n x)
, e(t) := e ^{2 }^{p}^{ i t}
 Since the a_{n} have unit
size on average, we have the following two trivial bounds:
 S(T,x) = O(T)
 s_{R}_{/}_{Z}
S(T,x)^{2} dx = å_{n}_{6}_{T}
a_{n}^{2} ~ cT
 Folklore Cancellation Conjecture: S(T,x) = O_{e}(T^{1/2+}^{e}), where the implied constant depends e but is uniform in x and T.
 In light of the L^{2}norm statement, this is the best possible
exponent.

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 Fix x 2 Q. S(T,x) can be smaller = O_{x}(T^{1/2}^{d}) (Landau).
 For example, the sum S(T,0) = å_{n}_{6}_{T} a_{n} is
typically quite small, because for example:
 L(s) = å_{n}_{>}_{1}
a_{n} n^{s}
is entire
 Smoothed sums behave even better:
decays rapidly in T (faster than any polynomial), for y say a Schwartz function on (0,1).
 [shift contour s to 1]
 Similar behavior at other rationals (related to Lfunctions twisted by
Dirichlet characters).
 However, uniform bounds over rationals x are still not easy.

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 First considered by Hardy and Littlewood for classical arithmetic
functions which are connected to degree 2 Lfunctions of automorphic
forms on GL(2).
 Typically for noncusp forms.
 E.g., for a_{n }= r_{2}(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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 Recall that we expect S(T,x) = å_{n}_{6}_{T} a_{n} e(nx) to
be O_{e}(T^{1/2+}^{e}) when a_{n} are
the coefficients of an entire Lfunction.
 according to the Langlands/Selberg/PiatetskiShapiro philosophy, these
are always Lfunctions of cusp forms on GL(2,A_{Q}).
 Main known result: S(T,x) = O_{e}(T^{1/2+}^{e}).
for cusp forms on GL(2) (degree 2 Lfunctions)
 For holomorphic cusp forms, this is classical and straightforward to
prove
 But for Maass forms this is much more subtle.
 Importance: used in HardyLittlewood’s seminal method to prove z(s) has infinitely many zeroes on
its critical line
(we will see this again later).

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 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 a_{n}
equal to the standard
Lfunction coefficients, S(T,x) = O_{e}(T^{3/4+}^{e}).
 This is halfway between the trivial O(T) and optimal O_{e}(T^{1/2+}^{e}) bounds.
 We will see that the full conjecture implies the correct order of
magnitude for the second moment of L(s)=å_{n}_{¸}_{ 1}a_{n} n^{s},
which beyond GL(2) is thought to be a problem as difficult as the
Lindelof conjecture.

30

 Definition of automorphic distributions and connection to representation
theory
 Applications to
 Constructing Lfunctions
 Summation Formulas
 Cancellation in sums with additive twists
 Implication to moments
 Existence of infinitely many zeroes on the critical line

31

 Recall the Mellin transform of the distribution t(x) = å_{n}_{¹}_{ 0} a_{n}n^{}^{n}e(nx) is
 Let y be an even, smooth
function of compact support on R^{*}. By Parseval
 for any s (integrand is entire,
so the contour may be shifted).
 If y_{e}(x) is an
approximate identity (near x = 1), My(1/2+it) approximates the (normalized) characteristic
function of the interval t 2 [1/e,1/e].
 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 tFy has Mellin transform Mt(s)*My(s). Its L^{2}norm
approximates the second moment of L(1/2+it), and is determined by the L^{2}norm
of tFy. The latter is controlled by the size
of smooth variants of S(T,x) = å_{n}_{·}_{T} a_{n} e(nx).
 Conclusion: cancellation in additive sums is related to moments.

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 Lindelöf conjecture: L(1/2+it) = O_{e}((1+t)^{e}) for any
e > 0.
 Fundamental unsolved conjecture in analytic number theory.
 Implied by GRH.
 Equivalent to moment bounds:
s_{T}^{T} L(½+it)^{2k} dt = O_{e}(T^{1+}^{e}) for each fixed k ¸ 1.
 The 2kth 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 noncusp
forms – implies the Lindelöf conjecture (next slide), and is thus a very
hard problem for d > 2.

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 Folklore theorem (known as early as the 60’s by Chandrasekharan,
Narasimhan, Selberg):
 If S(T,x) = O_{e}(T^{b}^{+}^{e}) for some ½ · b < 1, then
s_{T}^{T
}L(½ + it)^{2} dt
= O_{e}(T^{1 + }^{e }^{ +
(2}^{b}^{1) d}),
 Where d = the degree of the Lfunction
 E.g. Lfunction comes from GL(d,A_{Q}).
 Thus b = 1/2 is very hard to
achieve because it gives the optimal bound O_{e}(T^{1+}^{e}) .
 GL(3) result of O_{e}(T^{3/4+}^{e}) unfortunately does not
give new moment information.
 Voronoistyle summation formulas with Schmid give an implication
between:
 squareroot cancellation in sums of d1hyperkloosterman sums weighted
by a_{n}, and
 Optimal cancellation S(T,x) = O_{e}(T^{1/2+}^{e}) – and therefore Lindelöf also.

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 Definition of automorphic distributions and connection to representation
theory
 Applications to
 Constructing Lfunctions
 Summation Formulas
 Cancellation in sums with additive twists
 Implication to moments
 Existence of infinitely many zeroes on the critical line

35

 Suppose (for fictitious expositional simplicity) n = 0 for a cusp form on SL(2,Z). It is not difficult to handle
arbitrary n.
 Let H(t) = Mt(1/2+it). Then H(t) = H(t) is real.
 Let y_{1/T} be an
approximate identity such that My(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 t(x) vanishes
to infinite order at x=1 (y is concentrated near a point where t 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_{e}(T^{1/2+}^{e}) shows that the last
integral is still o(T) with room to spare.
 New phenomena: numerically that integral decays only like T^{1/2 }for
q=11.

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 Like the moment problem, nothing is known about infinitude of zeroes on
the critical line for degree d > 2 Lfunctions.
 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 t
on GL(4,Z)nGL(4,R) vanishes to infinite order at 1, then L(1/2+it,t) = 0 for infinitely many t 2 R.
