Integral moments. Note: in some applications a subconvex bound replaces Lindelöf. T ζ( it) 2k dt. ζ( it) 2k t w dt

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1 Garrett: Integral moments [Edinburgh, 4 Aug 8] Integral moments Paul Garrett garrett@math.umn.edu garrett/ (with Diaconu and Goldfeld) Idea: integral moments over families of L-functions arise as coefficients in automorphic spectral decompositions. Example: in several cases, extract subconvex bounds Note: in some applications a subconvex bound replaces Lindelöf Sample of moments and bounds: Euler-Riemann zeta (RH )Lindelöf ζ( + it) tε ε> (unknown) convexity ζ( + it) t 4 +ε ε> (true, easy) k th moment = Example of integral moments Z k (w) = T ζ( + it) k dt ζ( + it) k t w dt which has natural boundary for k 3 (Diaconu-PG-Goldfeld). Lindelöf T ζ( + it) k dt T +ε (for all k, ε > ) Suitable moment estimates yield subconvex bounds: T ζ( + it) dt T +ε subconvex bd T ζ( + it) 4 dt T +ε subconvex bd T ζ( + it) 4 dt = T P(log T) + O(T δ ) = subconvex bd T ζ( + it) k dt T +ε with k 6 = subconvex bd

2 Garrett: Integral moments [Edinburgh, 4 Aug 8] 98 Hardy-Littlewood: T ζ( + it) dt T log T ( subconvex bound) 9 Weyl (not by moments): ζ( + it) ( + t ) 6 +ε (= subconvex) 96 Ingham: T 979 Heath-Brown: T ζ( + it) 4 dt π T(log T) 4 ( subconvex bound) ζ( + it) 4 dt T P(log T) + O(T 7 8 +ε ) (= subconvex bound) Our prototype: 98, A. Good: holo cuspform f T L( + it,f) dt = at(log T +b) + O((T log T) /3 ) = subconvex Proof mechanism: () automorphic spectral decomposition () L-functions as decomposition coefficients. Examples of decomposition coefficients and L-functions: ( ) y s y f = Λ(s,f) (f on GL ) fg, Es = Λ(s,f g) (f,g on GL ) Decomposition example: SL ( )\H Φ Φ,F F,F F + 4πi F cfm Re(s)= Over a number field: Φ Φ,F F,F F + 4πiκ F cfm + χ = See sum over χ, integral over t. χ Φ,χ det χ det Φ,E s E s ds + Φ,, Re(s)= (χ det) Φ,E s,χ E s,χ ds

3 Garrett: Integral moments [Edinburgh, 4 Aug 8] Create integral kernel/poincaré series P α,w such that: Integral moment of cuspform f produced by integral P α,w f = L( π + it + α,f) L( it,f) [ t w] dt Spectral expansion of P α,w P α,w = π w Γ( w ) π w Γ( w ) E +α + with + 4πi Re(s)= G(s,α,w) = π (α+ w L( + α,f) G( F,F it F,α,w) F F on GL ζ(α + s)ζ(α + s) ξ( s) α+ s ) Γ( )Γ( α+s G( s,α,w) E s ds α s+w )Γ( )Γ( α+s +w ) Γ(α + w ) Continuous part cancels pole of leading term at α =. Evaluated at α = : P,w f = L( π + it,f) [ t w ] dt and P,w = ( ) pole at w= + 4πi Re(s)= + L(,F) F,F G( it F,,w) F F on GL ξ(s)ξ( s) ξ( s) Γ( w s w +s )Γ( ) Γ( w ) E s ds Over É(i) grossencharacters appear (Diaconu-Goldfeld 6) Similarly over number fields. With suitable data, Theorem: t-aspect subconvexity for GL over number fields (Diaconu-PG 6) 3

4 Garrett: Integral moments [Edinburgh, 4 Aug 8] Proof ingredients: First, over É: Spectral expansion of P α,w gives meromorphic continuation Obtain meromorphic continuation of generating function Z(w) Leading pole at w =. Trailing poles at µ = µ f with µ(µ ) eigenvalue for waveform f. Need polynomial vertical growth Need spectral gap (Kim-Shahidi): separate cuspidal poles from leading pole Need asymptotics of triple integrals F f of eigenfunctions (Sarnak, Bernstein-Reznikoff, Krötz-Stanton). exponentially decreasing, not merely rapidly. Namely, See also Goldfeld-Hoffstein-Lockhart, Hoffstein-Ramakrishnan for asymptotics of L(,F) (F cuspform) F,F Use positivity of moment sum-and-integral... Landau s lemma Complications over number field: Freeze parameters w at all but one place... thus, breaking t-aspect convexity at a single place, not hybrid Poles at eigenvalues of one Laplacian presumably accumulate,... requiring finesse proving polynomial vertical growth 4

5 Garrett: Integral moments [Edinburgh, 4 Aug 8] The diagonal/positivity property: The diagonal form L( + it,f χ)... χ rather than a smeared-out form L( + it,f χ ) L( it,f χ )... χ,χ is essential, or at least extremely convenient. Arises as deformation of diagonal distribution. Simpler example: Distribution u on S S integrating along the diagonal u(f g) = f g S Has diagonal Fourier expansion u = n e πinx e πiny since, by Plancherel, S f g = n f(n)ĝ( n) = m,n f g(m,n) û(m,n) Similarly for any diagonal integral. Clean-but-harsh pure diagonal distributions often usefully deformed to nearlydiagonal more-classical function. 5

6 Garrett: Integral moments [Edinburgh, 4 Aug 8] Something appealing that doesn t work: to obtain k th moments for GL cuspforms? Corresponding idea in more tangible situation of classical Fourier series: let u be the distribution on H = S... S }{{} k given by integration along the subgroup Θ = {(x,...,x k,y,...,y k ) : x... x k y... y k = } H The Fourier expansion is concentrated along a diagonal line: u = n e πin(x +...+x k y... y k ) For automorphic forms: let { ( ) ( x H = xk... ) ( y ) ( yk... ) } GL k and let u be integration along { Θ = elements in H with : x... x } k = y... y k For cuspform f on GL, restrict F = f... f f... f }{{} k to H k \H and evaluate u on it. Over É, produces Λ( + it,f) k dt Obstacle: spectral interpretation/expansion for k >? Can deform to soften exponential decay of gamma factors, but... 6

7 Garrett: Integral moments [Edinburgh, 4 Aug 8] Technicality: the weight in the integral moment At α =, finite-prime factors are literal Hecke-Jacquet-Langlands Mellin transforms of local Whittaker functions ( ) L v (s,f χ) = y s y χ(y) Wf,v k v and L v ( + it,f) = y/y it W f,v ( y ) W f,v ( y ) Deformation at archimedean places entangles this with integration over unipotent radical: with additive character ψ and ( ) ( ) ( ) y h = h y x = n = L v ( + it,f) is deformed into = = = ( ) local factor at archimedean v ϕ v (n) y /y s v W f,v (hn) W f,v (h n) ϕ v (n)ψ ( (y y )x ) W f,v (h) W f,v (h ) y /y s v ϕ v (y y ) W f,v (h) W f,v (h ) y /y s v Except for holomorphic discrete series, these seem not usefully expressible by classical special functions. The map from data ϕ v to these integrals is definitely not surjective to an elementary space of functions. ( ) x Nevertheless, asymptotically t w for data ϕ v = ( + x ) w/ 7

8 Garrett: Integral moments [Edinburgh, 4 Aug 8] Other features/possibilities for GL Over quadratic extensions of É, should break convexity in χ-aspect, with same proof. Full χ-aspect over number field subtler, like hybrid bounds. Allow deformation at finite place, should break convexity in depth: unlimited ramification of χ s at fixed place. (Delia Letang, work-in-progress) Replace cuspform by (packet of) Eisenstein series:... should break t-aspect convexity for Dedekind zetas of number fields, via fourth moments.... and should break t-aspect convexity for Hecke L-functions with grossencharacters, via fourth moments.... and should break χ-aspect convexity for quadratic fields.... and should break convexity in depth by deforming finite prime data. 8

9 Garrett: Integral moments [Edinburgh, 4 Aug 8] Higher rank example (Diaconu-PG-Goldfeld, 6) Create P α,w to produce moment expansion for GL 3 (É) cuspform f P,w f = F on GL L( + it,f F) F,F L( + + it it,f) L( + it + it,f) ζ( it ) M F dt M E dt dt More-continuous part is sort of integral moment of standard L(s,f) of f. Spectral expansion of GL 3 Poincaré series is induced from GL P α,w = ( part) E, α+ + + ( part) L(3α+ +,F) F,F F on GL Re(s)= E, α+,f ( part) ζ(3α+ + s) ζ( 3α+ + s) ζ( s) E,, α+, s α+, s α+ Only GL cuspforms appear in spectral expansion of P! No GL 3 cuspforms! It is good that no cuspidal data beyond GL appears. But this is also confusing: obscures easiest heuristics for computing and understanding spectral expansion of P. In fact, P α,w is a residue of overlying Poincaré series Q α,β,w with more balanced spectral expansion, and retaining meaningful moment expansion. More on this later. ds 9

10 Garrett: Integral moments [Edinburgh, 4 Aug 8] GL n (É) more generally Moment expansion: P,w f = F on GL n Re(s)= L(s,f F) M F,F F (s)ds +... Spectral expansion of P α,w + P α,w = ( part) E n, +α + ( part) L( nα+n +,F) F,F F on GL Re(s)= ( part) E n,, α+, s (n ) α+ E n, +α,f nα+n L( + s,χ) L( nα+n + s,χ) L( s,χ ), s (n ) α+ Only GL cuspforms appear in spectral expansion of P! No GL 3, GL 4, GL 5,... cuspforms! ds Construction of kernel P α,w ( ) n U = H = ( ) (n ) Z = center G, K v maximal compact in G v. Let ϕ = v ϕ v with ( ) A ϕ v ( K d v ) = (deta)/d n α (for v finite) v Extend by off H v Z v K v. For v require the same right left equivariance and K v -invariance, with ϕ v determined by values on U v. For example, take ( ) n x ϕ v = ( + x x n ) w/ Kernel is P α,w (g) = γ Z k H k \G k ϕ(γg)

11 Garrett: Integral moments [Edinburgh, 4 Aug 8] More-continuous terms: higher integral moments Most-continuous part of moment expansion for GL n where = Λ Λ L( + it,f Emin +i et ) M dtd t Π l n L( + it + it l,f) Π j<l<n ζ( it j + it l ) Λ = { t Ê n : t t n = } M dtd t More generally, let n = m k. For F on GL m, let In moment expansion have F = F... F on GL m... GL m }{{} k Re(s)= Λ L(s,f E F, +it ) M F,t,s dsdt Π l k L(s + it l,f F) = Π j<l k L( it j + it l,f F ) M dsdt a kind of higher moment of L( + it,f F)

12 Garrett: Integral moments [Edinburgh, 4 Aug 8] Wave-packets of Eisenstein series Replace cuspform f on GL n by (packet of) minimal-parabolic Eisenstein series E β with β n. Most-continuous part of the moment expansion Λ µ n, l n ζ(β µ + + it + it l) j<l<n ζ( it j + it l ) dtd t where, again, Λ = { t Ê n : t t n = } At β = n (ignore vanishing of E β!) Λ l n ζ( + it + it l) n j<l<n ζ( it j + it l ) M dtd t = GL n produces high moments of ζ. Example: for GL 3, where Λ = {(t, t )} Ê, Ê ζ( + it + it ) 3 ζ( + it it ) 3 ζ( it ) M dtdt Example: for GL 4 Λ ζ( + it + it ) 4 ζ( + it + it ) 4 ζ( + it + it 3) 4 ζ( it +it )ζ( it +it 3 )ζ( it +it 3 ) M dtd t Actually, these should be integrated as in corresponding packets.

13 Garrett: Integral moments [Edinburgh, 4 Aug 8] Caution: not all moments results bear on subconvexity: Assume Lindelöf, Weyl s Law to determine asymptotic-with-error for moment, see implied pointwise estimate. Failure to match convexity... does not imply corresponding asymptotic-witherror is useless. Outcome depends upon the parameters varied in the L-functions... upon the family averaged-over. Much more difficult to understand joint asymptotics-with-error. Do not expect to prove hybrid subconvexity. Essentially no result of this type is known! Thus, be cautious about reasonable forms of joint asymptotics-witherror, though Iwaniec-Sarnak formulation makes sense for hybrid estimates. Of special interest are second moments, most easily produced by spectral identities. Simplest failure to match convexity: T ζ( + it) dt = T P(log T) + O(T small ) implies, by standard methods, (known) and then But convexity is ζ( + it) ( + t ) small ζ( + it) ( + t ) small ζ( + it) ( + t ) 4 +ε Small shift in exponent is very small, so cannot help get the exponent below /4. Failed to match convexity. Success: T ζ( + it) 4 dt = T P(log T) + O(T small ) does break convexity. (known) 3

14 Garrett: Integral moments [Edinburgh, 4 Aug 8] Example: second moments of GL L-functions Convexity bound in terms of analytic conductor (Iwaniec-Sarnak) L( + it,f) ε ( + t + µ f ) 4 +ε ( + t µ f ) 4 +ε Success: t-aspect (as above). Fix f. L( + it,f) ε,f ( + t ) +ε (convexity) From T L( + it,f) dt = T P(log T) + O(T small ) (fixed f) Get pointwise L( + it,f) ε,f ( + t ) small breaking convexity in the t-aspect. Failure: eigenvalue-aspect Fix t. L( + it,f) ε,t ( + µ f ) +ε (convexity) From µ T L( + it o,f) = T P(log T) + O(T small ) since (Weyl) number of f with µ f T is of the order of T, would have failing to equalize convexity. L( + it,f) ε,t ( + µ f ) small 4

15 Garrett: Integral moments [Edinburgh, 4 Aug 8] Example: Rankin-Selberg convolutions Convexity bound in terms of analytic conductor L( + it,f g) ε ( signs ) 4 ( + t ± µ f ± µ g ) +ε t-aspect? Fix f, g. L( + it,f g) ε,f,g ( + t ) +ε (convexity) From T L( + it,f g) dt = T P(log T) + O(T small ) (fixed f) get pointwise L( + it,f g) ε,f ( + t ) small breaking convexity in t-aspect. Too good for existence of corresponding spectral family. g-aspect Fix t, f. L( + it,f g) ε,t,f ( + µ g ) +ε (convexity) From µ g T L( + it,f g) = T P(log T) + O(T small ) would have L( + it,f g) ε,t,f ( + µ g ) small breaking convexity. Plausible spectral family. 5

16 Garrett: Integral moments [Edinburgh, 4 Aug 8] Spectral identity, Rankin-Selberg for GL Avoiding conductor dropping considerations... Can produce spectral identity P,w f E +it = g cfm on GL L( + it,f g) wt(g) +... with positivity and diagonal properties, weight depends upon archimedean data of g (and of fixed t and f). P,w has good spectral expansion on GL GL. Apparently: Leading pole of P at w =. Trailing poles of P at µ = µ F with cuspform F eigenvalue µ(µ ). Known spectral gap should allow subconvex bound. Over number field, separate parameters at archimedean places. Deformation at finite set of finite places should allow averaging g in depth. 6

17 Garrett: Integral moments [Edinburgh, 4 Aug 8] Rankin-Selberg for GL 3? Avoiding conductor dropping considerations... A spectral identity exists, producing g-aspect moment P f E s = g cfm on GL 3 L(s,f g) wt(g) +... with positivity and diagonal properties, and weight function depending only upon archimedean data of g (and t and f). P itself has good spectral expansion on GL 3 GL 3. Convexity bound: L( + it,f g) ε,t,f ( ) 9 4 ( + µ )( + µ )( + µ 3 ) + ε Asymptotic with very good error L(s,f g) = T 5 P(log T) + O(T 9 small ) µ j T would break convexity in g-aspect. 7