Standard compact periods for Eisenstein series

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1 (September 7, 2009) Standard compact periods for Eisenstein series Paul Garrett garrett/m//. l GL 2 (k): CM-point alues, hyperbolic geodesic periods 2. B GL 2 (l) 3. (B k l), GSp (, ) 4. Degenerate Eisenstein series along anisotropic tori 5. Appendix: zeta functions of quaternion algebras, etc. For Γ SL 2 (Z), the spherical Eisenstein series E s (z) c,d y s cz + d 2s (c, d relatiely prime, z H) takes meaningful alues at CM-points, points z H such that k Q(z) is quadratic oer Q. Especially, when o Z[z] is the ring of algebraic integers in the quadratic field, and when o has class number one, E s (z) ζ k(s) ζq(2s) When o has larger class number, linear combinations of CM-point alues corresponding to the ideal classes yield the corresponding ratio. This example was understood in the 9th century, and is the simplest in well-known families of special alues and periods of Eisenstein series. The next case in order of increasing complexity is that of integrals of the same E s along hyperbolic geodesics in H that hae compact images in Γ\H. These were considered by Hecke and Maass. In fact, from a contemporary iewpoint the CM-point alue example and the hyperbolic geodesic periods example are identical, as is made clear in the first section, The first three examples are instances of periods of Eisenstein series on orthogonal groups O(n + ) with rational rank, along orthogonal groups O(n) with compact arithmetic quotients. Another family extending the small examples is periods of degenerate Eisenstein series along anisotropic tori. Periods of Eisenstein series are interesting prototypes for periods of cuspforms, often unwinding more completely than the corresponding integrals for cuspforms. In that context, periods attached to degenerate Eisenstein series are ineitably misleading to some degree. Neertheless, there is a correct indication that sharp estimates on periods are often connected to Lindelöf hypotheses and other ery serious issues. See the bibliography for background and some pointers to contemporary literature.. l GL 2 (k): CM-point alues, hyperbolic geodesic periods This example is essentially due to Hecke and Maass. Let l be a quadratic field extension of a global field k of characteristic not 2. Let G GL 2 (k), and H G a copy of l inside G, by specifying the isomorphism in l Aut k (l) Aut k (k 2 ) GL 2 (k)

2 Let P be the standard parabolic of upper-triangular elements in G, and E s (g) ϕ(γ g) γ P k \G k where ϕ is the eerywhere-locallly-spherical ector a ϕ( k) a s d d (for k in maximal compact in GA) With Z the center of G, we want to ealuate period of E s along H E s The subgroup P k is the isotropy group of a k-line k e for a fixed non-zero e k 2 l. The group G k is transitie on these k-lines, so P k \G k {k lines} The critical-but-triial point is that the action of l on l is transitie on non-zero elements. Thus, P k l GL 2 (k). That is, the period integral unwinds E s ZA(P k H k )\HA ϕ ZA\HA since H P Z. Since ϕ ϕ factors oer primes, the unwound period integral factors oer primes E s ZAHA ϕ Z \H ϕ ϕ The cleanest way to ealuate the local integrals is to use an integral presentation of the local spherical ector ϕ akin to better-known archimedean deices inoling the Gamma function. That is, present ϕ in terms of Iwasawa-Tate local zeta integrals ϕ (g) ζ k, (2s) det g s t 2s Φ (t e g) dt (with ζ k, (2s) k k t 2s Φ (t e) dt) for Φ a K -inariant Schwartz function on k 2 l. The leading local zeta factor gies the normalization ϕ () at g. Then Since Z \H ϕ ζ k, (2s) det h s k \l k t 2s det h k N l/k h k h l the local factor of the period becomes ζ k, (2s) h s l k \l t s l k Φ (t e h) dt dh ζ k, (2s) h s l l Φ (e h) dt dh Thus, the product of all the local factors of the period is E s ξ l(s) ξ k (2s) 2 ζ k, (2s) k \l k Φ (t e h) dt dh ζ k, (2s) ζ l,(s) (with E s on GL 2 (k)) t h s l Φ (t e h) dt dh

3 with ξ including the gamma factors. [.0.] Remark: In this normalization, the unitary line is Re(s) 2, and E 2 +it ξ l( 2 + it) ξ k ( + 2it) 2. B GL 2 (l) This is a Galois-twisted ersion of the diagonal imbedding GL 2 GL 2 GL 2 from the classical Rankin- Selberg situation. Let B be a quaternion diision algebra oer a number field k. By Fujisaki s lemma, k A B \B A is compact. Let l be a quadratic extension of k splitting B. Define a k-group by G GL 2 (l), and let H be the image of B inside G by choice of an isomorphism B Aut l (B) Aut l (l 2 ) GL 2 (l) Let P be the standard parabolic in G, and E s (g) where ϕ is the eerywhere-locally-spherical ector a ϕ( k) a s d d l γ P l \G l ϕ(γ g) (for k in maximal compact in GA) The norm is the product-formula normalization of the norm for l. With Z the center of G, we want to ealuate period of E s along H E s The subgroup P k is the isotropy group of an l-line l e for a fixed non-zero e l 2 B. The group G k is transitie on these l-lines, so P k \G k {l lines in B} The action of B on B is transitie on non-zero elements. Thus, P k B GL 2 (l). That is, the period integral unwinds partly E s ϕ ϕ ZA(P k H k )\HA k A l \B A since B P l. Since l is not central in B, this copy of l is not the central copy in GL 2 (l). We do hae the factorization ϕ ϕ oer primes, but the integral itself needs further unwinding. Further unwinding of the integral, as well as ealuation of the local integrals, is facilitated by an integral presentation of the spherical ectors ϕ, in terms of Iwasawa-Tate/Tamagawa/Godement-Jacquet local zeta integrals ϕ (g) ζ l, (2s) det g s l, t 2s l, Φ (t e g) dt (with ζ l, (2s) l l t 2s l, Φ (t e) dt) for Φ a K -inariant Schwartz function on l 2 B. The leading local zeta factor gies the normalization ϕ () at g. With Φ Φ, the period integral is ϕ det h s l t 2s l Φ(t e h) dt dh k A l \B A ξ l (2s) k A l \B A l A 3

4 ( ) det h s l t 2s l Φ(t e h) dt dh ξ l (2s) k A l A \B A kal \l A because the inner integral already produces a left l A -inariant function of h B A. What s left of the integral does now factor into local contributions: ol (k A l \l A ) ξ l (2s) Since the period integral is ZAl A \B A l A det h s l t 2s l Φ(t e h) dt dh ol (k A l \l A ) det h s l Φ(t e h) dh l ξ A l (2s) B A det h l N red h l N red h 2 k (for h B, central simple oer k) E s ol (k A l \l A ) ξ l (2s) ξ B (2s) (with E s on GL 2 (l)) The zeta function of the quaternion algebra B oer k is expressible in terms of that of k, at argument 2s as in the aboe, ξ B (2s) ξ k(2s) ξ k (2s ) rfd ζ k,(2s ) (product oer places ramified for B) Also, the olume is ol (k A l \l A ) Res sξ l (s) Res s ξ k (s) Λ k(, χ l/k ) (with χ l/k the quadratic character of l/k) Thus, the period is E s Λ k(, χ l/k ) ξ k (2s) ξ k (2s ) ξ l (2s) rfd ζ k,(2s ) Λ k (, χ l/k ) ξ k (2s) ξ k (2s ) ξ k (2s) Λ k (2s, χ l/k ) rfd ζ k,(2s ) Λ k (, χ l/k ) ξ k (2s ) Λ k (2s, χ l/k ) rfd ζ k,(2s ) (E s on GL 2 (l)) [2.0.] Remark: In this normalization, the unitary line is Re(s) 2, and E 2 +it Λ k (, χ l/k ) ξ k (2it) Λ k ( + 2it, χ l/k ) rfd ζ k,(2it) (E 2 +it on GL 2 (l)) 3. (B k l) GSp (, ) Let B be a quaternion diision algebra oer a number field k, not split by a quadratic extension l of k. Let l k(ω), where tr l/k (ω) 0. Let θ be the main inolution of B, and extend it l-linearly to C B k l. Define a B-alued quaternion-hermitian form on the two-dimensional B-ectorspace C by x, y ( tr l/k ) ( ω y θ x ) B (for x, y C) Choice of a right B-basis for C gies an isomorphism GL 2 (B) Aut B (C) 4

5 The similitude group G GSp (, ) of the form, is The subgroup of C imbeds in G, since for h H G {g GL 2 (B) : gx, gy ν(g) for ν(g) k } GSp (, ) H {h C : h θ h k } (as opposed to lying in l ) hx, hy ( tr l/k ) ( ω (hy) θ hx ) ( tr l/k ) ( ω y θ h θ h x ) (h θ h)( tr l/k ) ( ω y θ h θ h x ) since the trace is k-linear and h θ h is in k, not merely in l. Let P be a proper k-parabolic in G, characterized as the stabilizer of a fixed isotropic B-line B ein C. The simplest choice is e, so B B. Define an Eisenstein series E s (g) ϕ(γ g) γ P l \G l where ϕ is an eerywhere-spherical ector presented by an integral, ϕ(g) det g s det t 2s Φ(t e g) dt ξ B (2s) where, if necessary, det denotes a suitable reduced norm. The leading factor arranges that ϕ(). With Z the center of G, we want to ealuate period of E s along H B A E s To begin unwinding the period integral, H k must be transitie on isotropic B-lines. Indeed, for 0 x C such that x, x 0, then x θ x k, so x H. Thus, x x, proing the transitiity. Thus, E s ϕ ϕ (with C {h C : h θ h k }) ZA(P k H k )\HA The integral presentation of ϕ gies ξ B (2s) ξ B (2s) ZAB A \C A ol (Z AB \B A ) ξ B (2s) ZAB \C A ( C A ZAB \C A det h s ZAB \B A B A det t 2s Φ(t e h) dt dh ) det h s det t 2s Φ(t e h) dt dh B A det h s Φ(e h) dh ol (Z AB \B A ) ξ C (s) ξ B (2s) The zeta function of C is only slightly more complicated than the zeta function of a quaternion algebra, and, up to a finite product from ramified primes, E s ol (ZAB \B A ) (finite ) ξk(2s) ξ k (2s 2) ξ l (2s ) ξ k (2s)ξ k (2s ) Cancelling where possible, E s ol (ZAB \B A ) (finite ) ξ k (2s 2) Λ k (2s, χ l/k ) 5

6 [3.0.] Remark: In this normalization, the unitary line is Re(s) 3 2, and the period is E 3 2 +it ( const ) (finite ) ξ k ( + 2it) Λ k (2 + 2it, χ l/k ) 4. Degenerate Eisenstein series along anisotropic tori Let [l : k] n and let H be a copy of l in G GL n (k). Let P be the standard (n, ) parabolic in G, namely, fixing the line spanned by the n th standard basis ector e n under right matrix multiplication. Let χ s be the character on PA defined by Define ϕ s by A χ s det A s 0 d d n ϕ s (p k) χ s (p) and define an Eisenstein series by E s (g) (where A GL n and d GL ) (with k in maximal compact, p PA) γ P k \G k ϕ s (γ g) We will compute the period of E s along the copy of l in G. With Z the center of G, this is by unwinding, since ZAl \l A E s P k l G k ϕ s ZA\l A The function ϕ s factors oer primes, as does the unwound integral, giing th local factor k \l ϕ s, (h) dh To ealuate this, again use an integral representation of ϕ s, as ϕ s, (g) ζ k, (ns) det g s k t ns Φ(t e n g) dt where Φ is a suitable Schwartz function. Substituting this in the local integral, the local integral unwinds to ZA\l A ϕ s det h s Φ(e n g) dh ζ l(s) ζ k, (ns) l ζ A k, (ns) Thus, apart from modifications at finitely-many places, the period of this degenerate Eisenstein series oer a maximal anisotropic torus attached to the field extension l of k is ZAl \l A E s ξ l(s) ξ k (ns) 6

7 5. Appendix: zeta functions of quaternion algebras Let B be a quaternion algebra oer a number field k. For a Schwartz function Φ Φ on BA, at almost all places, we claim that det t s Φ (t) dt ζ k (s) ζ k (s ) B Indeed, we can suppose that B is split, and use an Iwasawa decomposition NMK with n In these coordinates, the Haar measure is Then B det t s Φ (t) dt x 0 and m d(mnk) dm dn dk M N det t s Φ (t) dt a 0 0 d M N ad s Φ a ax dm dn 0 d Almost eerywhere, Φ is the characteristic function of the local integers. Replace x by x/a and integrate in x, leaing a s d s a a Φ dm M 0 d This is a product of two Iwasawa-Tate integrals, namely ζ k (s ) ζ k (s) For l a quadratic extension of k not splitting B, let C B k l and C {h C : h θ h k } At a place of k splitting in l, the th local zeta integral for ξ C (s) becomes a ax a a Φ Φ x 0 t/a 0 t/a t 2s where Φ is the characteristic function of the integral-coordinate matrices. Replace x by x/a and x by x /a and integrate x, x away to obtain ( a a Φ 0 t/a ) a a Φ 0 t/a a a t 2s Thus, a and a are integral, and t/a and t/a are integral. Taking an outer integral oer t, this becomes (q j 2s qj+ ) q j 0 q j+ q [(q 2s ( q) 2 ) j 2q (q 2s ) j + q 2 (q 2 2s ) j] j 0 [ ( q) 2 q 2s 2q q 2s + q 2 ] q 2 2s q 2 4s ( q 2s ) ( q 2s ) 2 ( q 2 2s ) 7 + q 2s ( q 2s ) ( q 2s ) ( q 2 2s ) (at split place)

8 At a place of k inert in l, the th local zeta integral for ξ C (s) becomes a ax Φ t s a ax l Φ 0 t/a t 2s 0 t/a since t is integrated just in k. Replace x by x/a and integrate in x to obtain a a Φ t 2s 0 t/a a l This becomes (q j ) 2s (q2 ) j+ q 2 j 0 [ q 2 q 2s q 2 ] q 2 2s ( q 2s ) ( q 2 2s ) (at inert place) To compare the inert place computation with that for the split place, insert corresponding factors, giing q 2 4s ( q 2s ) ( q 2s ) ( + q 2 2s ) ( q 2 2s ) (at inert place) The sign is χ l/k. Thus, up to finitely-many factors corresponding to ramification, ξ C (s) ξ k(2s) ξ k (2s 2) ξ k (2s ) Λ k (2s, χ l/k ) ξ k (4s 2) ξ k(2s) ξ k (2s 2) ξ l (2s ) ξ k (4s 2) Bibliography M. Einsiedler, E. Lindenstrauss, Ph. Michel, A. Venkatesh, Distribution of periodic torus orbits and Duke s theorem for cubic fields, see michel/publi.html. P. Garrett, Fujisaki s lemma, garrett/m//fujisaki.pdf P. Garrett, Reduction theory, garrett/m//cptness aniso quots.pdf P. Garrett, Semi-simple algebras, garrett/m//algebras.pdf R. Godement and H. Jacquet, Zeta Functions of Simple Algebras, Lecture Notes in Mathematics no. 260 (972), Springer-Verlag, New York, 972. K. Iwasawa, Letter to J. Dieudonné, dated April 8, 952, in Zeta Functions in Geometry, editors N. Kurokawa and T. Sunada, Adanced Studies in Pure Mathematics 2 (992), G.D. Mostow and T. Tamagawa, On the compactness of arithmetically defined homogeneous spaces, Ann. of Math. 76 (962), T. Tamagawa, On the zeta functions of a diision algebra, On the Ann. of Math. (2) 77 (963), J. Tate, Fourier analysis in number fields and Hecke s zeta functions, thesis, Princeton (950), in Algebraic Number Theory, J. Cassels and J. Frölich, editors, Thompson Book Co., 967. A. Weil, Adeles and Algebraic Groups, Princeton, 96; reprinted Birkhauser, Boston, 982. A. Weil, Basic Number Theory, Springer-Verlag, New York,