Luận án tiến sĩ: Congruences among automorphic forms on the unitary group U(2,2)

In this thesis we prove that under certain conditions, if a special value of thesymmetric square L-function attached to a modular form f of appropriate weight, level and character vanish

Siegel and Klingen Eisenstein series

Siegel and Klingen Eisenstein series are induced from the maximal parabolic subgroups P and Q of G = U(2,2) respectively (For the definitions of P and Q see section 2.2.) Hence by Remark 3.1.9, they are functions of one complex variable,

27 which we denote by s Let dp: P(A) > R4 be the modulus character of P(A),

(At)-} with A € Resxjq GL2(A), u € Up(A), and dg: Q(A) — RL be the modulus character of Q(A)

\L c a with ứ € ResxjgGli(A), [25] € U(1,1)(A) and u € Ug(A) As before, let

Ko = KossKo¿ denote the maximal compact subgroup of G(A) Using the Iwasawa decomposition G(A) = P(A)Ko we extend both characters dp and dg to functions on G(A) and denote these extensions again by ốp and dg It is easy to check that ôp(g) = 1 for g € P(A)N Ko and similarly dg(g) = 1 for g € Q(A)NKo, hence the extensions are well-defined Moreover, it is clear that ốp (resp dg) is a P-automophic (resp Q-automorphic) form in the sense of Definition 3.1.1 with = 1 Fors ¢€C we have

(3.9) 6% € Ip(s—1/2,1) ={f: G(A) — Ơ | ƒ(muk) = ọp(m)(°~3)*3 ƒ(k), f is smooth and Ko-finite} and the local function ðb„ is the spherical vector for every place p of Q We similarly have dg € Ig(s — 1/2, 1).

P(Q)\G(Q) is called the Siegel Evsenstein series.

Remark 3.2.2 Note that Ep(g,s) = E(P,s— 1/2,1,ƒ,g), with E(P,s,1,f,g) as in Definition 3.1.7, where ƒ denotes the constant function with value 1.

The properties of #p(g, s) were investigated by Shimura in [Shi97] We summarize them in the following proposition.

Proposition 3.2.3 The series Ep(g,s) is absolutely convergent for Re(s) > 1 and can be meromorphically continued to the entire s-plane with only a simple pole at s = 1 Furthermore, the residue of Ep(g,s) at s = 1 is a constant function of g. The value of this constant is equal to

(3.10) ress) Ep(g, 3) = Aen where L{-,-) denotes the Dirichlet L-function.

Q(Q)\G(Q) is called the Klingen Eisenstein series.

Remark 3.2.5 In the case of Klingen Eisenstein series we also have Eg(g,s) E(Q,s—1/2,1, f,g), with E(Q,s,1, ƒ, g) as in Definition 3.1.7, where f denotes the constant function with value 1.

Proposition 3.2.6 The series Eg(g,s) converges absolutely for Re(s) > 1 and can be meromorphically continued to the entire s-plane The possible poles of Eo(g, s) are at most simple and are contained in the set {0,1/3, 2/3, 1}.

Proposition 3.2.6 follows from Lemma 1 in [RS91] The only difference is that instead of Eg(g,s), [RS91] uses an Eisenstein series that in section 4.3 we denote by E,(Z) The connection between Eg(g,s) and E,(Z) is provided by Lemma 4.3.3 In section 4.3 we will show that Eg(g,s) has a simple pole at s = 1 and calculate its residue.

Both Ep(g,s) and Eg(g,s) have their classical analogues, i.e., series in which g is replaced by a variable Z in the hermitian upper half-plane H Let g € G(R) be such that Z = gyi and set g = (gu, 1) € G(R) x G(A¿;) Define

We will show in Lemma 4.3.3 that

3eQ(2)\G(2) (in (1⁄))ss where for any 2 x 2 matrix M we denote its (i, j)-th entry by Ä;; It is also easy to show that

Ep(Z,s) = (detIm(yZ))*= ` — |det(c,Z+dy)|-*. yEP(Z)\G(Z) yEP(Z)\G(Z)Remark 3.2.7 Note that we use the same symbols #p(-, s) and Eg(-, s) to denote both the adelic and the classical Eisenstein series Which of the two kinds of Eisen- stein series are considered in a particular situation, can be recognized by looking at the argument (g or Z) We will continue this abuse of notation for other Eisenstein series we study.

Borel Eisenstein series 2 1 ee 30

In this section we give an example of an Eisenstein series induced from the Borel subgroup B of G As B is a minimal parabolic and G is of rank 2, such an Eisenstein series is a function of two complex variables which we denote by s and z To define it we will use the modulus functions ốp and dg defined in section 3.2.1 Note that B is contained in both P and Q, hence the following series

7€B(Q)\G(Q) is (at least formally) well defined Note that as the Levi subgroup of B is abelian (it is the torus 7'), the character 636% is a cusp form on T(A) The following proposition follows from [MW89], Proposition II.1.5.

Proposition 3.2.8 The series Eg(g,s,z) ts absolutely convergent for

It can be meromorphically continued to all of C x C and has poles along the lines z= 1/2 ands = 2/3.

Remark 3.2.9 It follows from the general theory (cf [Lan76], chapter 7) that by taking iterated residues of Eisenstein series induced from minimal parabolics one obtains Eisenstein series on other standard parabolics These series are usually referred to as residual Hisenstein series In fact Ep and Eg are residues of Hg taken with respect to the variable s and z respectively We will prove this fact in section4.3, but see also [Kim04], Remark 5.6.

Holomorphic Eisenstein series on SLz(2)

Siegel and Klingen Eisenstein series considered in section 3.2.1 are both defined using a section f which is the constant function with value 1 (cf Remarks 3.2.2 and bài

3.2.5) Classically this means that both series are of weight 0 and trivial character.

We will later need a variant of the Siegel Eisenstein series with positive weight and non-trivial character To motivate our definitions in section 3.2.4 we will now work out an example of the holomorphic Eisenstein series of weight m (for m an even positive integer) on SL.(Z) defined as

+eB;(Z)\SLa(Z) +€B›:(Z)\SLa(Z) Here z¡ denotes a variable in the complex upper half-plane H, By is the upper- triangular Borel subgroup of SLằ and j(y, 21) := Âyz1 + dự We will now redefine the series (3.11) in the context of Langlands theory developed in section 3.1 Let Kằ denote the maximal compact subgroup of SL2(A) with Ky = K2, K2+, where cos@ sing

—sing@ cos@ is the stabilizer of ¢ in SLo(R) Let By = T,U2 be the Levi decomposition of By with

1 Using the Iwasawa decomposition SL2(A) = B2{A) Ko we write g € SL2(A) as g = bk and define a Bj-automorphic form

#: Te(Q)U2(A) \ SLa(A) + C u(bk) = J (Koos 0) ”) where kyo € Koo is the co-component of k It is clear that y is well defined (i.e., trivial on Bo(R)M Koo) Let g = (goo, 1) € SLa(R) x SL¿(A;) with goo = dookoo € Bo(R) Koo Let 21 = 21 +0: := Goi € H Then boo yy”

Indeed, this follows from the fact that we can write ygu = bik, with

Let 62 : Bạ(A) — Rx denote the modulus character of SL2(A) defined by bp = lala * 2

As usual we extend 62 to a function on SLa¿(A) using the Iwasawa decomposition. Note that

We have € ùg,(—1/2, 1), with a local decomposition

Also 63 decomposes into a product of local factors in an obvious way Let f(g) := w(g)da(g)TM”.

We have f = Il, tp € Ip, (23, 1) It is clear that f, is spherical for p oo, but in contrast with examples in section 3.2.1, f, is not spherical In fact one has cos@ sing foo Joo = [lool Joo )e

—sing cosớ imo for g € SLe(R).

We define an adelic Eisenstein series by

7€B2(Q)\SL2(Q) Lemma 3.2.10 Write g = gqgooks € SL2(Q) SLo(R) Kor Then

Proof First note that the natural injection B,(Z) \ SLe(Z) — Bo(Q) \ SL2(Q) is a bijection, hence Em(g) = 3 ”„eứ;(zwsra(z) (Y9) It is enough to prove (3.14) for g = (Goo, 1) For such g, using (3.12), (3.13), and the definition of we get

Em(g) = Gasp jl Reo 1) = = Dean) “Tp aye 7 Foe) 7762.) Els) = J (Joo 1) (B00 2), Em (21):

Remark 3.2.11 In the notation of Definition 3.1.7 we have

Siegel Eisenstein series with positive weight

In this section we define a Siegel Eisenstein series with positive weight, level and non-trivial character For notation refer to chapter 2 Let m, N be integers with m >

0 and N > 0 Note that Ko is the stabilizer of iin G(R) Let ý: KX \ AZ — C* be a Hecke character of AX As explained in section 2.1, ~ has a local decomposition y= II, Wp, where p runs over all the places of Q Assume that and

Up(p) =1 iÍp # œ,z„ € OR,, and z,— 1€ NOKằ.

As before we set x = ]]jIyp Let dp denote the modulus character of P We define a P-automorphic form úp: Mp(Q)Up(A)\ G(A) + C by setting up(g)=0 ifg ¢ P(A)Ko(N) and yup (qh) = (det dạ)”túy (det dy) 2j(heo, i)" for ¢ € P(A), k € Ko(N).

Recall that for g € G(R), j(g, Z) := det(c,Z + đụ) Define a P-automorphic form f(g, 8) := up(g)Šp(g)°2.

We raise ốp to the power s/2 instead of s to make our notation consistent with [Shi97].

Note that f(g, 3) has a local decomposition f = |], fp, where ƒp(g, 8) = tps(8)ð/2(9)

(3.15) Hrp(qpEp) = 9 Wp(det dg,)'Up(detd.,) ifp| N,p # 00,

| Hoo (det dacs) i (Koo, i)7TM if p = œ and

(3.16) bpp uk | = | det Adet Ala,.A —

Thus fp € Ip((s — 1)/2,;ˆ) and if p{ Noo, fy is spherical.

Definition 3.2.12 The series yEP(Q)\G(Q) is called the Siegel Eisenstein series of weight m, level N and character w.

Remark 3.2.13 In the notation of Definition 3.1.7, we have

The series E(g,s, N,m,w) converges for Re (s) sufficiently large, and can be continued to a meromorphic function on all of C (cf [Shi97], Proposition 19.1) It also has a complex analogue E(Z, s,m, w, N) defined by for Z = gol, 9 = 9Q9Gakt € G(Q)G(R)Kos(N) It follows from Lemma 18.7(3) of [Shi97] and formulas (16.40) and (16.48) of [Shi00] that

E(Z,s,m,, N) = ằ wy (det d,)~1 det(c,Z + d,)"TMx (3.17)

Tp (tUU, ww) by

(3.18) (Mov.9)0)(6) = | f0 hug) where ƒ € Ip-(v,w), and (U’)~ denotes the unipotent radical of the parabolic Pa, opposite to P’ The operator M(w,v,w) factors into a product of local intertwining operators defined by an obvious modification of (3.18).

From now on we restrict our discussion to the case when P’ = P, with 6 = A—{a} for some a € A is a maximal parabolic Then a}, c is one-dimensional and the map

37 đc — C defined by 2pp +> 1 (cf Example 3.1.4) identifies a1; with C In this case we write À/(u, s,) instead of M(w,2spp,) There exists a unique element wo € W with wo(a) < 0 such that wo(@) c A ([Kim04], section 4.3).

Definition 3.3.1 Let Q’ be a standard parabolic subgroup of G’ with Levi decomposition Q’ = Mg:Ug: Then the integral

Ugr(Q)\Ugi (A) is called the constant term of E(Po,s,w, f,g) along the parabolic Q.

Theorem 3.3.2 (Langlands, cf [Kim04], Theorem 5.3) Eg: (Po, s,0, ƒ, g) = 0 unless Q! = Pg or Q’ = Py ye) Let ƒ, be as in Definition 3.1.5 If Po # Pua), then

Remark 3.3.3 As asserted before, the Eisenstein series E( Po, s, ý, ƒ, g) has a meromorphic continuation to the entire complex plane It is a theorem of Langlands that its poles are all simple and are the same as the poles of the various constant terms of E( Ps, s,v, f,g) In the next section we prove this fact for an Eisenstein series on the group U(1, 1) and calculate the residue explicitely.

The caseofU(1,l) 0.0.00 00000022 2G 37 4 The Petersson norm of a Maass lift 0 0.000 eee eee 43

Let B, denote the upper-triangular Borel subgroup of U(1,1) with Levi decomposition B, = TU), where

1 Let 6, : B,(A) — R, be the modulus character given by by u | = |aa|A qt for u € UẠ(A) Let K, = Ky denote the maximal compact subgroup of U(1, 1)(A) with a £6 =

F100 = € GLa(C) | |a|Ÿ + |B)? =1, ab ER being the maximal compact subgroup of U(1,1)(R) and Ais = ]],⁄„„ U(1,1)(2;).

As usually we extend 6, to a map on U(1,1)(A) using the Iwasawa decomposition. Set

Remark 3.3.4 The Eisenstein series (3.19) can also be written in the notation of Definition 3.1.7 In fact one has ụ‡ € Ip,(s — 1/2,1) and Euq,)(ứ,s) = E(ỡ,s— 1/2,1, f,g), where f is the constant function with value 1.

It again follows from the general theory of Langlands (but see also [Shi97] for a direct proof in our case) that the series Eu(ii(ứ, s) converges for Re(s) > 1 and continues meromorphically to the entire complex plane We now define a complex analogue of Eyiiiy(g,s) As SL2(R) acts transitively on H, so deos U(1,1)(R) D SL2(R) Hence for every z¡ € H there exists g € U(1,1)(R) such that z¡ = gol. Set g = (goo, 1) € U(1, 1)(R) x U(1, 1)(Ay).

Lemma 3.3.5 Let z, and g be as above Then 6;(g) = Im (2).

Proof Write go = [25] € U(1,1)(R) Then Im (z¡) = Im ((ai+b)(@+d)) |c + d|~? On the other hand [25] = XY, where cI2+ ldị2 |1 at+bd ore (|? + Ia?

Y =——- € te, vie?+Iđ? |e a after noting that |c? + d?| = |e|? + |d|? as cd € R Thus we have

Let z¡ and g be as in Lemma 3.3.5 Define the complex Eisenstein series corresponding to Eva,y (9; s) by

Lemma 3.3.5 together with the fact that the natural injection

By(Z) \ UG, 1)(Z) > By(Q) \ UMA, 1)(Q) is a bijection (cf [Shi97], Lemma 8.14) imply that

+€B:Z)\U4,1),) Let 2; = Re(z¡) and y, = Im(z) As Eua,p(2ìi + 1,5) = Eua,(2ì, s) the series Evy (415 3) possesses a Fourier expansion of the form

Arguing as in the proof of Theorem 1.6.1 in [Bum97] one sees that Ð ”„ o ca (ì, 5) for any fixed ¡ continues to a holomorphic function on the entire s-plane and for any fixed s decays exponentially as y; — oo In particular, for every 6 > 0, there exists a continuous function N(s) such that if y, > 4, then >? 49 |en(y1,8)| < N(s), where N(s) does not depend on y; Hence the poles of #u(¡,¡) (21, s) are the poles of co(yi, 3).

It is an easy exercise that co(yi,8) = Ep,(Bi,s — 1/2,1, f,g) with 2 = go2%, where

Ex, (Bi,s — 1/2,1, f,g) is the constant term of the series in Remark 3.3.4 Hence our conclusion is consistent with Remark 3.3.3.

Lemma 3.3.6 Let z‡ and g be as in Lemma 3.3.5 Then ¢(2s — 1) P (s — 5)

Note that in our case B; = Py is both a minimal and a maximal parabolic and we denote by a the positive root determined by Bj, ie.,

We have pz, = 5a Since the Weyl group of U(1, 1) has only two elements, it is clear that wo = [, T1], where wo is the element defined in section 3.3.1 and B, = Py Pup(@) a8 ệ = 0 Hence by Theorem 3.3.2 we have

We will now calculate M(wo, s,1)ð‡*1?, The operator M(wo,s,1) factors into a product of local intertwining operators

(plore 8.19889") (op) = fh đi)" (and,

=—1 Ấp for u € U;(Q,) It is easy to see (cf e.g., [Tan99], section 2) that

(3.27) Mp(wo, s, Lối, ” = Tope jp for p # oo For p = co we have

Note that since 61,.o(tuk) = 61, (t) for t € 71(R), u € U,(R) and k € Ki we can assume without loss of generality that 9 = [“g-:| € Ti(R) Then

Using formula (3.21) we see that

—1/ |1 2 1 la|~? ổn co Joo = ~ “Mh a+ ]sí8)-P 14 [ae as x € R Thus oo s+1/2 s+1/2 — |l„|—2s=1 !

(3.29) (Mao(wo,s, 1)6152/7) (Goo) = lal / (mm) "

Making a change of variable z +> |a| ”z we see that the right-hand side of (3.29) equals cai (P _ - T)2s+1 1 2\—s 1/2 =ỗ ss(0œ s+1/2 o9 [bate = blo) VE êm

Hence the above calculations together with (3.25), (3.26), (3.27), and (3.28) give

By (Bu,s,1, f9) = bilo) + VR EAL ĐÓ 5 (gyro (28+ 1)T (s+)

Using formula (3.24) we obtain the lemma L]

Corollary 3.3.7 The series Euq (2z, s) has a simple pole at s = 1 with residue the constant function with value 3.

Proof As remarked before Lemma 3.3.6, the difference Eyc1,1)(21, 8) — €o(ì, 8) continues to a holomorphic function on the entire s-plane Hence we have

The Petersson norm of a Maass lift

The goal of this chapter is to express the denominator of Cp, in formula (1.2) by an L-function of f We start out by defining the lifting procedure (section 4.1) and then carry out the calculation of the Petersson norm (#;, Fy) in the following sections.

Maass lifts 2 Q nu ng g g v v v v v kg kg va 43

Let H, as before, denote the complex upper half-plane The space H x C x C affords an action of the Jacobi modular group IY := SL2(Z) x Ó?, under which ([¢5],,) takes (7,z,w) €Hx Cx C to (44, 4), 4).

@:HxCxC3C is called a Jacobi form of weight k and index m if for every [2°] € SL¿(Z) and a b czw ar+6 # 10

P= ðlum c d = (er+4) -{ me) Pm (3555) and ở = ó|m|À, H] := e(mAA + Az + AW) bm(T, z + AT + pw + À£ + fi).

Let k be a positive integer divisible by 4 We denote by N the set of positive integers and put No := NU {0} Let F be a hermitian cusp form of weight k and full level By rearranging the Fourier expansion of F we obtain

(4.1) F(Z) = ` bm(r, 2, w)e(mr*) meN where Z = [7,2] € H and

1ENo t€ZOK tom tt fo(r) gives an injection of J¿¡, the space of Jacobi forms of weight k and index 1, into S;,_1 (4,(=*)) If we put = ¢; and define ó by Ớ|¿-i[¿ `] = fo, the composite F ++ ¢,(7,2,w) + fo(7) > ¢ gives the isomorphism alluded to in Theorem 4.1.3 We will refer to it as the Maass correspondence Denoting this isomorphism by 2, we can map any normalized Hecke eigenform f = Sasi b(n)q” € Ses (4, (=4)) to the element Fy := Q~!(ƒ — f?) € Sÿ22%(T7).

Here f? = ons O(n)qQ” The Maass correspondence is Hecke equivariant in a sense, which will be explained in section 5.2.4 Note that Fy = —Fyằ and Fy # 0 if and only if ƒ # f?.

Definition 4.1.4 If ƒ # f?, then Fy 4 0 is called the Maass lift of f or the CAP lift of ƒ.

Proposition 4.1.5 If f = 30,5, b(n)4" € Sp_1 (4, (=4)) is a normalized eigenform, then say 6(n) — bí) iÊn #1 (mod 4)

Proof This follows from formula (4) on page 670 in [Kri91] O

The Petersson normof Fp Q Q HQ HQ HQ ee, 45

Our goal in this section is to express the inner product (F;, Fy) in terms ofL-functions associated to f Let F(Z) = 3 m>o ÓmÍT, z,0)e(mT7”) and F(Z) mee Um(T, Z, w)e(mr*) be the Fourier-Jacobi expansions of two hermitian modular forms F and F” of weight k and full level Raghavan and Sengupta [RS91] consider the following Dirichlet series:

Drr(s) = Cx(s —k+ 3)c(2s — 2k+ 4) ằ (Om; Um) m Ÿ, m>0 where for two Jacobi forms ¡ and ta of weight k and index m, we put k —xm|z—1ứ12

(Úi, 92) = vr(7, 2, +U)2(T, Z, w)v € i du, gs with 7 = utiv, Z = Zo + 021, W = Wot 000, U,V, 2%, 21,Wo,Wi € R and du v~‘du do dzp dz; dứa dw; The integration is over a fundamental domain Z” for the action of TY on H x C x C defined in section 4.1 Here ¢(s) denotes the Riemann zeta function and ¢x(s) the Dedekind zeta function of K.

Theorem 4.2.1 (Raghavan, Sengupta, [RS91], Theorem 1) The Dirichlet series Dr p(s) can be meromorphically continued to the entire s-plane The function

Dp p(s) = (4n) ”°TÍ(3)I(s — k + 2)P(s — k + 3)Dr,r(s) is holomorphic in s except for possible simple poles ats = k— 3, k— 2, k— 1, k and satisfies a functional equation Dy p(s) = Dip (2k — 8 — 3).

To derive the desired formula for (Fr, Fr) we will use an identity proved in [RS91] that involves a variant of the Klingen Eisenstein series, which we now define The relevant properties of the Klingen Eisenstein series were discussed in section 3.2.1.

For a 2 x 2 matrix M, denote by Ä/;; the (2, 7)-th entry of M Let C be the subgroup of ['z consisting of all matrices whose last row is 0 0 0 1) We will later relate

C to the Klingen parabolic subgroup of G Set

The series converges for Re (s) > 3 ([RS91], Lemma 1) In the course of the proof of Theorem 4.2.1 Raghavan and Sengupta establish the following identity

They also prove that E*(Z) can be analytically continued in s to the entire complex plane except for possible simple poles at s = 0, 1,2, 3.

The goal of the next two sections is to prove the following theorem:

Theorem 4.2.2 The function E* has a simple pole at 3 with residue (independent of Z) equal to +c@).

Assuming Theorem 4.2.2, we can relate (F;,F;) to the symmetric square L- function L(Symm? f,s) of f using the following formula due to Gritsenko [Gri90al:

Here we define L(Symm? ƒ, s) for a normalized eigenform f = 5>°2, a(n)g” as an n=l Euler product:

_ —g —sv1—1 x I] [q — 21p *)(1 = apiapap ` ®)(1 — Q2 2p )| pz⁄2 where the complex numbers a, and apằ are defined by the equation

By Fact 2.3.1, we have a(p) = (=) a(p) for p+ 2 This implies that a(p) € R if p splits in K and a(p) € iR if p is inert in K It follows that @j7 = ap, for split p and đp1 = —Qp.2 for inert p We will call ay and œ; › the classical p-Satake parameters of f to distinguish them from the “representation-theoretic” Satake parameters which we will define in chapter 9.

Combining formulas (4.5) and (4.7) with Theorem 4.2.2 we obtain:

Finally, to relate (ở, 61) to (f, f), in section 4.5 we will prove the following lemma.

Lemma 4.2.3 The following identity holds:

Combining Lemma 4.2.3 with formula (4.9) we finally obtain:

Theorem 4.2.4 The following identity holds:

Residue of the Klingen Eisenstein §@Ties 0.2 eee eens 48

This section will be devoted to proving Theorem 4.2.2.

Proof of Theorem 4.2.2 We will now calculate the residue of the Klingen Eisenstein series defined in section 3.2.1, and at the end relate it to the residue of E,(Z) Once we have the residue of F(Z), the residue of E(Z) can be determined by the formula (4.6) For notation refer to chapter 2.

Let Ko = Kooko, be the maximal compact subgroup of G(A) First note that

Bc Q where B = TUg denotes the Borel subrgoup of G and Q = MgUg the Klingen parabolic This embedding on the level of Levi subgroups and unipotent radicals is explicitely given by:

We now define four characters on Mp(A) and Mg(A) The first two are: the modulus characters ốp and dg defined by formulas (3.7) and (3.8) respectively The other two are defined as follows: Let l “ , € Mo(A) Since [25] € U(,1)(A), we can use the Iwasawa decomposition for U(1,1)(A) with respect to the upper- triangular Borel to write [¢5] = [* 2,]k with k € Ki, where Kj is as in section

3.3.2 Note that if k = [i 2 I, then tr BỊ € Ko We define 1 kạ ka by

1 aol Finally the fourth character gp: Mp(A) — Ry is defined by:

DL * ky ko y ks kg _ù—— op = |zy (zy )la, zt kị ks xa ký ki | j where we used the Iwasawa decomposition for GL2(Ax) = Resx/q GL›(A) with respect to its upper-triangular Borel Br, and its maximal compact subgroup KR U(2) [Trico GL2(Ox,) to write A € GLo(Ax) as ol ki ko

We again have Ra ka kk | € Ko. kl, ki,

It is easy to check that all four characters can be extended to the adelic points of the corresponding parabolics by setting them to be trivial on the parts of the unipotent radicals Then using the Iwasawa decompositions

(4.16) G(A) = B(A) Ko = P(A) Ko = Q(A) Ko the characters can be extended to functions on G(A) One also verifies that

(4.17) 535% = 6o o = 64740) for any complex numbers s and z Let

+ 2/3 and Re (2z) > 1/2 and admits meromorphic continuation to all of C2 Using identity (4.17) and rearranging terms we get:

Es(g.s,z):= ` ôq(xd) "7 ` da(eaxg)”= 2 2

+eP(Q)\G(Q) aeB(Q)\P(Q) Let Eu(,;(ứ, s) be the Eisenstein series on U(1,1) defined by formula (3.19) We also define an Eisenstein series on Resx/q GL2(A) by:

+€Bn(Q)\Resx/a GLa(Q) where dp denotes the modulus character on Bp defined by:

On the A-points we can extend 7g to a map G(A) — U(1,1)(A)/K¡ and mp toa map G(A) — Resx/q GL2(A)/Ke by declaring them to be trivial on Ko Hence we can rewrite formulas (4.19) as

(4.24) Es(g.sz):= ` ụa(xứ)°*fEvu(ma(xe).2z) =

As remarked in section 3.3.2 the series Eyii1)(g,s8) is absolutely convergent for Re(s) > 1 and admits meromorphic continuation to the entire s-plane By Corollary 3.3.7 it has a simple pole at s = 1 with residue `,

Proposition 4.3.1 (i) For any fixed s € C with Re (5) > 2/3 the function Ep(g, s, z) has a simple pole at z = 1/2 and

3 where Eo(g, s} is the adelic Klingen Eisenstein series (cf Definition 3.2.4).

(ii) For any fixed z € C with Re(z) > 1/2 the function Ep(g, s, z) has a simple pole at s = 2/3 and

(4.26) TeS,_2 B(g, 8,2) 602) 7 (0 5 + :) where Ep (g, z) is the adelic Siegel Eisenstein series (cf Definition 3.2.1).

We will prove this proposition in section 4.4.

We are now going to show that Eg(g,s) has a pole at s = 1, determine its residue and relate Eo(ứ, s) to E;(Z), which will allow us to calculate the residue of the latter.

Lemma 4.3.2 The Eisenstein series ERsy,o GL2 (g,s) is absolutely convergent for Re(s) > 1, admits meromorphic continuation to the entire s-plane and has a simple pole at s = 1 with residue z2

Proof The proof is very similar to the proof of the analogous statement in the case of the group U(1,1), so we omit it here The latter proof has been carried out in detail in section 3.3.2 O

Using identity (4.24) together with Proposition 4.3.1 and interchanging the order in which we take residues we obtain:

(4.29) res, _ and thus we finally get

All that remains now is to relate Eo(ứ, s) to E;(2).

Lemma 4.3.3 Let g = (goo,1) € G(A) and Z = gx„ì Then

Proof First note that det Im yZ )

E,(Z)=4 ——=—]›:4) > (Gmtin where C’ is the subgroup of Ứz consisting of matrices whose last row is of the form

0 0 0 a with a € OF Moreover we have C’ = wQzw7! with tu = | and 1

——_—_ = = = yeC\Pz (Im +Z2)in +eQz\Tz (Im 0u Z)in

+€Qz\Tz (Im yw! Z) 2.2 +€Q9z\Tz (Im 1)ằ2 as tu € Ủy.

Now for y € Tz we have ổo(y9) = ôổo(qg), where g = (goo, 1) and Goo = qookoo c with do € Q(R), ko € Kow If do = um with m = _— i € Mg(R) and d u € Ug(R), then ðo(9) = 5q(um) = ôq(m(m~”um)) = ôq(m) = |aj.

A direct calculation shows that det Im u(mi) = detImmi and that (Im u(mi))o2 (Im mi)o2 On the other hand

Immi ee _ (ci+d)(ci-+d) hence we have det Im yZ

The lemma now follows from the fact that the natural injection

Qz \Tz — Q(Q) \ G(Q) is a bijection This is a consequence of the identity Q(A) = Q(Q) Q(R) Q([ptoo 2). which follows from Lemma 8.14 of [Shi97] O

Using Lemma 4.3.3 and formula (4.30) we conclude that E,(Z) has a simple pole( at s = 3 with residue - on mat {= `) and hence that E7(Z) has a simple pole at )L(3,(=) ) you) - +,¢(3) This finishes the proof of Theorem

Eo(g,s) as a residual Eisenstein series 0-00 200004 56

This section is devoted to proving Proposition 4.3.1 We will only present a proof of part (i) of the proposition as the proof of (ii) is completely analogous In what follows Z will denote a variable in the hermitian upper half-plane H, and z¡ a variable in the complex upper half-plane H Otherwise we use notation from the previous sections As before we denote by Ko the maximal compact subgroup of G(A) Write g = ứqứk € G(A) with ga € G(Q), go € G(R) and k € Ky We have Èp(g,s,z) = Ep(go,8,2) and Eo(g,s) = Eg(go,8), hence it is enough to prove (4.25) for g = (ứứs, 1) € G(R) x G(A¿;) Let Ay denote the maximal compact subgroup of U(1,1)(A) and let mg : G(A) — U(1,1)(A)/K be as in formula (4.22).

Lemma 4.4.1 If g = (goo, 1) € G(A), then Im (zo(g)2) = Im (g+Ì)2a.

Remark 4.4.2 Note that for any 2 x 2 matrix M with entries in C one has

Im (M22) = (Im(M))o2 Hence the conclusion of Lemma 4.4.1 can also be written as Im (ro(g)?) = Im ((gooi)2.2) (cf the proof of Lemma 4.3.3).

Proof of Lemma 4.4.1 Using the Iwasawa decomposition for G(R) we can write

Jo = umk € Ug(R)MQ(R)Ko,00 with m = _ ) As Ua(R) Mg(R) = Mo(R) Ue(R) we have 79(g)i = (ai + e d b)(ci + d)~", hence Im (779(g)i) = |ei + d|-?.

On the other hand gi = umki = umi A direct calculation now shows that

(Im (umi))o2 = (Im (mi))22 = |ci + dị”.

Lemma 4.4.3 For any Z € H, there exists y € Q(Z) such that (Im7Z)oo > š. hư

Proof Since SL2(Z) C U(1,1)(Z), for every Z € H there exists +' € U(1,1)(Z) such that 7'(Z.2) belongs to the standard fundamental domain of SL2(Z) on the complex upper half-plane H In partcular Im(7(Zo2)) > 3 Suppose 7 = [5] Then 2 d

+:= | a, | € Q(Z) and a simple calculation shows that 1 d e

The lemma follows by Remark 4.4.2 LÌ

Lemma 4.4.4 For every Z € ?(, we have sup Im(yZ) < co.

+€Tz Proof This is just an easy adaptation to the case of hermitian modular forms of the proof of Hilfsatz 2.10 of [Fre83] L1

Proposition 4.4.5 Let 6 > 0 and g = (0+, 1) € G(R) x G(A¿) For everys EC with Re(s) > 1 + ỗ and every z € C with |z — 3| < 6, the series

(4.33) D:=|z-1/2| SY) lóa(xứ)**”8Euu(na(9).22)|

Proof It follows from [Shi97], Lemma 8.14 that the natural injection

Q(Z) \ G(Z) = Q(Q)\ G(Q) is a bijection Hence in (4.33) we can replace >) cgay c(qy With dD yeq¢z)\ cz): Moreover, as shown in the proof of Lemma 4.3.3 det Im(yZ) _ 1/3 dim Z))aa 0909) where Z = gai, y € G(Z) For g’ = (7,1) € U(1,1)(R) x U(1,1)(A¿) let z¡ Joot € H For z’ € C define a complex Eisenstein series by setting

The series EZu(i,+)(2¡, 2’) was investigated in section 3.3.2 where we showed that

In our present case (4.34) leads to

Eua,b(ma(39).2z) = Eu (te(Vg) ot, 22).

Hence using the same arguments as in the proof of Lemma 4.3.3 we see that b= X (qaieaZ)\""

As 9 = (Goo, 1) and y € G(Z) C Ko, we have 7o(9)s = o((Yứỉs;1))s By

Lemmas 4.4.1 and 4.4.3 we can find a set S of representatives of Q(Z) \ G(Z) such that for every y € Š we have

As discussed in section 3.3.2 the series Ey(i,1)(21, 22) has a Fourier expansion of the form

Eva, (21,22) = › e„(2z, Im (z 2minRe (1) neZ and Ey1,1)(21,2z) — co(2z,Im (z¡)) for every fixed z¡ continues to a holomorphic

? function on the entire z-plane and for every fixed z is rapidly decreasing as Im (z¡) — oo It follows that for any given N > 0 there exists a constant M(N) (independent of z¡ and independent of z as long as |z ~ 1/2] < ở) such that |Euqa,(2ì, 22) — co(2z, Im (z¡))| < M(N) as long as Im (z¡) > N Set

Taking N = 1/2, we see by formula (4.35) that there exists a constant M (independent of +) such that | Bui) (ty + iyy,2z)| < M + |co(2z, yy)| By Lemmas 3.3.5 and 3.3.6 there exists a positive constant C' independent of z and of + such that

|z — 1/2|lco(2z, yy)| < dys? + Cy, for jz — 1/2| < 6 Since y, > 1/2 and |z ~ 1/2} < 6 we have two cases: either 1/2 < yy < l or y, > 1 In the first case |z — 1/2||co(2z, x)| < C’ for some positive constant C’ and in the second case |z — 1/2||co(2z, wx„)| < |y,|!*?° Hence in either case we have

Thus we conclude that there exists a positive constant A such that

For s’ € C lying inside the region of absolute convergence of F(Z) let det Im (yZ) \*

|E|e(Z) := ằ (= ae) yEQ(Z)\G(Z) denote the majorant of #;(Z) By formula (4.36) we have det Im (y2) Sete

Note that |E|3,42.(Z) is well-defined (i.e 3s + 2z is in the region of absolute convergence of #„(Z)) by our assumption on s and z Denote the second term of the right-hand side of formula (4.37) by Do Then det Im (+Z))!??.

By Lemma 4.4.4 there exists a constant M(Z) such that detIm(yZ) < M(Z) and hence

Dạ < AM(Z)'*” | E]gs492-(1428) < 00 as Re (3s + 2z — (1 + 26)) > 3 by our assumptions on z and s This finishes the proof L]

Proof of Proposition 4.3.1 We need to show that for a fixed s € C with Re (s) > 2/3 and for every € > 0 there exists ở > 0 such that |z — 1/2| < 6 implies

As remarked at the beginning of the section we can assume without loss of generality that g = (go, 1) € G(R) x G(A¿) We first show that (4.38) holds for s with Re(s) > 1 Fix s € C with Re(s) > 1 and 6’ > 0 such that 0 < ở < Re(s) — 1. From now on assume |z — 1/2] < ở Fix a set S of representatives of Q(Q) \ G(Q).

By Proposition 4.4.5 and the fact that Eo(g,s’) converges absolutely for s with

Re (3) > 1, there exists a finite subset S; of S such that the following two inequali- ties:

Z— 1 | ụa(%ứ9)°†?3Euu,(a(ứ),2z)| < mo are simultaneously satsfied Here Sz denotes the complement of S, in S We have D(z) < D,(z) + Do(z), where

D;(z) := (: — 5) À ` ụo(xứ)°”?!8Eva(v(39),2z) — 5 À ` ụo(xg)°?18|,

Note that if we replace 6’ with a smaller 6” > 0, then estimates (4.39) and (4.40) remain true as long as |z — 1/2] < 6” for the same choice of S; Hence we find 6 > 0 with 6 < ở such that D¡(z) < § This is clearly possible as D,(z) is a finite sum and 3/27 is the residue of EZu(q,(mo(ứ), 2z) at z = 1/2 by Corollary 3.3.7 On the other hand Do(z) < Ds(z) + Da(z), where

D3(z) = oo ủ | ụa(+9)°°?'3Euaa(ma(x3).22)| 1 and

D„(z) := = ằ |(đa(sứ))°?%I : yES2 Formulas (4.39) and (4.40) imply now that D3(z) < e/4 and ¿(z) < €/4 Hence

D(z) < Dy(z) + Do(z) < Di(z) + D3(z) + Da(z) < < as desired.

We have thus established the equality resz~1/2#/g(g, s, z) = 2 EQ(9, s+ 1/3) for s with Re(s) > 1 However, both sides are meromorphic functions in s and since the right-hand side is holomorphic for Re(s) > 2/3, so must be the left-hand side.Hence they agree for Re(s) > 2/3 O

Inner product formula for Jacobi forms 00 0002 ee eae 61 5 Hecke operators 1 1 HQ ng Quà ng kg gà gi kg sa 65

This section is devoted to proving Lemma 4.2.3.

Proof of Lemma 4.2.8 Let \ and w2 denote two Jacobi forms of weight k and index m It is easy to show that

(4.41) (wr, ve) = [ *( w1(7, z, 10)2(T, Z, we = dz dz, đua dus) du dù,

F Fy where F is the standard fundamental domain for the action of SL2(Z) on the complex upper half-plane and Z; C {7} x C x C is a fundamental domain for the action of the matrices F 0 | and h d ul (A, € Ox) on C x C After performing a change of variables on C x C (keeping 7 fixed) zZ=z+U w=z-u, and denoting by Z7; the fundamental domain Z7; in the new variables, the integral over F, in (4.41) becomes

8 Je tì, 2’, w')pa(7, 2/, w’)ore D dz dz, dwy dw max.

Set v1 = wo = ị, where ở is the first Fourier-Jacobi coefficient of the CAP form F,; Using formula (4.3) we can write:

(4.42) (di, 1) = sob [ae )f#u(r)u~*®1(t,,T) du du teA cA with

(4.43) Fr `aet+O bet’ +OxK xe 9 (lm (Re wy) đạo dz duc dwy.

Changing variables again we get

(4.44) I(t,t.r)= So Soe N(d)7) Ly Ip a€t+OxK bet’ +OK with h= | e(2x’Re (a) — 22’Re(b)) e52*1) đa? dri,

Hi where (] is the parallelogram in C spanned by the two R-linearly independent complex numbers 1 and 7, and zÍ = zạ +Zz¡ € C, with 2,7, € R Before we define

Iz we note that J; can be written as

(4.45) = | cv 41) (| e(2z “Re (a) — 2zRe (b)) ax, dr’.

Now the integral inside the parantheses in (4.45) equals e-#”Re (9): if Re (a) = Re (b) and 0 otherwise Hence

4 [im c~45@¡)? e-87Re (4) #1 de! if Re (a) = Re (b)

0 if Re (a) # Re (0) The integral

Q2 where {22 denotes the region in the complex plane spanned by the two R-linearly independent complex numbers 1 and —7 and y’ = ¿ + iy, € C with yg, y, € R, can be handled in a similar way In fact one gets:

0 if Im (a) # Im (6) Substituting (4.46) and (4.47) into (4.44) one sees that /(f,f,7) = 0 if t # t’, and that after rearranging terms

(4.49) (ởì, dị) = / ` #ứ )#,(} U*~* du du du.

F teA From this it follows that 7,0, N >0 be integers Let ý : A} — CX be a Hecke character, which we think of as a character : (Resx/q GL) (A) — CX We have a factorization = I], tứ; into local characters Wp : (Resx/q GL¡) (Q,) + CX Assume that w is such that and

Up(p)=1 ifp#o, zp€ểX,, andz,—1€ NOxằ.

As before we put Yy := [],)1 Up We will sometimes abuse notation and write wy for the character qy : (Z/NZ)* — C* which makes the diagram

Recall that in section 2.2 we defined

TẠ(N):= cTz|A,B,De Mạ(Oy), C € M;(NOy)A B

C D Denote by M=>(N,w) the space of functions F : H — C such that

F\|my = wn (det a,)F = wn (det d,) 1 F for all y € 2(N) Note that we do not require F to be holomorphic.

Example 5.2.1 Let (2, s,m, ú, N) be as in section 3.2.4 Then E(Z,s,m,w,N) €

We now proceed as in section 5.1 For details, see [Gri90a] Let

For a € Ab(N), the double coset space Tầ(N)aIÿ(N) decomposes into a finite disjoint union of right cosets

ThN)ar BN) = ]]Tš(09)4, with a; € AR(N) One could define R'(N) as the Z-module generated by the double cosets Tộ(V)a[(N), but this does not give the correct Z-module structure on the Hecke algebra We will return to this issue later Instead we define R®(N) to be the C-vector space generated by distinct double cosets [2(N)aI2(N) There exists a C-linear map

Rh(N) = Endc(MẸP(N,9)) given by

(Fầ(M)aTi(M)]|Ƒ = F|m[T8(N)aT8(CM)| = À ` bw (det das) F|maj. j

We denote the image of this map by T®(m, N, w).

Definition 5.2.2 A function F € M2®(N, 0) is called a Hecke eigenform or simply an eigenform if it is a simultaneous eigenfunction for all T € Thứn, N,v).

For a rational prime p we define operators

Lemma 5.2.3 Ifp{N we have the following decompositions:

(5.7) tt xuðeZ/p pả bEOK /p h ploy h KT

+€Z/p - 1 deOx /p " P yEZ/p for any prime p, re=tyn) | r, | TRÚM 1 la ỉ8 * T Trụ

58) = JJ ủnm|?? "fu Tp nw[ 7l o,Ô,+€Z/p TTA TS ãccZ/p P

U H TẠ(N) ng LITR(M) | ": $€Z/p 7 for p= TT split or ramified in K and

(5.9) Pl œéOy/p -@1 la B PS PỊ

U JT | ? 2 Ju Jf n0) | 7| œ€Okr/p —ðp p +eOkyjp P

BEZ/p? EZ /p for p inert in K.

Proof For the decomposition of 7” see [Kri91], section 7 The decompositions of 7? and of T?, can be obtained in a similar way We omit the calculations.

We now describe the action of the operators T?, TÌ, and T® on the Fourier coefficients of hermitian modular forms From now on we deal exclusively with holomorphic forms.

Lemma 5.2.4 Let FE Mn(N,w) with Fourier expansion

(5.10) crpr(B) =p" “*cr(p *B) + (Np)p ”ep(pB)+

3 | Q — where Np denotes the norm of the ideal (p) Moreover, for p = 1T split or ramified ink, pt N,

TT T7 ocZ/p Hi Wœ 7 and for p inert in K, pt N,

Dp 1 aeOry/p —a@ Ì Pp Proof This follows directly from Lemma 5.2.3 L]

Remark 5.2.5 Note that it follows from Lemma 5.2.4, that the Fourier coefficients of T?F and of TF for p split or ramified need not be algebraic integers even if the

Definition 5.2.6 Whenever the weight m is understood from the context, we set T® to be the image of T"(m,1,1) inside the space of C-linear endomorphisms of Sm(Tz), the (holomorphic) hermitian cusp forms of weight m and level 1.

Proposition 5.2.7 (Gritsenko, [Gri90a]) For any split or ramified prime p = ni set) := {12, Tệ, Tạ} and for any inert prime p, set Xi, := {Tp,TP,} The Hecke algebra T® is generated as a C-algebra by the set U, =p.

Satake parametrization 2 2 2 ee ee 74

From now on we fix m and assume that N = 1 We will use the Satake homomorphism (1.e., the map Sat, defined below) to realize T® as a subring of a polynomial ring over C This leads to a notion of classical Satake parameters We emphasize the word classical to distinguish them from the representation-theoretic Satake parameters which will be introduced in Chapter 9 For details we refer the reader to[Gri90b] and [HS83].

First let p be a prime inert or ramified in kK We define the Hecke algebra TEằ for G,,(Z,) to be the C-vector space consisting of finite sums }7 caG,(Zp)aG,(Z,p) with cy € C and the usual law of multiplication (cf [HS83]) We define a C-linear map

Sat;: TÔ „ => CÍZo,1,2,g ,21 ,Z2 | as follows: write Œ„(2g)aG,(Z„) as a disjoint union of right cosets ||, G,(Zp)a; with p Thẻ pes

Satp(G,(Zp)aG.(Zp)) = ằ xq” (é ứ\) “2 (p 2;)°22

(cf [HS83]) Every eigenform F € S,(I'z) induces a C-linear map Arc : Tệ„ > C sending to T € TEằ to the eigenvalue of T corresponding to F’ This map extends to a C-linear map

The complex numbers À;;(F) := Àpò(Z;), 7 = 0,1,2 are called the classical p- Satake parameters of F If F is clear from the context we sometimes write A); instead of Ap j(F).

Now consider a split prime p = 77 Let 7, and ig denote the two different embeddings of K into Q, We identify G,,(Q,) with the group Go = {(91,92) € GLi(Q,) x GLI4(Qp)|lứJứ) = ủ((:.92))J} via the map a @ 1 + (4:(a), (a), a € K We then identify the group Go with the group GL4(Q,) x GL1(Q,) via

(91,92) > (91, 0((ứỡ, 92))) Similarly G(Q,) is identified with GL4(Q,) (It is con- venient to introduce a change of variable, which makes the image of G,,(Q,) in GL4(Q,) x GL4(Q,) take the form (91, ((ỉi, 92))(g{)~") (cf [Gri90b}).

In this case the Hecke algebra TZ, , of G„(Q;) = GL4(Q,) x GL¡(Q;) is generated by double cosets Tal, with a = (ai,ứs) € GL4(Q,) x GLi(Q,) and ẽ' = ẽỡ xT, with [; = GL4(Z,) and T, = GL¡(Z¿) Inside TỀ,, is the Hecke algebra TG, for G(Q,) = GL4(Q,) generated by double cosets [ứĂ['; In fact we may identify

TEằ with a polynomial ring over TS, in indeterminates zp and zp‘, by mapping

TY, x Pe(ai, @2)Ty x Dạ = (TiaiP,)(T2a2F2) to the element TiaTirp re),

On the other hand there is a ring injection TS, — Clx1, v2, 23,24] defined by Daly = [[Tib¿ ye, Ti (p7'21)*3, where 8; € GL4(Q,) has the form pe * * * | pers * * ps

As before each eigenform F € S,,(I'z) determines a set of classical p-Satake parameters of F which we denote by Apj, 7 = 0, ,4.

Remark 5.2.8 A similar description holds for the elliptic Hecke algebra and one obtains in this way (for every p not dividing the level of the elliptic eigenform in question) the classical Satake parameters, which we denote by œ„ and apằ They agree with the Satake parameters defined in section 4.2 We omit the details.

Integral structure of the hermitian Hecke algebra

From now on we take m = k, a positive integer divisible by 4 We now define an integral structure on Th, Let F € S,(['z) be an eigenform For every rational prime

77 p, let Àp;(#') denote the classical p-Satake parameters of F’ Let p be a prime of Ox lying over p Set

Apu(F) = (Np) OP a" (ppg (F), where w is the unique Hecke character of K unramified at all finite places with infinity type Woo(Zoo) = 2)”, See also Remark 6.2.4 Zz

Definition 5.2.9 The elements Aằ;(F) will be called the Galois-Satake parameters of F atp.

By Theorem 10.1.3 (which will be discussed in section 10) to every eigenform

F € &(Tz), one can attach a Galois representation pr: Gx — GL,(Q,), such that pr is unramified at primes p which do not divide 2ý and for such p the set of eigenvalues of pr(Frob,) coincides with the set of Galois-Satake parameters of F at p Let órg(X) be the characteristic polynomial of pr(Froby,).

The coefficients a; can be determined from the explicit form of the map Sat, (cf. [Gri90a], section 2) One has

Ga = ],where Àrc(7) denotes the eigenvalue of T corresponding to the eigenform #'.

Proposition 5.2.11 The space Š„(Úz) has a basis consisting of eigenforms.

Proof This is a standard argument, which uses the fact that T® is commutative and all T € T® are self-adjoint L]

Example 5.2.10 and Proposition 5.2.11 motivate the following definition of the integral structure on T® For a split or ramified prime p = Z7 set and for an inert prime p set up := {77T}.

Definition 5.2.12 We set Tỷ (resp ToT) to be the Z-subalgebra of T” generated by U, 2p (respectively by U4 Up) For any Z-algebra A, we set Th := Tỳ @z A and Tt := Tht @; A.

Note that by Proposition 5.2.11, T® is the direct product of finite number of copies of C, and hence torsion-free Thus Tỷ is also torsion-free Note that for a split or ramified p the elements of 3„ differ from the elements of 3, defined in Proposition 5.2.7 only by multiplicative constants (but it is these constants that will make the action of Tỳ on the space of hermitian modular forms preserve the integrality of Fourier coefficients) In particular, one has T2, = T" Hence Tỷ is a finite free Z-algebra.

Lemma 5.2.13 Let £ be a rational prime, E a finite extension of Q¿ and O the valuation ring ring of E Suppose that F(Z) = })scr(B)e?ntr(P?) c S,(Tz) with cr(B) € O for all B Let T € TS Then TF(Z) = })pcrr(B)e?H(52) with crr(B) € O for every B.

Proof This follows directly from Lemma 5.2.4, Proposition 5.2.11 and Example 5.2.10 oO

Let AN denote a basis of eigenforms of S; (Tz).

Theorem 5.2.14 Let F € NTM There exists a number field Lp with ring of integers O,, such that Arc(T) € O1, for all T € Tô, Here Àrc(T) ts the eigenvalue of

Proof This is similar to the Eichler-Shimura isomorphism in the case of elliptic modular forms LÌ

Let £ be a rational prime and E a finite extension of Q¿ containing the fields Dr from Theorem 5.2.14 for all F € N* Denote by Ó the valuation ring of E and by À a uniformizer of O.

Definition 5.2.15 Let F € N° Then F gives rise to an O-algebra homomorphism

TỀ, — O assigning to T the eigenvalue of T corresponding to the eigenform F We denote this homomorphism by Ap Furthermore, we denote by Àz the composition of Ar with the natural projection O + F := Ó/À, and set mp := ker Àz.

Theorem 5.2.14 implies that we have

The O-algebra T}, also decomposes as a product of local rings This is completely analogous to the ellitpic modular case discussed in section 5.1 In particular the ideals mp (for F €.N*) exhaust the set of maximal ideals of T3, We have

To = [[ 73.0: hw h m where the product runs over the maximal ideals of TZ and T3 „ denotes the local- ization of TỀ, at m A similar description holds for TR, This proves the following corollary.

Corollary 5.2.16 If F € N® then there exists T € 'TÈ such that TF = F andTF’ = 0 for all eigenforms F” such that mp 2# mp, 0€, with not all eigenvalues congruent to those of F modulo À.

Action on the Maassspace 0000 cee eee eae 80

Theorem 5.2.17 (Gritsenko, [Gri90a], section 2) The action of the Hecke algebra T® respects the decomposition of S;,(Uz) into the Maass space and its orthogonal complement.

Theorem 5.2.18 (Gritsenko, [Gri90a], section 3) There exists a C-algebra map

Desc : Th > Tt such that for every T € T® the diagram

SMeass (Dz) —+— S55 (Tz) all [mrs Sear (4, (9) 45,1 (4, (=4)) commutes In particular one has

Desc(7.,) = p* “(1+ ứ?)T;z + p”~Š(p° + p?+p— 1) if pis inert in K, (5.14) Desc(T) = p*-?a-*(1+p)T, ifp= z7 is split in K,

Here T,, denotes the elliptic Hecke operator (5.1), and TY Tạ denotes the operator from Definition 5.1.9.

Corollary 5.2.19 If f € Š;_¡ (4, (=*)) is an eigenform, then so is Fy.

Proof This is immediate from Theorem 5.2.18 L]

Lifting Hecke operators to the Maass space 81 5.38 Adelic Hecke operators 2 ee va ga ga kg va 84 5.38.1 Localtheory HH gà ee 84

Let E and O be as before We will now prove a result regarding the map Desc, which will be used in section 7.4 Let Tz and T7 be as in Definition 5.1.9 For a discussion of the relevant properties of TZ, refer to section 8.1 It is clear from

Theorem 5.2.18 and the definition of Tz” that Desc(T')*) = T’, for any O-algebra

A Moreover, we have the following diagram

[n+ Tone — Tw Tom with the lower horizontal arrow defined so that the diagram commutes Let NV, as before, denote the basis of normalized eigenforms of S,_1 (4, (=*)) Recall that for f ¢N, f & ƒ? if and only if Fy # 0 The set VTM := {F, | g € N} is a basis of eigenforms of S)#5([z) by Corollary 5.2.19 Here, the superscript ‘M’ stands for ‘Maass’ Fix ƒ € MN with ƒ # f? Note that for g € N, m¿ = m¿ if and only if mp, = mỹ Here m; C To is the kernel of À¿|r„ (cf Definition 5.1.6), and mp, is the kernel of Àrlrs+ (cf Definition 5.2.15) The ideals m, and mp, are defined similarly All four are maximal ideals Hence Desc respects the direct product decomposition in diagram (5.15) In particular, Desc : T+ -ằ Tớ, factors through Tot, > Tomi, Let T3* be the image of TỆ” in Endoc(6}22%(Tz)).The horizontal arrows in diagram (5.15) factor through TB & ]]a Tàn and the following diagram

| | ⁄ | Met Toit —— la TO — Tw To, commutes All the horizontal arrows in diagram (5.16) are surjections and the lower ones are induced from the upper ones, which respect the direct product decompositions The maximal ideals m7, m and m’ are defined in an obvious way In particular we have

Let NM! := {F € NTM | up, = tir} The goal of this section is to prove the following proposition.

Proposition 5.2.20 If fe N, ƒ # f? is ordinary at £, and £ > k, then for every split prime p = 17, pt £, there exists Th(p) € Toa such that Desc(T*(p)) € Tí, mở , Ff 3 equals the image of Tp € 'Tọẹ under the canonical projection To —> Tomi,’

As remarked before to every eigenform #` € Šx(ẽz) one can attach a 4-dimensional é-adic Galois representation pp As will be explained in section 10.1, if F = Fy, for some g € N, then the Galois representation has a special form

(Dg @ €) lơ where pg is the Galois representation attached to g (cf Theorem 10.1.2) and e is the f-adic cyclotomic character Let ƒ be as in Proposition 5.2.20 Set R’:= | [pc NM O and let R be the O-subalgebra of R’ generated by the tuples (Ar(T)) re AM for all

Te TS Here Àz : Tt — © is the O-algebra homomorphism sending T to the eigenvalue of 7' corresponding to the eigenform #' Then R is a complete Noetherian

83 local O-algebra with residue field F = O/X It is a standard argument to show that

Proof of Proposition 5.2.20 As remarked before, for ƒ,g € N, mà, = Mp, if and only if m, = mj, hence if mp, = mp, then m; = mj This implies that since f is ordinary at ý, so are all g € N such that Fy € NV 7! Let I, denote the inertia group at É Since g is ordinary at £, we have

(cf Theorem 10.1.2), and hence ek-2 x ĐF;ÌL, = 1 eel x

IfÊ > k it is easy to see that there exists ơ € J, such that the elements đị := e°~?(ứ),

By := 1, 83 := (0), By := €(c) are all distinct mod ý For every g as above, we choose a basis of the space of p, so that x = 0 Set

We extend p to an R-algebra map p’ : R[G| — M,(R’) Note that ỉứ(Erobze) = I] ỉr,(Erobz)/ứp, ( = [| 0g(Erobx)

FgeNM FyeNM and tr p'(Frob, e) = (a9(P)) ewe củ, where g = È}°.au(n)qg" Define 7°(p) to be the image of tr p/(Frob, e) under the O-algebra isomorphism R > Tore, O

Corollary 5.2.21 If ƒ € N, f # fe is ordinary at È, and Ê > k, then for every split prime p = TT, p†£, there exists Th(p) € TS + such that Desc(T*(p)) € Tạ mi

Mp ’ equals the image of Tp € To under the canonical projection To —> Tow,’

Proof We can choose TẦ(p) € Te to be any preimage of T*(p) € Tổ, from

Proposition 5.2.20 under the canonical Ó-algebra map Te — Tet, Oj

In this section we study the action of Hecke operators on the space of automorphic forms and relate them to the classical operators defined so far We begin by considering the local situation The results of this and the following section are not needed in the sequel.

Let p be a finite prime of Q, and (V,7) an irreducible admissible representation of G(Q,) For a summary of the theory of such representations, see e.g [Bum97], section 4.2 Recall that Ko, := G(Z,) is the maximal (open) compact subgroup of G(Q,) Fix a left-invariant Haar measure dg on G(Q,) such that ƒ Kop 49 = 1 We denote by Hx,,, the algebra of compactly supported, smooth (i.e., locally constant), bi-Ko,-invariant functions from G(Q,) into C The multiplication in Hy,,, is defined to be the convolution

For any subset H of G(Q,) denote by [H} the characteristic function of H The function [Kp] is the identity element of Hx, ,.

Definition 5.3.1 The algebra Hx,,, is called the spherical Hecke algebra of G(Q,).

Lemma 5.3.2 The algebra Hx, 1s commutative.

Proof The proof is the same as the proof of Theorem 4.6.1 in [Bum97] O

The representation 7 defines an action v +> 7(¢)u of Hx,,, on V by m(d)u =| Ó(9)(g}0 dg.

G(Qp) Lemma 5.3.3 Let a € G(Q,) and assume that the double coset space Ko paKo,p decomposes as Ky paKoy = []j_, Kop with a; € G(Q,) Then for 0u €V we have x([KopaKog])u = À ` x(a)u ¡=1

Proof We have z(Ko,eKoy)s= | Ko,pakKo,p stovdg=S>f g)u dg. iKo,p

Making a change of variable g > aj‘g and using the fact that dg is a left-invariant measure we get

T([Kopao„] "NI (aig yoy = Yo aj)v

For any subgroup H of G(Q,), we denote by V” the subspace of V consisting of vectors fixed by H The elements of V*°ằ are called spherical vectors.

Lemma 5.3.4 The space V9? is at most one-dimensional.

Proof Since (V,7) is irreducible, it follows from Proposition 4.2.3 in [Bum97] that

VỀ is a simple Hx,,,-module Since Hx,, is commutative by Lemma 5.3.2, the claim follows L

Lemma 5.3.5 Letd€ Hx, and o € Vr Then r(ề})u € V9ằ,

Proof We need to show that 7(k)x(ở)0 = 1(ó)0 for k€ Kop We have x(k)r(ð} = m(k) [ ơ"

By Lemma 5.3.3 the integral over G(Q,) is just a finite sum, so we can interchange m(k) and Jứ(q,: Then x(R)r()ứ= | d(g)r(kg)odp

Making a change of variable g +> k”g and using the fact that dq is left-invariant we see that the last integral equals Je(a,) ó(k~1g)(g)u dg Since ¢ is left Ko „-invariant, the lemma follows L]

Lemma 5.3.6 Let a € G(Q,) If KopaKop = Lj, Kopai with a; € G(Qp), and v € V9z, then 3 m(a;`)u € V0,

Proof Note that if KopaKopy = []j_, Kopai, then Kopa Xeằ = |lÊ_Ăd;'Kop.

Hence by Lemma 5.3.3 demas" e = t([KopaTM’ Kopl)v. ¡=1

The claim now follows by Lemma 5.3.5 C]

In this section v will denote a (finite or infinite) place of Q Let g = gaqguk €G(Q)G(R)Ko¿(N) with Kor(N) = ]l¿ Kow(N) as before Every F € M3?(N, v)

87 gives rise to a function yr : G(A) — C defined as

@r(9) = (FlmGoo) (i) * (det dx), where Z = gui € H Let p # oo be a rational prime For a € G(Q) let ay) denote the element of G(A) whose p-component is a and all the other components are 1. Moreover, we will say that a is p-significant if the image of a under the natural embedding G(Q) — G(Q,) lands in Ko,,(.N) for all v # p, œ.

Lemma 5.3.7 Let p be a (finite) rational prime with p{ N Let F € MTM (N,v) and suppose a € G(Q) is p-significant Let g = (go, 1) € G(R) x G(At) be such that

(F ma) (Z) = 5 (Goo, i)" vn (det da)or (gaz):

(Flma) (2) = 5 (4s Goo)" F (aGooi) = 7(ags, i)" F (Goo: i)” F (Agee: 1) an = J (Goo i)TM Yr ((4g9o0s 1)) = F (Goo, i)” PF ((Goo a7", a7", -))

= j(goos i)" Yr ((Goo.1,1, -)(1,1, ,1,a4,1, )x x (lra7t,a7t, ,a7", 1,a7, )), where in the last expression a~! in the second factor and 1 in the third factor appear on the p-th place Note that (1,a7!,a71, ,a7+,1,a71, ) € Ko(N) as a is p- significant Thus we have

(F|ma) (Z) = 5 (Goo, i) "Yr (gag) Vn (det da).

Let a € G(Q) be p-significant Then [o(W)aTo(N) = []j_, Po(N)ai with a; €G(Q), p-significant In section 5.2.1 we defined an action of the double coset on MfPh(N,) Lemma 5.3.7 provides a tool to translate this action to an action on functions of the form yr as defined above.

Corollary 5.3.8 Let p be a (finite) rational prime with pt N Let F € MUỤR(N,0) and let a € G(Q) be p-significant Let g = (gx.1) € G(R) x G(Ai) be such that

(Flm[fo(N)aTo(W)]) (Z) = 9 (Goo, 1) "Dvr ga;,))-

Proof The corollary follows immediately from Lemma 5.3.7 and the definition of the action of Fe(W)aTo(N) on MIP(N,ð) Oo

5.4 Hecke Eigenvalues of Siegel Eisenstein series

In this section we compute the eigenvalue of the operator T® defined by (5.6) on the Siegel Eisenstein series of positive weight, level and non-trivial character

(cf section 3.2.4) The eigenvalues of TP, for p inert, on Siegel Eisenstein series of level 1 were computed in [Kri91] Even though we will not make use of these calculations later on, they would be useful if one would like to generalize the theory of Maass lifts to hermitian modular forms of level higher than one We will use notation from section 3.2.4 In this section v will denote a place of Q First note that E(Z,s,m,v,N) € M2®(N, 71), hence it makes sense to talk about the action of T? on E(Z,s,m, wv, N).

Proposition 5.4.1 Let p= a7 be a prime of Q which splits in K and assume that ĐỊN Then the series E(Z,s,m,t,N) defined by formula (3.17) is an eigenform for T® with eigenvalue p?m~$-m[3—m (0 4 1)2x(a) 1 (by (p)*p?$ +7°)

1 Hence by (5.8) we obtain a right coset decomposition of 7o of the form

We now study the action of T?') on E(Z, s,m,, N) Let g = (goo, 1) € G(R)xG(A;) be such that Z = gyi Then E(g,s,N,m,v) = j(g0,i1) "E(Z, s,m,v, N) Note that if o1 ẤN, then Ko,(N) is the maximal compact subgroup of G(Q,) We also have (cf section 3.2.4)

E(g,s,N,mv)= So) tp(g)ụp(ag)°)2é= So dz+ 39)ụps(xứ,)°” yEP(Q)\G(Q) yEP(Q)\G(Q) ¥ with pp, and dp, as in (3.15) and (3.16) For g’ € G(A) set f(g’, 8) = Upo(9')OPa(g')”.

Denote by A the set of right coset representatives from (5.19) Note that allae A are p-significant, hence by Corollary 5.3.8 we have

Thus we are reduced to studying the right-hand side of (5.20) For g € G(A) we have ằ 90, ,8, í,m, Ú) = S]]z (2e(2(ứ))x ` 8) = acA acA v

~ =>, 1I2‹( 9v:8) | Íứ(0p8”`,8 = |]H] 0.5) À ` /p(gpa~",8) acA \ up utp acA

Note that we have Ko(N) = Koy, the maximal compact subgroup of G(Q,), and (by the same method as (5.8) was obtained) we have

Kop 1 z 1 Kop = I] Ko pa na} acA

Since f,(-,s) is right Ko,-invariant, Lemma 5.3.6 implies that >,.4 fp(gpa7", 8) is also right Ko,-invariant Thus by Lemma 5.3.4 there exists À € C such that À ` folGpa*, 8) = Afp(9p; 8). acA

Since A is independent of gp, we have A = ƒp;(1,s) '5 ).

Note that Np = 2 Now, taking the product over all the finite places we see that we obtain the Euler factors for the right-hand side of (6.3) L

So far we have studied the numerator (Fy, E(-,8,m, T*)6,) (Chapter 6), and the denominator (Fy, F;) (Chapter 4) of the coefficient Cr, in formula (1.2) We ex- pressed them by L-functions associated to ƒ To use these formulas for studying congruences between Fy and other hermitian modular forms, we need to first investi- gate the integrality of Fourier coefficients of E(-,s,m,I"") and of 6, This is done in sections 7.1 and 7.2 We state the first main theorem of this thesis (Theorem 7.4.5) in section 7.4.

7.1 Fourier coefficients of Eisenstein series

We keep the notation from section 6.1 and assume b = 1 Consider the set X of Hecke characters x’ of K, such that gm

(7.2) X;(Zp) = 1 if p Joo, tp € OF, and zp — 1 € cOxy.

Here m= k—1=-t—2> 0 (since ý < —6) denotes the weight of the Eisenstein series E(Z,s,m,I") defined in section 6.1 For g € G(A), let E(g, s,¢,m, x’) denote the Siegel Eisenstein series defined in section 3.2.4 We put, as before,

101 where Z = gyi and g = (gs,l) Recall from section 6.1 that we made use of a congruence subgroup I? of G(Q) such that 0, € A,(F}) and F}nK* = {1} In this section we fix a particular choice of ẽ*, namely, we set [> := I}(c) Note that as long as c† 2, we have [?(c) M K* = {1} and since (cond wv”) | c, where wv’ is the character of 6y, we have 6, € ,M1,(F}(c)) Lemma 17.2 in [Shi00] provides a connection between E(Z,s,m,x',e) and E(Z,s,m,T2(c)) Because the lemma in [Shi00] is stated in a context slightly different from ours, we quote it here in a form which is applicable to our situation.

Lemma 7.1.1 Let X be the set of all Hecke characters of K satisfying (7.1) and

= ằ ằ E(Z2,s,m,x,€) |m @. x'€X aeTh(e)\F} (e) Note that the map

T2) \TỆ() = Ma(Ox/cỉx) given by y+ b„ (mod cOx) is a bijection In fact we can choose a set of representatives for I*(e) \T2(c) in the form [7%] with b, € M2(Ox) and thus get

(7.4) ằ E(Z,s,m,x',Â) |m œ =| Ox/cOx |* E(Z,s,m,x',c) = CE(Z,s,m,x’,c), aéTh (e)\TB(e) since E(Z,s,m,x',c) |m a = E(Z, s,m, x’, c)x’(det d.) for all a € To(c).

Definition 7.1.2 Let M be a non-zero integer For a Hecke character / : Q*\A* > C”* we set

Lu(s,¥) = L(s,9) ][ — 0 pn, p|M where E(s, ý) denotes the Dirichlet £-function.

Recall that for any Hecke character ý : #Z* \ AX => C* we denote by vq its restriction to A*.

It has been shown in [Shi00] (Theorem 17.12(11)) that D(Z, s,m, x’,c) is holomorphic in the variable Z for s = 2— 5 as long as m > 2 In our case m = —t —2 > 4 as t < —6 The transformation property

(7.5) E(agw, s,c,m, x’) = x¿(det dy)71j(w, i) 1 E(g, 8,¢,m, x’), where a € G(Q) and w € Ko(c), proved in [Shi97] (formula (18.6.2)) ensures that

Instead of looking at D(Z, s,m, x’, c) we will study the Fourier expansion of a transform D*(Z, s,m, x’,c) defined by

(7.7) D*(Z,8,m,x',c) = D(Z, s,m, x’, C6) |md, where J = E “| First note that since D is holomorphic at s = 2 — 4, so is ~1

D*(Z,2 —m/2,m,x’,c) = Sok (2 — =) e(trhZ) hes 2 for the Fourier expansion of D* More generally we can write

D'(Z,s,m,x,e) = ằ cX (Y,s)e(trhX), hes where X = Re Z and Y = Im Z If for a given s, D* is holomorphic in Z, then

One can also write a “Fourier expansion” for the adelic transform of

D(g, 8,¢,m, x’) = L(28, XQ) Lc (2s —1Xx@ (=)) E(g,8,¢,m, x’) defined by

D*(g,8,c,m,x’) = D(gJy",¢,8,m, x’), where J, is the matrix in G(A) which is 1 at the infinite place and equals J at all the finite places The Fourier expansion takes the form:

D* ,8,€,Tn, X | = À `c(h, qg, s)ea(tr (hứ)),

(ứ) 1 hes where ơ € S(A) := {œ € 4⁄;(A) | (@)' = a} and g € GL2(K) with detg = 1 For the definition of the exponential eA see [Shi00], section 1.6 We will not need it here.

(7.9) en(Y,s) = (det ¥)7"2c(h, Y/?, 8), where Y1⁄2 is a matrix in S(A) whose oo-component is the positive-definite matrix whose square is Y and all the finite components are 1 So, we are reduced to studying c(h,Y1⁄2, s).

1—rank(h) —4 j-l x J] L (2 —m— jx’ (=) II j=0 pec where fi, y1/2,, 1s 6 polynomial with coefficients in Z and constant term 1, and c is a certain finite set of primes Ifn WO’) ) (det ay) =

+€T?(e)\Tÿ(e) 0 otherwise, we conclude that (7.16) equals

AX SO BZ, s,m, ve) T5(c) THO] = £[T(e) : T1(e)]E(2, s,m, vc) = x'.ex

(E(., Đ,TM, T2(c)) 4, FF) pe (6) — CPS (c) , T?(e)] (EC, 8; mv’, c)ỉx, FF) peo) ,

(7.18) (DG,s,m,v’, €)Ox; Ft) re =c “TẠ() : Te(o)]7 (2s, Hq) x

Moreover, by |Shi00], formula (17.5) and Remark 17.12(2), we have

(7.19) ŒC, s,m, TY (€)) Ox; Fp) mu =[f():P! (E(-,8,m, D2 (c)) Ox: Fp) rn =

= [T}(e) : TP? l ằ E(Z, s,m, an) OF) œ€T\TẺ (c) rh

Since 6,|;@ = 6, and Ƒ;|¿œ = Fy for œ € I}(c) we finally have

TROT 2, (om Mina) li) Ahan =

(7.21) (D(:,s,m,,c)9x, Fs) pe (6) = (c'AB(2 - m/2)) © x L¿(2s, ỦQ) Le (2s —1,vQ (=)) (det r)~* x cr, (7) Le (Fy, 3 + 1, x)

= c~*B(2 — m/2)~}(det 7)” (4n)? T (m') F (mĩ — 1) x cr, (T)La(F,3—m/2,x) lim L(s),

We have (úq)"” = xq Hence ¥q = (xq)”” = ỦQ as all Hecke characters are assumed to be unitary Thus L(s) = 1 Moreover

(723) (D(,2~ mỊ2,m, Wc), Ft)p(¿ = Ra ? 3D + k + 2) +k + 1)x x Ly (F, 3 — m/2, x),where R := er,(7)c"*B(2 — m/2)~} (8 detr)~?~*~2.

In this section we prove the first main result of this thesis We will show that é-divisibility of L(Symm? ƒ, k) implies the existence of a non-Maass cusp form congruent to Fy modulo 2 We keep the notation from previous sections.

Fourier coefficients of Eisenstein SeTÌS HQ ee 101