Siegel-Eisenstein Series and Triple Products of Coleman’S Families (A Joint Work with S.Boecherer)

Siegel-Eisenstein series and triple products of Coleman’s families (a joint work with S.Boecherer)

A. A. Panchishkin∗ http://www-fourier.ujf-grenoble.fr/˜panchish e-mail : [email protected], FAX: 33 (0) 4 76 51 44 78

Abstract

We start with classical examples of modular form, including elliptic modular forms, Siegel-Eisenstein series, and triple modular forms. For a prime number p ≥ 5, we discuss congruences mod pv between modular form, and explain their relation with p-adic integration. Slide 1 Consider three classical modular cusp eigenforms ∞ X fj (z) = an,j exp(2πinz) ∈ Skj (Nj , ψj ), (j = 1, 2, 3) of n=1 weights kj , conductors Nj , and of nebentypus characters

ψj mod Nj (j = 1, 2, 3), see [Ma-Pa05]. According to H.Hida

[Hi86] and R.Coleman [CoPB], one can include each fj (under

suitable assumptions on p and on fj ) into a p-adic analytic ∞ 0 X n family k 7→ {f 0 = an(f 0 )q } of cusp eigenforms f 0 of j j,kj j,kj j,kj n=1 0 weights kj in such a way that fj,kj = fj , and that all their

∗A talk for the Kinki University mathematics seminar. Date: 20 September (16:30-17:30) 0 0 Fourier coeﬃcients are p-adic analytic functions kj 7→ an,j (kj ). In the second part of this talk we describe p-adic measures attached to Garrett’s triple product of three Coleman’s families and extend our previous results in [PaTV]. We consider the product of three Satake parameters:

0 0 0 (1) 0 (1) 0 (1) 0 λ = λp(k1, k2, k3) = αp,1(k1)αp,2(k2)αp,3(k3). We consider the p-adic weight space

Slide 2 ∗ X = Homcont(Y, Cp).

This is an analytic space over Cp, which consists of all ∗ ∗ continuous characters of the proﬁnite group Y = (Z/NZ) × Zp, 0 containing all (kj , ψj ). Assume that the slope of this product

0 0 0 0 0 0 σ = ordp(λ(k1, k2, k3)) = σ(k1, k2, k3) = σ1 + σ2 + σ3

0 0 0 0 is constant and positive for all triplets k = (k1, k2, k3) in a 3 3 p-adic neighbourhood B ⊂ X of k = (k1, k2, k3) ∈ X . We use the theory of p-adic integration with values in an

algebra A over the Tate ﬁeld Cp.

We obtain a p-adic L-function of four variables as the p-adic Mellin transform of a measure µ with values the Banach algebra A = A(B) associated with an analytic space B ⊂ X3.

We obtain the function in question of four variables Lµ(x, s) 3 by evaluation at s = ((s1, ψ1), (s2, ψ2), (s3, ψ3)) ∈ B ⊂ X : this Slide 3 is a p-adic analytic function in four variables (x, s) ∈ X × B ⊂ X × X × X × X:

Lµ˜(x, s) := evs(Lµ˜)(x)(x ∈ X, s ∈ B1 × B2 × B3, Lµ˜(x) ∈ A).

We construct such a measure from higher twists of classical Siegel-Eisenstein series, which produce distributions with values in certain Banach A-modules M = M(N; A) of triple modular forms with coeﬃcients in the algebra A.

0 Contents

1 Classical modular forms and L-functions 5

2 Generalities on triple products 17

3 Statement of the problem 22 Slide 4

4 Arithmetical nearly holomorphic modular forms 29

5 Siegel-Eisensten series 32

6 Main results 43

7 Criterion of admissibility 77

1 Classical modular forms and L-functions

Our purpose is to extend to Garrett’s triple products our previous results in [PaTV], Two variable p-adic L functions attached to eigenfamilies of positive slope, Invent. Math. v. 154, N3 (2003), pp. 551 - 615. Slide 5 We view modular forms as: where q = exp(2πiz), 1) power series z ∈ H, and deﬁne ∞ X f = a qn ∈ [[q]] the L-function n C ∞ n=0 X −s Lf (s, χ) = χ(n)ann 2) holomorphic functions n=0 on the upper half plane for a Dirichlet character H = {z ∈ C | Im z > 0} χ :(Z/NZ)∗ → C∗.

1 A famous example: the Ramanujan function τ(n)

The function ∆ (of the variable z) τ(1) = 1, τ(2) = −24, is deﬁned by the formal expansion τ(3) = 252, τ(4) = −1472 P∞ n ∆ = n=1 τ(n)q τ(m)τ(n) = τ(mn) Q∞ m 24 = q m=1(1 − q ) for (n, m) = 1, is a cusp form of weight k = 12 |τ(p)| ≤ 2p11/2 (Ramanujan- Slide 6 with respect to Γ = SL2(Z). Deligne) for all primes p . gp > h6=sum(k=1,20,k^5*q^k/(1-q^k)+O(q^20)) gp > h4=sum(k=1,20,k^3*q^k/(1-q^k)+O(q^20) gp > Delta=((1+240*h4)^3-(1-504*h6)^2)/1728

q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 - 16744*q^7 + 84480*q^8 - 113643*q^9 - 115920*q^10 + 534612*q^11 - 370944*q^12 - 577738*q^13 + 401856*q^14 + 1217160*q^15 + 987136*q^16 - 6905934*q^17+ 2727432*q^18 + 10661420*q^19 + O(q^20)

The L-function attached to a primitive form f of weight k and Dirichlet character ψ mod N admits an Euler product (Hecke)

X −s where H (X) = Lf (s, χ) = χ(n)ann , p,f k−1 2 Slide 7 n≥1 1 − apX + ψ(p)p X Y −s −1 = Hp,f (χ(p)p ) = (1 − α(1)X)(1 − α(2)X) p compare with Euler’s product is the Hecke polynomial, X Y (1) (2) ζ(s) = n−s = (1 − p−s)−1 α and α are called n≥1 p the Satake parameters of f

(for any variable Dirichlet character χ :(Z/NZ)∗ → C∗).

2 Let Γ be a subgroup of ﬁnite index in the modular group SL2(Z). Recall: a holomorphic function f : H → C is called a modular form of weight k with respect to Γ iﬀ the conditions a) and b) are

satisﬁed (notation f ∈ Mk(Γ)): a) Automorphy condition

k f((aγ z + bγ )/(cγ z + dγ )) = (cγ z + dγ ) f(z) (1.1) Slide 8 for all elements γ = aγ bγ ∈ Γ; cγ dγ b) Regularity at cusps: f is regular at cusps z ∈ Q ∪ i∞ (the cusps can be viewed as ﬁxed points of parabolic elements of Γ); this a b means that for each element σ = c d ∈ SL2(Z) the function −k az+b f|kσ(z) := (cz + d) f cz+d admits a Fourier expansion over 1/N X 1/N non–negative powers of q : f|kσ = an,σq , N = Nσ ≥ 1. n=0

A function f : H → C is called a cusp form if the constant term an,σ = 0 (i.e. if f vanishes at all cusps σ(∞), notation f ∈ Sk(Γ)), see T.Miyake ([Miy], 1989), G. Shimura ([Shi71], 1971) and Yu.Manin-A.Panchishkin ([Ma-Pa05], 2005). Another known example of a modular form is given by the Eisenstein series 0 X −k Gk(z) = (m1 + m2z) (1.2) Slide 9 m1,m2∈Z

(with respect to SL2(Z) of weight k ≥ 4, prime denoting (m1, m2) 6= (0, 0)). One has Gk(z) ≡ 0 for odd k and

 ∞  2(2πi)k ζ(1 − k) X X G (z) = + dk−1qn , (1.3) k (k − 1)!  2  n=1 d|n

∞ Bk P −s where ζ(1 − k) = − k , ζ(s) = n is the Riemann zeta function, n=1

3 th Bk is the k Bernoulli number. One can also deﬁne the Eisenstein series using the action of the group Γ = SL2(Z) as

X X −k Ek(z) = 1|kγ = (cz + d) = γ∈P \Γ (c,d)=1,c>0 c=0,d=1 ∞ 2 X X (k − 1)! 2 1 + dk−1qn = · G , ζ(1 − k) 2(2πi)k ζ(1 − k) k n=1 d|n Slide 10 n a b o −k az+b where P = γ = 0 d ∈ Γ , f|kσ(z) := (cz + d) f cz+d . gp > e4=1+240*sum(n=1,20,n^3*q^n/(1-q^n)+O(q^20)) %1 = 1 + 240*q + 2160*q^2 + 6720*q^3 + 17520*q^4 + 30240*q^5 + 60480*q^6 + 82560*q^7 + 140400*q^8 + 181680*q^9 + 272160*q^10 + 319680*q^11 + 490560*q^12 + 527520*q^13 + 743040*q^14 + 846720*q^15 + 1123440*q^16 + 1179360*q^17 + 1635120*q^18+ 1646400*q^19 + O(q^20)

The graded algebra M(SL2(Z)) = ⊕k≥0 evenMk(SL2(Z)) is isomorphic to the polynomial ring of the (independent) variables P∞ n P∞ n E4 = 1 + 240 n=1 σ3(n)q and E6 = 1 − 504 n=1 σ5(n)q . In 3 2 particular, ∆ = (E4 − E6 )/1728 (see the above example).

1.1 Congruences bewteen modular forms Slide 11 A new chapter in arithmetic was opened by Serre and Deligne, who explained a whole series of mysterious facts concerning various arithmetical functions. Examples of these facts are: the conjecture of Ramanujan–Petersson |τ(p)| < 2p11/2 for the Ramanujan function τ(p), and the congruence of Ramanujan X τ(n) ≡ d11 mod 691 : (1.4) d|n

4 gp > h12=sum(n=1,20,n^11*q^n/(1-q^n)+O(q^20)) %9 = q + 2049*q^2 + 177148*q^3 + 4196353*q^4 + 48828126*q^5 + 362976252*q^6 + 1977326744*q^7 + 8594130945*q^8 + 31381236757*q^9 + 100048830174*q^10 + 285311670612*q^11 + 743375541244*q^12 + 1792160394038*q^13 + 4051542498456*q^14 + 86498048 64648*q^15 + 17600780175361*q^16 + 34271896307634*q^17 Slide 12 + 64300154115093*q^18 + 116490258898220*q^19 + O(q^20)

gp > (Delta-h12)/691 %10 = -3*q^2 - 256*q^3 - 6075*q^4 - 70656*q^5 - 525300*q^6 - 2861568*q^7 - 12437115*q^8 - 45414400*q^9 - 144788634*q^10 - 412896000*q^11 - 1075797268*q^12 - 2593575936*q^13 - 5863302600*q^14 - 12517805568*q^15 - 25471460475*q^16 - 49597544448*q^17 - 93053764671*q^18 - 168582124800*q^19 + O(q^20)

Classical Kummer congruences

for Bernoulli numbers can be sated and proved in terms of ∗) Eisenstein series. Let us normalize the series Gk by (see [Se73]): (k − 1)! G∗(z) = G (z) − pk−1G (pz), with the q-expansion: k 2(2πi)k k k ∞ ζ∗(1 − k) X X G∗ = + σ∗ (n)qn, σ∗ (n) = dk−1, k 2 k−1 k−1 n=1 d|n Slide 13 (d,p)=1 X ζ∗(s) = ζ(s)(1 − p−s) = n−s is ζ with p-Euler factor n=1 (p,n)=1 removed. Theorem 1.1 a) Let k ≡ k0 mod (p − 1)pN−1 then ∗ ∗ N Gk ≡ Gk0 mod p in Q[[q]] for all k 6≡ 0 mod (p − 1). This example illustrates of a general phenomenon: congruences and algebraic properties of special L-values can be deduced from the corresponding congruences and algebraic properties of the Fourier coeﬃcients of modular forms

5 b) Let k ≡ k0 mod (p − 1)pN−1 then for any c ∈ Z, (c, p) = 1, k ∗ k0 ∗ N c > 1 we have that (1 − c )Gk ≡ (1 − c )Gk0 mod p (without restriction on k).

Remark. We obtained a p-adic family of modular forms: Slide 14 ∗ X n k 7→ Gk = an(k)q , N → Q[[q]], n≥0

with certain p-adic analytic functions an(k). The constant term is the Kubota-Leopoldt p-adic zeta function, see [Ku-Le, Iw]. Proof of Theorem 1.1 uses the classical Kummer congruences:

let us ﬁx any c ∈ Z, (c, p) = 1, c > 1. (c) k+1 k Theorem (Kummer). Put ζ(p)(−k) = (1 − c )(1 − p )ζ(−k), P i N k ≥ 0, and let h(x) = αix ∈ Z[x] such that h(a) ≡ 0 mod p for i ∗ P (c) N all a ∈ Zp. Then αiζ(p)(−i) ≡ 0 mod p . i M−1 Slide 15 P k Proof (see [Ka78]) uses the sums of powers Sk(M) = n , the n=1 Bernoulli numbers Bk, and and the Bernoulli polynomial Pk k k−i Bk(x) = i=0 i Bix :

M−1 X 1 S (M) = nk = [B (M) − B ]. k k + 1 k+1 k+1 n=1

6 Corollary 1.2 (on p-adic continuity of zeta values) If 0 h(x) = xk − xk , k ≡ k0 mod (p − 1) pN−1 then

Corollary 1.3 (Mazur) There exists a unique measure µ(c) on ∗ Zp such that for all k ≥ 1 Z k (c) (c) k k−1 Slide 16 x dµ = ζ(p)(1 − k) = (1 − c )(1 − p )ζ(1 − k). ∗ Zp

Proof of Corollary 1.3: the existence of µ(c) is equivalent to abstract Kummer congruences for the values Z k+1 (c) (c) ∗ x dµ = ζ(p)(−k), (on the group Y = Zp): ∗ Zp

∗ N X (c) N ∀a ∈ Zp, h(a) ≡ 0 mod p =⇒ αiζ(p)(−i) ≡ 0 mod p . i

A family of slope σ > 0 of cusp eigenforms fk0 of weight k0 ≥ 2 containing f

∞ 0 X 0 n k 7→ fk0 = an(k )q n=1 0 1) the Fourier coeﬃcients an(k ) of fk0 ∈ Q[[q]] ⊂ Cp[[q]] 0 and the Satake p-parameter αp(k ) Slide 17 are given by certain A model example 0 0 p-adic analytic functions k 7→ an(k ) of a p-adic family for (n, p) = 1 (not cusp and σ = 0): 2) the slope is constant and positive: Eisenstein series 0 ord(αp(k )) = σ > 0 X k0−1 an = d , fk0 = Ek0 d|n

7 2 Generalities on triple products

Consider three primitive cusp eigenforms

∞ X fj(z) = an,je(nz) ∈ Skj (Nj, ψj), (j = 1, 2, 3) (2.1) n=1

Slide 18 of weights k1, k2, k3, of conductors N1,N2,N3, and of nebentypus characters ψj mod Nj (j = 1, 2, 3), and let χ denote a Dirichlet character. The triple product twisted with Dirichlet characters χ is deﬁned as the following complex L-function (an Euler product of degree eight):

Y −s L(f1 ⊗ f2 ⊗ f3, s, χ) = L((f1 ⊗ f2 ⊗ f3)p, χ(p)p ), (2.2)

p-N1N2N3

−1 where L((f1 ⊗ f2 ⊗ f3)p,X) = (2.3) (1) ! (1) ! (1) !! αp,1 0 αp,2 0 αp,3 0 det 18 − X (2) ⊗ (2) ⊗ (1) 0 αp,1 0 αp,2 0 αp,3 Y (η(1)) (η(2)) (η(3)) = (1 − αp,1 αp,2 αp,3 X), η : {1, 2, 3} → {1, 2}, and Slide 19 η

kj −1 2 (1) (2) 1−ap,jX−ψj(p)p X = (1−αp,j (p)X)(1−αp,j (p)X), j = 1, 2, 3,

are the Hecke p–polynomials of forms fj. We always assume that

k1 ≥ k2 ≥ k3, and k1 ≤ k2 + k3 − 2 (“balanced” weights) (2.4)

8 We use the corresponding normalized L function (see [De79], [Co], [Co-PeRi]), which has the form:

Λ(f1 ⊗ f2 ⊗ f3, s, χ) = (2.5)

ΓC(s)ΓC(s − k3 + 1)ΓC(s − k2 + 1)ΓC(s − k1 + 1)L(f1 ⊗ f2 ⊗ f3, s, χ), where Γ (s) = 2(2π)−sΓ(s). The Gamma-factor determines the Slide 20 C critical values s = k1, ··· , k2 + k3 − 2 of Λ(s), which we explicitely π2 evaluate (like ζ(2) = ). 6 A conjectural) functional equation of Λ(s) has the form:

s 7→ k1 + k2 + k3 − 2 − s.

According to H.Hida [Hi86] and R.Coleman [CoPB], one can

include each fj (j = 1, 2, 3) (under suitable assumptions on p and

on fj) into a p-adic analytic family

∞ 0 X n Slide 21 f : k 7→ {f 0 = a (f 0 )q } j j j,kj n j,kj n=1

0 of cusp eigenforms f 0 of weights k in such a way that f = f j,kj j j,kj j (j = 1, 2, 3), and that all their Fourier coeﬃcients a (f 0 ) are n j,kj 0 0 given by certain p-adic analytic functions kj 7→ an,j(kj).

9 3 Statement of the problem

Given three p-adic analytic families fj of positive slope σj > 0, to construct a four-variable p-adic L-function attached to Garrett’s triple product of these families

(interpolating the special values L(f 0 ⊗ f 0 ⊗ f 0 , s, χ) at 1,k1 2,k2 3,k3 0 0 0 critical points s = k1, ··· , k2 + k3; we prove that these values are Slide 22 algebraic numbers afters dividing by certain “periods”). ∗ We consider the p-adic weight space X = Homcont(Y, Cp). This is an analytic space over Cp, which consists of all continuous ∗ ∗ characters of the proﬁnite group Y = (Z/NZ) × Zp, containing all 0 classical weights (kj, ψj), j = 1, 2, 3. The classical analogue of the weight space is the whole complex plane

∗ ∗ s C = Homcont(R+, C ), s 7→ (y 7→ y ).

Consider the product of the Satake parmeters

(1) (1) (1) 0 0 0 λp = αp,1αp,2αp,3 = λp(k1, k2, k3)

(1) (2) We assume: 1) |αp,j |p ≤ |αp,j |p, j = 1, 2, 3, 2) the slope 0 0 0 σ = ordp(λp(k1, k2, k3) is constant and positive for all triplets 0 0 0 3 (k1, k2, k3) in a p-adic neighbourhood B ⊂ X of the ﬁxed triplet of weights (k1, k2, k3). Slide 23 The existence of families of slope σ > 0: R.Coleman, [CoPB]

A program in PARI for computing He gave an example with such families is contained in [CST98] p = 7, f = ∆, k = 12 (see also the Web-page of W.Stein, a7 = τ(7) = −7 · 2392, σ = 1. http://modular.fas.harvard.edu/ )

10 Our method

is a combination of Garrett’s integral representation for the triple 0 0 0 L-functions of the form: for r = 0, ··· , k2 + k3 − k1 − 2, 0 0 Λ(f1,k0 ⊗ f2,k0 ⊗ f3,k0 , k2 + k3 − r, χ) = ZZZ1 2 3 Y dxjdyj f˜ 0 (z )f˜ 0 (z )f˜ 0 (z )E(z , z , z ; −r, χ) ( ) 1,k1 1 2,k2 2 3,k3 3 1 2 3 2 yj 2 2v 3 j (Γ0(N p )\H) 0 0 0 Slide 24 where E is the triple modular form of weight (k1, k2, k3), and of ﬁxed character (ψ1, ψ2, ψ3), obtained from a nearly holomorphic Siegel-Eisenstein series by applying Boecherer’s higher twist and Ibukiyama’s diﬀerential operator. the theory of p-adic integration with values in A-modules M(A) of triple nearly holomorphic modular forms over p-adic Banach algebras A, which allows to view E as an element of M(A), and the spectral theory of Atkin’s U-operator allows to evaluate the integral using a projection of M(A) to the λ-part M(A)λ.

Here A = A(B) is a certain p-adic Banach algebra of functions on an open analytic subspace B ⊂ X3 in the product of three copies of ∗ the weight space X = Homcont(Y, Cp). This is an analytic space over Cp, which consists of all continuous characters of the proﬁnite group Slide 25 ∗ ∗ Y = (Z/NZ) × Zp. The classical analogue of the weight space is the whole complex plane

∗ ∗ s C = Homcont(R+, C ), s 7→ (y 7→ y ). The weights k0 correspond to certain points in the weight space X.

11 Let Bj ⊂ X denote an open analytic subspace conaining kj ∈ X,

and let Aj = Aj(Bj) be the aﬃnoid algebra attached to Bj. Any P n series fj = n≥1 anq ∈ Aj[[q] produces a family of q-expansions X n f 0 = ev 0 (f ) = ev 0 (a )q ∈ [[q]] , which happen to be j,kj kj j kj n Cp Slide 26 n≥1 classical modular forms in Q[[q]] under a ﬁxed embedding i : ,→ and the specialization maps ev 0 : A → . p Q Cp kj j Cp

• We may assume that B = B1 × B2 × B3, and consider the 0 0 0 0 specialization maps evk0 : A → Cp, where k = (k1, k2, k3) ∈ B.

• We construct an analytic function Lµ : X → A = A(B) as the p-adic Mellin transform Z ∗ Lµ(x) = x(y) dµ(y)(where x ∈ X = Homcont(Y, Cp), x = x(y)), Y Slide 27 of a certain measure µ with values in A on the proﬁnite group Y .

We obtain the function in question Lµ(x, s) by evaluation at

s = ((s1, ψ1), (s2, ψ2), (s3, ψ3)) ∈ B: this is a p-adic analytic function

in four variables (x, s) ∈ X × B1 × B2 × B3 ⊂ X × X × X × X:

Lµ(x, s) := evs(Lµ(x)) (x ∈ X, s ∈ B1 × B2 × B3, Lµ(x) ∈ A).

12 • We check an equality relating the values Lµ(x, s) at the r arithmetical chracters x = ypχ ∈ X, and at triple weights 0 0 0 s = (k1, k2, k3) ∈ B, with the normalized critical special values

∗ 0 0 0 0 0 L (f 0 ⊗f 0 ⊗f 0 , k +k −2−r, χ)(r = 0, ··· , k +k −k −2), Slide 28 1,k1 2,k2 3,k3 2 3 2 3 1 for certain Dirichlet characters χ mod Npv, v ≥ 1, where the normalisation of L∗ includs at the same time certain Gauss sums, 0 0 0 Petersson scalar products, powers of π and of λp(k1, k2, k3), and a certain ﬁnite Euler product.

4 Arithmetical nearly holomorphic modular forms

Let A be a commutative ring (a subring of C or a normed O-algebra A where O is the ring of integers in a ﬁnite extension K of Qp). Arithmetical nearly holomorphic modular forms (in the sense of Slide 29 Shimura, [ShiAr] are certain formal series ∞ X g = a(n; R)qn ∈ A[[q]][R], with the property n=0

that for A = C, z = x + iy ∈ H,R = (4πy)−1, the series converges to a C∞-modular form on H of a given weight k and Dirichlet character ψ. The coeﬃcients a(n; R) are polynomials in A[R]. If

degR a(n; R) ≤ r for all n, we call g nearly holomorphic of type r ∂ r+1 (i.e., annihilated by ( ∂z ) , see [ShiAr]).

13 ˜ We use the notation Mk,r(N, ψ, A) or M(N, ψ, A) for A-modules of such forms. (In our constructions the weight k varies). A known example (see the introduction to [ShiAr]) is given by the series ∞ X n − 12R + E2 := −12R + 1 − 24 σ1(n)q n=1 0 Slide 30 3 s X −k −2s −1 = lim y (m1 + m2z) |m1 + m2z| , (R = (4πy) ) π2 s→0 m1,m2∈Z P where σ1(n) = d|n d. There is the action of the Shimura diﬀerential operator

δk : Mk,r(N, ψ, A) → Mk+2,r+1(N, ψ, A), 1 ∂ k given over by δ (f) = ( − )f. C k 2πi ∂z 4πy

This operator is a correction of the Ramanujan operator ∞ ∞ ∞ ∞ X X 1 ∂ X ∂ X θ( a qn) = na qn = ( a qn) = q ( a qn), n n 2πi ∂z n ∂q n n=0 n=1 n=0 n=0

which does not preserve the modularity. For example θ∆ = E2∆, where E2 is a quasimodular form (in the sense of Kaneko and Zagier, see [Ka-Za]).

Slide 31 Notice that δkf = (θ − kR)f, and that Serre’s operator k f 7→ θf − 12 E2f takes Mk to Mk+2. Note that that the arithmetical twist operator ∗) ∞ ∞ X n X n θχ( anq ) = χ(n)anq n=0 n=1 is an obvious analog of the Ramanujan operator. In our work we use Boecherer’s higher twist operator acting on Siegel modular forms which is analogous to Ibukiyama’s operator

14 The tensor product over A

Mk,r,T (N, ψ, A) = Mk1,r(N, ψ1, A)⊗Mk2,r(N, ψ2, A)⊗Mk2,r(N, ψ3, A)

produce triple arithmetical modular forms as certain formal series of the form ∞ X n1 n2 n3 g = a(n1, n2, n3; R1,R2,R3)q1 q2 q3 Slide 32 n1,n2,n3=0 −1 ∈ A[[q1, q2, q3]][R1,R2,R3], where zj = xj + iyj ∈ H,Rj = (4πyj) ,

with the property that for A = C, the series converges to a ∞ 3 C -modular form on H of a given weight (k1, k2, k3) and character (ψ1, ψ2, ψ3), j = 1, 2, 3. The coeﬃcients a(n1, n2, n3; R1,R2,R3) are

polynomials in A[R1,R2,R3]. Non trivial examples of such modular forms come from the restriction to the diagonal of Siegel moduar forms of degree 3.

5 Siegel-Eisensten series

Recall some deﬁnitions concerning Siegel modular forms.   0m −1m Let J2m =  . The symplectic group 1m 0m

t Slide 33 Spm(R) = g ∈ GL2m(R)| g · J2mg = J2m , acts on the Siegel upper half plane

t Hm = z = z ∈ Mm(C)|Imz > 0 by g(z) = (az + b)(cz + d)−1, where we use the bloc notation a b g = c d ∈ Sp2m(R), and the congruence subgroup m ∗ ∗ Γ0 (N) = {γ ∈ Spm(Z) | γ ≡ 0 ∗ } ⊂ Spm(Z).

15 m A Siegel modular form f ∈ Mk(Γ0 (N), χ) of degree m > 1, weight k and a Dirichlet chracter χ mod N is a holomorphic function a b m f : Hm → C such that for every γ = c d ∈ Γ0 (N) one has

f(γ(z)) = χ(det d) det(cz + d)k.f(z).

The Fourier expansion of such f uses the semi-group Slide 34 t Bm = {T = T ≥ 0|T half-integral} : P T Bm Bm f(z) = a(T)q ∈ C[[q ]](a formal q-expansion ∈ C[[q ]]), T∈Bm √ T1 T2 T1+T2 where qij = exp(2π( −1zi,j)), q · q = q , and qT = exp(2πitr(Tz)) m Y Tii Y 2Tij −1 = qii qij ⊂ C[[q11, . . . , qmm]][qij, qij ]i,j=1,··· ,m i=1 i

Example 5.1 (Siegel-Eisenstein series) (studied by Prof. S.Nagaoka in [Nag2], p.408):

(2) 2 −2 −1 E4 (z) =1 + 240q11 + 240q22 + 2160q11 + (240q12 + 13440q12 Slide 35 2 2 30240 + 13440q12 + 240q12)q11q22 + 2160q22 + ... (2) 2 −2 −1 E6 (z) =1 − 504q11 − 504q22 − 16632q11 + (−540q12 + 44352q12 2 2 166320 + 44352q12 − 504q12)q11q22 − 16632q22 + ....

16 Arithmetical nearly holomorphic Siegel modular forms

Consider a commutative ring A, the formal variables

q = (qi,j)i,j=1,...,m, R = (Ri,j)i,j=1,...,m, and the ring of formal arithmetical Fourier series ( ) X Bm T Slide 36 A[[q ]][Ri,j] = f = a(T,R)q a(T,R) ∈ A[Ri,j] (5.1) T∈Bm (over the complex numbers this notation corresponds to qT = exp(2πitr(TZ)), R = (4πIm(Z))−1). The formal Fourier expansion of a nearly holomorphic Siegel modular form f with coeﬃcients in A is a certain element of

Bm A[[q ]][Ri,j].

5.1 Algebraic diﬀerential operators of Maass and Shimura

Let us consider the Maass diﬀerential operator (see [Maa]) ∆m of ∞ degree m, acting on complex C -functions on Hm by: Slide 37 ˜ ˜ −1 ∆m = det(∂ij), ∂ij = 2 (1 + δij)∂/∂ij, (5.2)

its algebraic version is the Ramanujan operator of degree m: 1 1 Θ := det( ∂˜ ) = det(θ ) = ∆ , (5.3) m 2πi ij ij (2πi)m m

−1 ∂ T T where θij = 2 (1 + δij)qij , Θm(q ) = det(T)q . ∂qij

17 The Shimura diﬀerential operator (see [Shi76, ShiAr]):

k+1− −1−k −1 δkf(z) = det(R) κΘm det(R)κ f , where R = (4πy) ,

acts on arithmetic nearly holomorphic Siegel modular forms, and the composition is deﬁned Slide 38 (r) m m δk = δk+2r−2 ◦ · · · ◦ δk : Me k (N, ψ; Q) → Me k+2r(N, ψ; Q), (5.4) where −1m δ f(z) = det(y)−1 det(z − z¯)κ−k∆ det(z − z¯)k−κ+1f(z) k 4π m

(r) We show that the Fourier expansion of δk f, where f = P c(T)qT ∈ Mm(N, ψ), is given by the formula Bm k

(r) X T δk f = Q(R, T; k, r)c(T)q . T∈Bm Here we use a universal polynomial, explicitely described in [CourPa], Theorem 3.14 as follows: Slide 39 Q(R, T) = Q(R, T; k, r) (5.5) r X r X = det(T)r−t R (κ − k − r)Q (R, T), t L L t=0 |L|≤mt−t Q (R, T) = tr tρ (R)ρ? (T) · ... · tr tρ (R)ρ? (T)), L m−l1 l1 m−lt lt r m were we use the natural representation ρr : GLm(C) −→ GL(∧ C ) r m (0 ≤ r ≤ m) of the group GLm(C) on the vector space Λ C . Thus m m ρr(z) is a matrix of size r × r composed of the ? t −1 subdeterminants of z of degree r. Put ρr(z) = det(z)ρm−r( z) .

18 ? Then the representations ρr and ρr turn out to be polynomial representations so that for each z ∈ Mm(C) the linear operators ? ρr(z), ρr(z) are well deﬁned. In (5.5), L goes over all the multi-indices 0 ≤ l1 ≤ · · · ≤ lt ≤ m, such that Slide 40 |L| = l1 + ··· + lt ≤ mt − t, and RL(β) ∈ Z[1/2][β] in (5.5) are polynomials in β of degree (mt − |L|). For example, if m = 1, r X r Γ(β + t) Q(R, T; β, r) = Tr−t Rt, and for m = 2 we refer to t Γ(β) t=0 [Cour], Ch.6.

Note the diﬀerentiation rule of degree m (see [Sh83], p.466): m X t ˜ ? ˜ ∆(fg) = tr ρr(∂/∂z)f · ρm−r(∂/∂z)g , r=0 ˜ tr(uz) tr(uz) ρr(∂/∂z)e = ρr(u)e , ˜ α α−1 ? ρr(∂) (det(z) ) = cr(α) det(z) ρm−r(z). If r = 1, m arbitrary, one has (see [Maa]): m X X m−l t ? ξ Slide 41 δkf(z) = c(ξ) (−1) cm−l(k+1−κ)tr ρm−l(R) · ρl (T) q T∈Bm l=0

−1 Γm(α + κ) where R = (4πy) = (Ri,j) ∈ Mm(R), cm(α) = , Γm(α + κ − 1) m−1 m(m−1)/4 Y Γm(s) = π Γ(s − (j/2))). j=0 r (r) X r Γ(k + r) If m = 1, r arbitrary, δ = (−1)r−j Rr−jθj. k j Γ(k + j) j=0

19 Example 5.2 (Siegel-Eisenstein series of higher level)

G∗(Z, s; k, χ, N) (5.6) X = det(y)s χ(det c) det(cZ + d)−k| det(cZ + d|−2s · c,d

[m/2]  Y · Γ(˜ k, s)L (k + 2s, χ) L (2k + 4s − 2i, χ2) , where Slide 42 N  N  i=1

(c, d) runs over all “non-associated coprime symmetric pairs” with det(c) coprime to N, κ = (m + 1)/2, and for m odd the Γ-factor has the form: (2.134) mk −m(k+1) −m(s+k) Γ(˜ k, s) = i 2 π Γm(k + s). m + 1 We use this series with m = 3, κ = = 2, [m/2] = 1. 2

Theorem 5.3 (Siegel, Shimura [Sh83], P. Feit [Fei86]) Let m be an odd integer such that 2k > m, and N > 1 be an integer, then: For an integer s such that s = −r ≤ 0, 0 ≤ r ≤ k − κ, there is the following Fourier expansion X G?(Z, −r) = G?(Z, −r; k, χ, N) = a(T,R)qT, (5.7)

Am3T≥0 Slide 43 where for s > (m + 2 − 2k)/4 in (5.7) the only non-zero terms occur for positive deﬁnite T > 0, polynomials Q(R, T; k − 2r, r) are

given by (5.5) , and for all T > 0, T ∈ Am, where a(T,R) = M(T, χ, k − 2r) · det(T)k−2r−κQ(R, T; k − 2r, r), (5.8)

Y −k+2r M(T, k − 2r, χ) = M`(T, χ(`)` ) (5.9) `| det(2T)

is a ﬁnite Euler product, in which M`(T, x) ∈ Z[x].

20 6 Main results

Let us describe p-adic admissible measures attached to Garrett’s triple L-function of three families of pimitive cusp eigenforms.

6.1 Distributions and admissible measures Slide 44

For a ﬁxed positive integer N ∈ N consider the proﬁnite group Y = Y = lim Y , where Y = ( /Npv )×. N,p ← v v Z Z v

× There is a natural projection yp : Y → Zp . Let us ﬁx a normed O-algebra A where O is the ring of integers in a ﬁnite extension K of Qp.

Definition 6.1 (a) For h ∈ N, h ≥ 1 let Ph(Y, A) denote the A-module of locally polynomial functions of degree < h of the × × variable yp : Y → Zp ,→ A ; in particular, P1(Y, A) = Cloc−const(Y, A) (the A-submodule of locally constant functions). We adopt the

notation Φ(U) := Φ(χU) for the characteristic function χU of an loc−an Slide 45 open subset U ⊂ Y . Let also denote C (Y, A) the A-module of locally analytic functions and C(Y, A) the A-module of continuous functions so that P1(Y, A) ⊂ Ph(Y, A) ⊂ Cloc−an(Y, A) ⊂ C(Y, A). (b) For a given positive integer h we deﬁne an h-admissible measure on Y with values in an A-module M as a certain homomorphism A-modules: Φ:˜ Ph(Y, A) → M,

21 such that for all a ∈ Y and for v → ∞

Z j ˜ −v(j−h) (y − a) dΦ = o(p ) for all j = 0, 1, ··· , h − 1, a+(Npv ) p,M This condition allows to integrate locally-analytic functions on Y along Φ˜ using Taylor’s expansions!

Slide 46 6.2 Up–Operator and the method of canonical projection

We construct an h-admissible measure Φeλ : Ph(Y, A) → M(A) out 1 of a sequence of distributions Φr : P (Y, A) → M(A) on local r monomials yp of each degree r by Z r λ yp dΦe = πλ(Φr((a)v)), where Φr((a)v)) ∈ M = M(A). (a)v

Here Φr take values in an A-module

M = M(A) ⊂ A[[q1, q2, q3]][R1,R2,R3]

of nearly holomorphic triple modular forms over A (for

0 ≤ r ≤ h − 1, h = [2ordpλp] + 1) × A is an Cp-algebra, and λ ∈ A is a ﬁxed non-zero eigenvalue of Slide 47 triple Atkin’s operator UT = UT,p, acting on M(A),

λ πλ : M(A) → M(A)

is the canonical projection operator onto the maximal A-submodule λ M(A) over which the operator UT − λI is nilpotent (we call M(A)λ the λ-characteristic submodule of M(A)).

The projector πλ is deﬁned by its kernel: T n Ker πλ := n≥1 Im (UT − λI) .

22 Triple modular forms are certain formal series

∞ X n1 n2 n3 g = a(n1, n2, n3; R1,R2,R3)q1 q2 q3 n1,n2,n3=0 −1 ∈ A[[q1, q2, q3]][R1,R2,R3], where zj = xj + iyj ∈ H,Rj = (4πyj) ,

Slide 48 with the property that for A = C, the series converges to a ∞ 3 C -modular form on H of a given weight (k1, k2, k3) and character (ψ1, ψ2, ψ3), j = 1, 2, 3. The coeﬃcients a(n1, n2, n3; R1,R2,R3) are

polynomials in A[R1,R2,R3], and the triple Atkin’s operator is given by ∞ P n1 n2 n3 UT (g) = a(pn1, pn2, pn3; pR1, pR2, pR3)q1 q2 q3 . n1,n2,n3=0

∗ Eigenfunctions of Up and of Up .

P∞ n Recall that for a primitive cusp eigenform fj = n=1 an(f)q , P∞ n there is an eigenfunction fj,0 = n=1 an(fj,0)q ∈ Q[[q]] of U = Up (1) with the eigenvalue α = αp,j ∈ Q (U(f0) = αf0) given by p 0 f = f − α(2)f |V = f − α(2)p−k/2f | (6.10) j,0 j p,j j p j p,j j 0 1 ∞ ∞ Slide 49 X −s X −s (1) −s −1 an(fj,0)n = an(fj)n (1 − αp,j p ) . n=1 n=1 p-n

0 ∗ Moreover, there is an eigenfunction fj of Up given by 0 −1 ∞ 0 ρ ρ X n fj = fj,0 , where fj,0 = a(n, f0)q . (6.11) k Np 0 n=1

Therefore, UT (f1,0 ⊗ f2,0 ⊗ f3,0) = λ(f1,0 ⊗ f2,0 ⊗ f3,0).

23 Let us describe in more detail critical values of the L function

L(f1 ⊗ f2 ⊗ f3, s, χ). For an arbitrary Dirichlet character χ mod Npv consider the following Dirichlet characters:

Slide 50 v v ¯ χ1 mod Np = χ, χ2 mod Np = ψ2ψ3χ, (6.12) v ¯ 2 χ3 mod Np = ψ1ψ3χ, ψ = χ ψ1ψ2ψ3; later on we impose the condition that the conductors of the

corresponding primitive characters χ0,1, χ0,2, χ0,3 are Np-complete (i.e. have the same prime divisors as resp. those of Np).

Theorem A (algebraic properties of the triple product)

Assume that k2 + k3 − k1 ≥ 2, then for all pairs (χ, r) such that the

corresonding Dirichlet characters χj have Np-complete conductors,

and 0 ≤ r ≤ k2 + k3 − k1 − 2, we have that ρ ρ ρ Λ(f1 ⊗ f2 ⊗ f3 , k2 + k3 − 2 − r, ψ1ψ2χ) Slide 51 ρ ρ ρ ρ ρ ρ ∈ Q hf1 ⊗ f2 ⊗ f3 , f1 ⊗ f2 ⊗ f3 iT where

ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ hf1 ⊗ f2 ⊗ f3 , f1 ⊗ f2 ⊗ f3 iT := hf1 , f1 iN hf2 , f2 iN f3 , f3 iN

= hf1, f1iN hf2, f2iN f3, f3iN .

24 Fix a positive integer N, a Dirichlet character ψmodN and consider m ∗ the commutative proﬁnite group Y = YN,p = lim( /Np ) and ←− Z Z m × its group XN,p = Homcont(Y, Cp ) of (continuouos) p-adic characters (this is a Cp-analytic Lie group). The group XN,p is isomorphic to a ﬁnite union of discs U = {z ∈ Cp | |z|p < 1}. A p-adic L-function L : X → is a certain meromorphic Slide 52 (p) N,p Cp function on XN,p. Such a function often come from a p-adic

measure µ(p) on Y (bounded or admissible in the sense of

Amice-Vélu, see [Am-V]). The p-adic Mellin transform of µ(p) is given for all x ∈ XN,p by Z L(p)(x) = x(y)dµ(p)(y),L(p)(x): X → Cp YN,p

Theorem B (on admissible measures attched to the triple product). Under the assumptions as above there exist a -valued measure µ˜λ on Y , and a -analytic function Cp f1⊗f2⊗f3 N,p Cp D(p)(x, f1 ⊗ f2 ⊗ f3): Xp → Cp, given for all x ∈ XN,p by the R λ integral D (x, f1 ⊗ f2 ⊗ f3) = x(y)d˜µ (y), and having (p) YN,p f1⊗f2⊗f3 the following properties: tors (i) for all pairs (r, χ) such that χ ∈ XN,p , and all three Slide 53 corresonding Dirichlet characters χj have Np-complete conductor (j = 1, 2, 3), and r ∈ Z is an integer with 0 ≤ r ≤ k2 + k3 − k1 − 2, the following equality holds:

4(k2+k3−2−r) r (ψ1ψ2)(2)Cχ D(p)(χxp, f1 ⊗ f2 ⊗ f3) = ip 2v G(χ1)G(χ2)G(χ3)G(ψ1ψ2χ1)λp ρ ρ ρ Λ(f1 ⊗ f2 ⊗ f3 , k2 + k3 − 2 − r, ψ1ψ2χ) 0 0 0 hf1 ⊗ f2 ⊗ f3 , f1,0 ⊗ f2,0 ⊗ f3,0iT,Np where v = ordp(Cχ), G(χ) denotes the Gauß sum of a primitive Dirichlet character χ0 attached to χ (modulo the conductor of χ0),

25 (ii) if ordpλp = 0 then the holomorphic function in (i) is a bounded Cp-analytic function; (iii) in the general case (but assuming that λp 6= 0) the holomorphic h function in (i) belongs to the type o(log(xp )) with h = [2ordpλp] + 1 and it can be represented as the Mellin transform of the h-admissible -valued measure µ˜λ (in the sense of Cp f1⊗f2⊗f3 Slide 54 Amice-Vélu) on Y

(iv) if h ≤ k − 2 then the function D(p) is uniquely determined by the above conditions (i).

Remark 6.2 It was checked by B.Gorsse and G.Robert that

0,ρ 0,ρ 0,ρ ρ ρ ρ hf1 ⊗f2 ⊗f3 , f1,0⊗f2,0⊗f3,0iT,Np = β·hf1, f1iN hf2, f2iN f3, f3iN ∗ for some β ∈ Q (see [Go-Ro]).

Let us describe now a p-adic measures attached to Garrett’s triple product of three Coleman’s families ∞ 0 X 0 n k 7→ {f 0 = a (k )q }(j = 1, 2, 3). (6.13) j j,kj n,j n=1 Consider the product of three eigenvalues:

0 0 0 (1) 0 (1) 0 (1) 0 λ = λp(k1, k2, k3) = αp,1(k1)αp,2(k2)αp,3(k3) Slide 55 and assume that the slope of this product

0 0 0 0 0 0 σ = ordp(λ(k1, k2, k3)) = σ(k1, k2, k3) = σ1 + σ2 + σ3

0 0 0 is constant and positive for all triplets (k1, k2, k3) in an appropriate p-adic neighbourhood of the ﬁxed triplet of weights (k1, k2, k3). Let A = A(B) denote an aﬃnoid algebra A associated with an

analytic space B = B1 × B2 × B3, a neighbourhood around 3 (k1, k2, k3) ∈ X (with a given k and ψ mod N).

26 Theorem C (on p-adic measures for families of triple

products) Put H = [2ordp(λ)] + 1. There exists a sequence of

distributions Φr on Y with values in M = M(A) giving an H-admissible measure Φ˜ λ with values in Mλ ⊂ M with the following properties:

λ There exists an A-linear form ` = `f1⊗f2⊗f3,λ : M(A) → A (given by (6.19), such that the composition Slide 56 ˜ λ µ˜ =µ ˜f1⊗f2⊗f3,λ := `f1⊗f2⊗f3,λ(Φ )

0 0 0 is an H-admissible measure with values in A, and for all (k1, k2, k3) in the aﬃnoid neighborhood B = B1 × B2 × B3, as above, satisfying 0 0 0 k1 ≤ k2 + k3 − 2 we have that the evaluated integrals

λ r ev 0 0 0 (` )(Φ˜ )(y χ) (k1,k2,k3) f1⊗f2⊗f3,λ p

r on the arithmetical chracters ypχ coincide with the critical special

values ∗ 0 0 Λ (f 0 ⊗ f 0 ⊗ f 0 , k + k − 2 − r, χ) 1,k1 2,k2 3,k3 2 3 0 0 0 for r = 0, ··· , k2 + k3 − k1 − 2, and for all Dirichlet characters Slide 57 χ mod Npv, v ≥ 1, with all three corresonding Dirichlet characters

χj given by (6.12), having Np-complete conductors. Here the normalisation of Λ∗ includes at the same time certain Gauss sums, 0 0 0 Petersson scalar products, powers of π and of λ(k1, k2, k3), and a certain ﬁnite Euler product.

27 The p-adic Mellin transform and four variable p-adic analytic functions

Any h-admissible measure µ˜ on Y with values in a p-adic Banach

algebra A can be caracterized its Mellin transform Lµ˜(x) R Lµ˜ : X → A, deﬁned by Lµ˜(x) = Y x(y)dµ˜(y), where Slide 58 x ∈ X, Lµ˜(x) ∈ A,

Key property of h-admissible measures µ˜: its Mellin transform Lµ˜ is analytic with values in A.

Let A = A(B) = A1⊗ˆ A2⊗ˆ A3 = A(B1)⊗ˆ A(B2)⊗ˆ A(B3) denote again the Banach algebra A where B is an aﬃnoid neighbourhood 3 around (k1, k2, k3) ∈ X (with a given integer k and Dirichlet character ψ mod N).

Theorem D (on p-adic analytic function in four variables

Put H = [2ordp(λ)] + 1. There exists a p-adic analytic function in

four variables (x, s) ∈ X × B1 × B2 × B3 ⊂ X × X × X × X:

Lµ˜ :(x, s) 7−→ evs(Lµ˜(x))(x ∈ X, Lµ˜(x) ∈ A).

0 0 0 with values in Cp, such that for all (k1, k2, k3) in the aﬃnoid neighborhood as above B = B1 × B2 × B3, satisfying 0 0 0 Slide 59 k1 ≤ k2 + k3 − 2, we have that the special values Lµ˜(x, s) at the r 0 0 0 arithmetical chracters x = ypχ, and s = (k1, k2, k3) ∈ B coincide with the normalized critical special values

∗ 0 0 0 0 0 Λ (f 0 ⊗f 0 ⊗f 0 , k +k −2−r, χ)(r = 0, ··· , k +k −k −2), 1,k1 2,k2 3,k3 2 3 2 3 1 for Dirichlet characters χ mod Npv, v ≥ 1, such that all three

corresonding Dirichlet characters χj given by (6.12), have Np-complete conductors where the same normalisation of Λ∗ as in Theorem C.

28 0 0 0 Moreover, for any ﬁxed s = (k1, k2, k3) ∈ B the function

x 7−→ Lµ˜(x, s)

is p-adic analytic of type o(logH (·)).

Slide 60

Indeed, we obtain the function in question Lµ(x, s) by evaluation at

s = ((s1, ψ1), (s2, ψ2), (s3, ψ3)) ∈ B: this is a p-adic analytic function

in four variables (x, s) ∈ X × B1 × B2 × B3 ⊂ X × X × X × X:

Lµ˜(x, s) := evs(Lµ˜)(x)(x ∈ X, s ∈ B1 × B2 × B3, Lµ˜(x) ∈ A).

This is a joint work in progress with S.Boecherer, we use: 1) the higher twists of the Siegel-Eisenstein series, studied in [PaSE], 2) Ibukiyama’s diﬀerential operators (see [Ibu], [BSY]), producing a

sequence of distributions Φr on a proﬁnite group Y with values in nearly holomorphic triple modular forms M over the algebra A. Slide 61 Then we apply the above admissibility criterion of Theorem 7.1 with κ = 2), in order to construct A-valued admissible measures.

v v ¯ Remark. Recall that χ1 mod Np = χ, χ2 mod Np = ψ2ψ3χ, v ¯ χ3 mod Np = ψ1ψ3χ, so that the assumption on χj is quite restrictive. Note however, that the existence of our A-valued admissible measure µ˜λ = `(Φ˜ λ) established without this technical assumption.

29 6.3 Scheme of the Proof

In the case of ﬁxed balanced weights (k1, k2, k3), we construct

certain Q-valued distributions denoted by µf1⊗f2⊗f3,r on the Slide 62 proﬁnite group YN,p, and attached to the special values at s = k2 + k3 − 2 − r with 0 ≤ r ≤ k2 + k3 − k1 − 2 of the triple ρ ρ ρ product L(f1 ⊗ f2 ⊗ f3 , s, ψ1ψ2χ) twisted with a Dirichlet v character ψ1ψ2χ mod Np . We use an integral representation of ∞ this special value in terms of a C -Siegel-Eisenstein series Fχ,r (to

be speciﬁed later), of degree 3 and of weight k2 + k3 − k1.

For an arbitrary Dirichlet character χ mod Npv we consider the Dirichlet characters (6.12):

v v ¯ χ1 mod Np = χ, χ2 mod Np = ψ2ψ3χ, v ¯ 2 χ3 mod Np = ψ1ψ3χ, ψ = χ ψ1ψ2ψ3.

We use the Siegel-Eisenstein series Fχ,r which depends on the 2 character χ, but its precise nebentypus character is ψ = χ ψ1ψ2ψ3, ? v 2 Slide 63 and it is deﬁned by Fχ,r = G (Z, −r; k, (Np ) , ψ), where Z denotes a variable in the Siegel upper half space H3, and the normalized series G?(Z, s; k, (Npv)2, ψ) is given by (5.6). This series depends on s = −r, and for the critical values at integral points s ∈ Z such that 2 − k ≤ s ≤ 0, it represents a (nearly) holomorphic Siegel modular form in the sense of Shimura [ShiAr]:

X k−2r−κ T Fχ,r = det(T) Q(R, T; k − 2r, r)aχ,r(T)q . T

30 Here we use the universal polynomial (5.5).

Our construction consists of the following steps

1) We consider the proﬁnite ring A = lim( /Npv ). Starting N,p ←− Z Z v from any nearly holomorphic Siegel modular form X F = a(T,R)qT, we consider ﬁrst the following partial Fourier Slide 64 T series

v X T ΨF (ε + Np S) := a(T,R)q , (6.14) v T,t12≡ε12 mod Np v t13≡ε13,t23≡ε23 mod Np   t11 t12 t13   in which T = t t t  runs over half integral symmetric  12 22 23 t13 t23 t33 non negative matrices. The series (6.14) deﬁnes a modular

distribution ΨF on the additive proﬁnite group     0 ε ε  12 13      S = SN,p := ε = ε 0 ε  ε12, ε13, ε23 ∈ AN,p  12 23    ε13 ε23 0  Slide 65 as the value of this distribution on the open subset ε + NpvS ⊂ S. This distributions takes values in C∞-(nearly holomorphic) modular forms on the Siegel half plane H3. This construction generalizes the higher twist of Fr, already utilized in the work [Boe-Schm], in a simpler situation.

31 The integral Z ϕ(ε)dΨF (ε) S is deﬁned over a commutative ring A for any periodic function ϕ : S → A as the following formal series: P T ΨF (ϕ) := T ϕ0(T)a(T,R)q , where   Slide 66 0 t12 t13   ϕ0(T) = ϕ t 0 t . The higher twist is given then by  12 23 t13 t23 0 the following choice of the function:

ϕ(T) =χ ¯1(t12)¯χ2(t13)¯χ3(t23),

with Dirichlet charachers χj as above.

2) Next we consider a sequence of the (real analytic)

Siegel-Eisenstein series Fχ,r viewed as formal (nearly holomorphic) Fourier series over Q. Their coeﬃciens admit explicit polynomial X k−2r−κ T expressions: Fχ,r = det(T) Q(R, T; k − 2r, r)aχ,r(T)q . T A crucial point of our construction is the higher twist . We deﬁne

the higher twist of the series Fχ,r by the characters (6.12) as the Slide 67 following formal nearly holomorphic Fourier expansion:

χ¯1,χ¯2,χ¯3 Fχ,r = (6.15) X k−2r−κ T χ¯1(t12)¯χ2(t13)¯χ3(t23) det(T) Q(R, T; k − 2r, r)aχ,r(T)q . T This is a C∞-Siegel modular form, which belongs to the ring of formal Fourier expansion whose coeﬃcients are polynomials in −1 R = (4πIm (Z)) over the ﬁeld Q (embedded into Cp).

32 The coeﬃcient aχ,r(T) is a ﬁnite Euler product

Y −k+2r M(T, k − 2r, ψ) = M`(T, ψ(`)` ), (6.16) `| det(2T)

2 in which M`(T, x) ∈ Z[x], and ψ = χ ψ1ψ2ψ3. It follows that

Y −k 2 2r aχ,r(T) = M`(T, (ψ1ψ2ψ3)(`)` χ(` )` )) Slide 68 `| det(2T) r ∗ This is a ﬁnite linear combination of terms χ(y)y with y ∈ Zp, with algebraic integer coeﬃcients, satisfying generalized Kummer congruences:

X ri v for any ﬁnite linear combination αiχi(y)y ≡ 0 mod p on Y i X v =⇒ αiaχ ,r (T) ≡ 0 mod p O . i i Q i

3) We consider the diagonal embedding

diag : H × H × H → H3, and an algebraic version of the “pull-back”: starting from any P T nearly holomorphic Siegel modular form F = T a(T,R)q , we notice that

∗ X X t1 t2 t3 diag F = F ◦ diag = a(T,R)q1 q2 q3 , (6.17) Slide 69 t1,t2,t3≥0 T:t11=t1, t22=t2,t23=t3 where the sum on the right is ﬁnite. We use an algebraic version of Ibukiyama’s diﬀerential operator, which generalizes the algebraic “pull-back”: starting from any P T nearly holomorphic Siegel modular form F = T a(T,R)q , this operator has the form

0 0 λ ,ν X X 0 0 0 t1 t2 t3 Lk0+s(F ) = P(k1, k2, k3, T)a(T,R)q1 q2 q3 , t1,t2,t3≥0 T:t11=t1, t22=t2,t23=t3

33 0 0 0 0 0 0 0 0 0 where λ = k1 − k3 ≥ µ = k1 − k2 ≥ 0, and P(k1, k2, k3; r; T) is certain Ibukiyama’s polynomial . λ0,ν0 Notice that the operator Lk0+s transforms a nearly holomorphic Siegel modular form of weight k0

0 0 0 to a nearly holomorphic triple modular form of weight (k1, k2, k3) 0 0 0 0 (s = −r, k = k2 + k3 − k1). Slide 70 As a result we obtain a sequence of triple modular distributions

Φr(χ) given by

0 0 r λ ,ν χ¯1,χ¯2,χ¯3 Φr(χ) = 2 · Lk0−r(Fχ,r ) = (6.18) X χ¯1(t12)¯χ2(t13)¯χ3(t23) T k−2r−κ 0 0 0 T det(T) Q(R, T; k − 2r, r)P(k1, k2, k3; r; T)aχ,r(T)q .

Thus we obtain a sequence Φr of distributions on Y with values in

the tensor product Mr∗ (A) ⊗ Mr∗ (A) ⊗ Mr∗ (A) of three certain A-modules of arithmetical nearly holomorphic modular forms (the normalizing factor 2r is neeeded in order to prove certain

congruences between Φr). The important property of these distributions is that the Slide 71 nebentypus character of the triple modular form Φr(χ) is ﬁxed and 0 0 is equal to (ψ1, ψ2, ψ3) (for all (k1, k2, k3) and χ). Using this property we compute the canonical projector

λ πλ : M(A) → M (A)

onto the λ-characteristic A-submodule of the triple Atkin’s

operator UT,p to the triple modular form Φr(χ).

34 We prove that the resulting sequence of modular distributions

πλ(Φr) on the proﬁnite group Y produces a certain p-adic admissible measure Φ˜ λ (in the sense of Amice-Vélu, [Am-V]) with values in a certain locally free A-submodule of ﬁnite rank

λ M (A) ⊂ M(A), M(A) = Mk,r∗ (A) ⊗ Mk,r∗ (A) ⊗ Mk,r∗ (A)

S v of the A-module M(A) = v≥0 Mk,r∗ (Np , ψ1, ψ2, ψ3; A) of formal Slide 72 nearly holomorphic triple modular forms of levels Npv and the

ﬁxed nebentypus characters (ψ1, ψ2, ψ3). We use congruences

between triple modular forms Φr(χ) ∈ M(A) (they have explicit formal Fourier coeﬃcients), and a general admissibility criterion.

4)We use a Q-valued linear form of type D ˜ ˜ ˜ E f1 ⊗ f2 ⊗ f3, h L : h 7−→ D ˜ ˜ ED ˜ ˜ ED ˜ ˜ E f1, f1 f2, f2 f3, f3

deﬁned on the ﬁnite dimensional Q-vector characteristic subspace λ(k0) h ∈ Mk0 (Q) ⊂

∗ ∗ ∗ Mk1,r (Np, ψ1; Q) ⊗ Mk2,r (Np, ψ2; Q) ⊗ Mk3,r (Np, ψ3; Q). This map is then extended to an A-linear map

λ ` = `f1⊗f2⊗f3,λ : M(A) → A; (6.19) Slide 73 on the locally free A-module of ﬁnite rank M(A)λ. λ This map produces a sequence of A-valued distributions µr (χ) ∈ A in such a way that for all suitable weights k0 ∈ B one has

λ × 0 × evk0 (µr (χ)) = L(evk0 (πλ(Φr)(χ))), λ ∈ A , λ(k ) ∈ Q , 0 0 0 0 where k = (k1, k2, k3) ∈ B, evk0 : B → Cp denotes the evaluation map with the property

evk0 : M(A) → Mk0 (Cp).

35 More precisely, we consider three auxilliary families of modular forms

f˜ 0 (z) = (6.20) j,kj ∞ X ν a˜ 0 e(nz) ∈ S 0 (Γ (N p j ), ψ ), (1 ≤ j ≤ 3, ν ≥ 1), n,j,kj kj 0 j j j n=1 with the same eigenvalues as those of (6.13), for all Hecke operators Slide 74 T , with q prime to Np. In our construction we use as f˜ 0 certain q j,kj “easy transforms” of primitive cusp forms in (2.1). In particular, we ˜ ˜ 0 choose as fj certain eigenfunctions fj,k0 = f 0 of the adjoint j j,kj ∗ Atkin’s operator U , in this case we denote by f 0 the p j,kj ,0 corresponding eigenfunctions of Up.

The Q-linear form L produces a Cp-valued admissible measure µ˜λ = `(Φ˜ λ) starting from the modular p-adic admissible measure ˜ λ Φ of stage 3), where ` : M(Cp) → Cp denotes a Cp-linear form, interpolating L.

5) We show that for any appropriate Dirichlet character χ mod Npv the integral

λ µr (χ) = L(πλ(Φr(χ))) ∈ A

Slide 75 0 0 0 evaluated at (k1, k2, k3) ∈ B = B1 × B2 × B3, coincides (up to a normalisation) with the special L-value

∗ ρ ρ ρ 0 0 D (f 0 ⊗ f 0 ⊗ f 0 , k2 + k3 − 2 − r, ψ1ψ2χ) 1,k1 2,k2 3,k3 under the above assumptions on χ and r).

36 We use a general integral representation of Garrett’s type. The basic idea how a Dirichlet character χ is incorporated in the integral representation [Ga87, BoeSP] is somewhat similar to the Slide 76 one used in [Boe-Schm], but (surprisingly) more complicated to carry out. Note however that the existence of a A-valued admissible measure µ˜λ = `(Φ˜ λ) established at stage 4), does not depend on this technical computation.

7 Criterion of admissibility

Theorem 7.1 Let 0 < |α|p < 1 and Suppose that there exists a positive integer κ such that the following conditions are satisﬁed: for all r = 0, 1, ···, h − 1 with h = [κordpα] + 1, and v ≥ 1, v v Φr(a + (Np )) ∈ M(Npκ )(the level condition) (7.1)

Slide 77 and the following p-adic congruence holds: for all w ≥ max(κv, 1) and for all t = 0, 1, ···, h − 1 t X t U w (−a )t−rΦ (a + (Npv)) ≡ 0 mod p−vt (7.2) r p r r=0 (the divisibility condition)

˜ α r v Then the linear form given by Φ (δa+(Npv )yp) := πα(Φr(a + (Np )) on local monomials (for all j = 0, 1, ···, h − 1), is an h-admissible measure: Φ˜ α : Ph(Y, Q) → Mα ⊂ M

37 Proof uses the commutative diagram:

π M(Npv+1, ψ; A) −→α,v Mα(Npv+1, ψ; A)   v  v U y yo U Slide 78 M(Np, ψ; A) −→ Mα(Np, ψ; A) = Mα(Npv+1, ψ; A). πα,0

The existence of the projectors πα,v comes from Coleman’s Theorem A.4.3 [CoPB]. On the right: U acts on the locally free A-module Mα(Npv+1, A)

via the matrix:   α · · · · · · ∗   0 α · · · ∗    ×   where α ∈ A  ..  0 0 . ···   0 0 ··· α v Slide 79 =⇒ U is an isomorphism over Frac(A), and one controls the denominators of the modular forms of all levels v by the relation:

−v v πα,v(h) = U πα,0(U h) =: πα(h) (7.3)

The equality (7.3) can be used as the deﬁnition of πα. The growth

condition (see section 6.1) for πα(Φr) is deduced from the congruences (7.2) between modular forms, using the relation (7.3).

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