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.