Constructions of p-adic L-functions and admissible measures for Hermitian modular forms

HAL Id: hal-01875160

https://hal.archives-ouvertes.fr/hal-01875160

Preprint submitted on 17 Sep 2018

HAL

is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire

HAL, est

destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Constructions of p-adic L-functions and admissible measures for Hermitian modular forms

Alexei Pantchichkine

To cite this version:

Alexei Pantchichkine. Constructions of p-adic L-functions and admissible measures for Hermitian

modular forms. 2018. �hal-01875160�

Constructions of p -adic L -functions and admissible measures for Hermitian modular forms

Alexei PANTCHICHKINE

Institut Fourier, University Grenoble-Alpes

September 20-22, 2018 Sarajevo

To Dear Mirjana Vukovic with admiration

For a primepand a positive integern, the standard zeta functionLF(s)is considered, attached to an Hermitian modular formF =P

HA(H)q^H on the Hermitian upper half planeHnof degreen, whereHruns through semi-integral positive denite Hermitian matrices of degreen, i.e. H ∈Λn(O) over the integers Oof an imaginary quadratic eldK, whereq^H = exp(2πiTr(HZ)). Analyticp- adic continuation of their zeta functions constructed by A.Bouganis in the ordinary case (in [Bou16]

is presently extended to the admissible case via growingp-adic measures. Previously this problem was solved for the Siegel modular forms, [CourPa], [BS00]. Present main result is stated in terms of the Hodge polygonP_H(t) : [0, d]→Rand the Newton polygonP_N(t) = P_N,p(t) : [0, d]→Rof the zeta functionL_F(s)of degreed= 4n. Main theorem gives ap-adic analytic interpolation of the Lvalues in the form of certain integrals with respect to Mazur-type measures.

p-adic zeta functions of modular forms

Since the p-adic zeta function of Kubota-Leopoldt was constructed by p-adic interpolation of zeta-valuesζ(1−k) =−Bk/k(k≥1)[KuLe64], alsop-adic zeta functions ofvarious modular forms were constructed, such asp-adic interpolation of the special values

L∆(s, χ) =

∞

X

n=1

χ(n)τ(n)n^−s, (s= 1,2,· · ·,11), ∆ =

∞

X

n=1

τ(n)qⁿ,

for the Ramanujan functionτ(n)twisted by Dirichlet charactersχ: (Z/p^rZ)^∗→C^∗. Interpolation done in the elliptic and Hilbert modular cases by Yu.I.Manin and B.Mazur, via modular symbols andp-adic integration, see [Ma73], [Ma76]).

In the Siegel modular case Sp(2n,Z) the p-adic standard zeta functions were constructed in [Pa88], [Pa91] viaRankin-Selberg Andrianov's identity(neven), and [BS00] viadoubling method.

In 2000-2001 Mirjana Vukovíc eected an important scientic visit to the Fourier Institut (France) supported by a regional cooperation program, together with Professors A.N.Andrianov (from Sankt-Petersburg, Russia, and Siegfried Boecherer (Mannheim, Germany).

Mirjana Vukovíc prepared and presented in the Seminar of Number Theory of the Fourier Institute the work Structures gradués et paragradués., see [Vu] developping vastly the ideas of Marc Krasner ([Kr44]-[KrVu87]).

Happy Birthday Dear Mirjana!

Best Wishes from Grenoble, Moscow, and from all friends came to the conference in Dubrovnik in September 2016!

Figure 1: Mirijana Vukovic with friends in Dubrovnok, September 2016

Hermitian modular group Γ_n,K and the standard zeta function Z(s,f)(denitions)

Letθ=θK be the quadratic character attached toK, n⁰=_n

2

.

Γn,K=

M = A

C B D

∈GL2n(OK)|M ηnM^∗=ηn

, ηn=

0n

I_n

−In

0_n

,

Z(s,f) =

2n

Y

i=1

L(2s−i+ 1, θⁱ⁻¹)

! X

a

λ(a)N(a)^−s, (dened via Hecke's eigenvalues: f|T(a) =λ(a)f,a⊂OK)

=Y

q

Zq(N(q)^−s)⁻¹(an Euler product over primesq⊂OK, with degZq(X) = 2n, the Satake parametersti,q, i= 1,· · ·, n), D(s,f) =Z(s−`

2+1

2,f) (Motivically normalized standard zeta function with a functional equations7→`−s; rk = 4n, and motivic weight`−1).

Main result:

^p-adic interpolation of all critical valuesD(s,f, χ)normalized by ×Γ_D(s)/Ωf, in the critical stripn≤s≤`−nfor all χmodp^r in both bounded or unbounded case , i.e. when the productαf =

Q

q|p

Qn i=1tq,i

p^−n(n+1)is not ap-adic unit.

The idea of motivic normalization: Ikeda's lifting [Ike08]

The standard Gamma factor of Ikeda's lifting, denoted byf, of an elliptic modular formf extends to a general (not necessarily lifted) Hermitian modular formf of weight`, used as a pattern, namely

S2k+1(Γ0(D), θ)3f f =Lif t(f)∈S2k+2n⁰(ΓK,n), ifn= 2n⁰ is even(E) S2k(SL(Z))3f f =Lif t(f)∈S2k+2n⁰(ΓK,n), ifn= 2n⁰+ 1is odd(O) the standardL− function off =Lif t⁽ⁿ⁾(f) isZ(s,f) =

n

Y

i=1

L(s+k+n⁰−i+ (1/2), f)L(s+k+n⁰−i+ (1/2), f, θ) [Ike08]

=

n−1

Y

i=0

L(s+`/2−i−(1/2), f)L(s+`/2−i−(1/2), f, θ).

because in the lifted casek+n⁰ =`/2, and the Gamma factor of the standard zeta function with the symmetrys7→1−sbecomes (see p.14)Γ_Z(s) =Qn−1

i=0 Γ_C(s+`/2−i−(1/2))<sup>2</sup>. This Gamma factor suggests the following motivic normalizationD(s) =Z(s−(`/2) + (1/2)) with theGamma factor

Γ_D(s) = Γ_Z(s−(`/2) + (1/2)) =

n−1

Y

i=0

Γ_C(s−i)²,

and theL-functionD(s)satises the symmetry s7→`−s of motivic weight`−1 with theslopes 2·0,2·1, . . .2·(n−1),2·(`−n),· · ·,2·(`−1), so thatDeligne's critical valuesare ats=n, . . . , s=`−n.

General zeta functions: critical values and coecients More general zeta functions are Euler products of degreed

D(s, χ) =

∞

X

n=1

χ(n)ann^−s=Y

p

1

Dp(χ(p)p^−s), Λ_D(s, χ) = Γ_D(s)D(s, χ), wheredegDp(X) =dfor all but nitely manyp, andDp(0) = 1.

In many cases algebraicity of the zeta values was proven as D^∗(s₀, χ)

Ω^±_D ∈Q({χ(n), a_n}_n), where D^∗(s, χ)is normalized byΓ_D,

at critical pointss0∈Zcritas linear combinations ofcoecientsandividing outperiodsΩ^±_D, where D^∗(s0, χ) = Λ_D(s0, χ)ifh^`,`= 0.

In p-adic analysis, the Tate eld is used C^p = ˆ¯

Q^p, the completion of an algebraic closure ¯ Q^p, in place ofC. Let us x embeddings

( ip: ¯Q,→Cp

i_∞: ¯Q,→C, and try to continue analytically these zeta values tos∈Zp,χmodp^r.

Main result stated with Hodge/Newton polygons ofD(s)

The Hodge polygonPH(t) : [0, d]→Rof the functionD(s)and the Newton polygonPN,p(t) : [0, d]→Ratpare piecewise linear:

The Hodge polygon of pure weight w has the slopes j of lengthj = h^j,w−j given by Serre's Gamma factors of the functional equation of the form s 7→ w+ 1 −s, relating Λ_D(s, χ) = Γ_D(s)D(s, χ)andΛ_Dρ(w+ 1−s,χ)¯ , whereρis the complex conjugation ofa_n, andΓ_D(s) = Γ_Dρ(s) equals to the productΓ_D(s) =Q

j≤^w₂ Γ_j,w−j(s), where Γ_j,w−j(s) =

(Γ_C(s−j)^hj,w−j, ifj < w,

Γ_R(s−j)^hj,j+ Γ_R(s−j+ 1)^hj,j− , if2j=w, where Γ_R(s) =π^−s2Γs

2

,Γ_C(s) = Γ_R(s)Γ_R(s+ 1) = 2(2π)^−sΓ(s), h^j,j =h^j,j₊ +h^j,j₋ ,X

j

h^j,w−j =d. The Newton polygon atp is the convex hull of points(i,ordp(ai)) (i= 0, . . . , d); its slopes λ are thep-adic valuations ordp(αi) of the inverse rootsαi of Dp(X) ∈Q¯[X]⊂Cp[X]: lengthλ = ]{i|ordp(αi) =λ}.

p-adic analytic interpolation of D(s,f, χ)

The result expresses the zeta values as integrals with respect to p-adic Mazur-type measures.

These measures are constructed from the Fourier coecients of Hermitian modular forms, and from eigenvalues of Hecke operators on the unitary group.

Pre-ordinary case: PH(t) = PN,p(t) at t = ^d₂ The integrality of measures is proven by T.Bouganis [Bou16], representing D^∗(s, χ) = Γ_D(s)D(s, χ) as a Rankin-Selberg type integral at critical points s = m. Coecients of modular forms in this integral satisfy Kummer-type con- gruences and produce certain bounded measures µ_D from integral representations and Petersson product, [CourPa]. For the case ofpinert in K, see [Bou16].

Admissible case: h=PN(^d₂)−PH(^d₂)>0 The zeta distributions are unbounded, but their sequence produceh-admissible (growing) measures of Amice-Vélu-type, allowing to integrate any continuous characters y ∈ Hom(Z^∗p,C^∗p) = Yp. A general result is used on the existence of h- admissible (growing) measures from binomial congruences for the coecients of Hermitian modular forms. Their p-adicMellin transforms LD(y) = R

Z^∗py(x)dµ_D(x), LD : Yp → Cp give p-adic an- alytic interpolation of growthlog^h_p(·) of theL-values: the values LD(χx^m_p) are integrals given by ip

D^∗(m,f, χ) Ωf

∈Cp.

A Hermitian modular form of weight ` with characterσ

is a holomorphic function f on Hn (n ≥ 2) such that f(ghZi) = σ(g)f(Z)j(g, Z)^` for any g ∈ Γn,K. Here σ be a character of Γ⁽ⁿ⁾_K , trivial on n₁

n

0 B 1_n

o

, and for Z ∈ Hn, put ghZi = (AZ+B)(CZ+D)⁻¹,j(g, Z) = det(CZ+D).

Fourier expansions: a semi-integral Hermitian matrix is a Hermitian matrixH ∈(√

−DK)⁻¹Mn(O) whose diagonal entries are integral.

Denote the set of semi-integral Hermitian matrices byΛ_n(O), the subset of its positive denite elements isΛ_n(O)⁺, withO=OK.

A Hermitian modular formf is called a cusp formif it has a Fourier expansion of the formf(Z) = X

H∈Λn(O)⁺

A(H)q^H.Denote the space of cusp forms of weight`with characterσ byS`(Γn,K, σ).

The standard zeta function of a Hermitian modular form

For all integral idealsa⊂OletT(a)denotes the Hecke operator associated to it as in [Shi00], page 162, using the action of double cosetsΓξΓwithξ= diag( ˆD, D),(det(D)) = (α),Dˆ = (D^∗)⁻¹, α∈a.

Consider a non-zero Hermitian modular formf ∈M`(Γ), for a (congruence) subgroupΓ⊂Γn,K, and assumef|T(a) =λ(a)f withλ(a)∈Cfor all integral idealsa⊂O. Then

Z(s,f) =

2n

Y

i=1

L(2s−i+ 1, θⁱ⁻¹)

! X

a

λ(a)N(a)^−s, the sum is over all integral ideals ofOK.

This series has an Euler product representationZ(s,f) =Q

q(Zq(N(q)^−s)⁻¹, where the product is over all prime ideals ofOK,Zq(X)is the numerator of the seriesP

r≥0λ(q^r)X^r∈C(X), computed by Shimura as follows.

Euler factors of the standard zeta function, [Shi00], p. 171

The Euler factors Zq(X)in the Hermitian modular case at the prime idealqofOK are

(i)Zq(X) =

n

Y

i=1

(1−N(q)ⁿ⁻¹tq,iX)(1−N(q)ⁿt⁻¹_q,iX)−1

, ifq^ρ=q andq6 |c,(the inert case outside levelc), (ii)Zq1(X₁)Zq2(X₂) =

2n

Y

i=1

(1−N(q₁)²ⁿt⁻¹_q

1q2,iX₁)(1−N(q₂)⁻¹t_q1q2,iX₂)−1

, ifq16=q2,q^ρ₁=q2 andqi 6 |cfori= 1,2 (the split case outside level), (iii)Zq(X) =

n

Y

i=1

1−N(q)ⁿ⁻¹tq,iX⁻¹

, ifq^ρ=qandq|c(inert level divisors ), (iv)Zq₁(X1)Zq₂(X2) =

n

Y

i=1

(1−N(q1)ⁿ⁻¹t⁻¹_q

1q₂,iX1)(1−N(q2)ⁿ⁻¹tq₁q₂,iX2)⁻¹ , ifq16=q2,qi|cfori= 1,2 (split level divisors).

where thet_?,iabove for ? =q,q₁q₂, are theSatake parameters of the eigenformf.

The standard motivic-normalized zeta D(s,f, χ)

The standard zeta function of f is dened by means of thep-parameters as the following Euler product:

D(s,f, χ) =Y

p 2n

Y

i=1

1−χ(p)α_i(p)

p^s 1−χ(p)α_4n−i(p) p^s

−1

,

whereχ is an arbitrary Dirichlet character. Thepparametersα1(p), . . . , α4n(p)ofD(s,f, χ)forp not dividing the levelC of the formf are related to the the4ncharacteristic numbers

α1(p), · · ·, α2n(p), α2n+1(p), · · ·, α4n(p)

of the product of allq-factorsZq(Nq^(`−1)/2)X)⁻¹ for allq|p, which is a polynomial of degree 4nof the variableX=p^−s(for almost allp) with coecients in a number eldT =T(f).

There is a relation between the two normalizationsZ(s−₂^`+¹₂,f) =D(s,f)explained in [Ha97]

for general zeta functions.

Description of the Main theorem

Let Ωf be a period attached to an Hermitian cusp eigenform f, D(s,f) = Z(s− ^`₂+¹₂,f)the standard zeta function, and

αf =αf,p=



 Y

q|p n

Y

i=1

tq,i



p^−n(n+1), h= ordp(αf,p), The numberαf turns out to be an eigenvalue of Atkin's type operatorUp:P

HAHq^H 7→P

HApHq^H on somef0, andh=PN(^d₂)−PH(^d₂).

Denition.

^{Let M be a} O-module of nite rank where O ⊂ Cp. For h ≥ 1, consider the followingCp-vector spaces of functions onZ^∗p : C^h⊂C^loc−an⊂C. Then

- a continuous homomorphismµ:C→M is called a(bounded) measureM-valued measure on Z^∗p.

-µ:C^h→M is called anhadmissible measureM-valued measure onZ^∗pmeasure if the following growth condition is satised

Z

a+(p^v)

(x−a)^jdµ _p

≤p^−v(h−j)

for j = 0,1, ..., h−1, and et Yp = Hom_cont(Z^∗p,C^∗p) be the space of denition of p-adic Mellin transform

Theorem

([Am-V], [MTT]) For anh-admissible measureµ, the Mellin transformLµ:Yp→C^p exists and has growtho(log^h)(with innitely many zeros).

Main Theorem.

Let f be a Hermitian cusp eigenform of degree n ≥2 and of weight ` >4n+ 2. There exist distributionsµ<sub>D,s</sub> fors=n,· · ·, `−nwith the properties:

i) for all pairs(s, χ)such thats∈Zwithn≤s≤`−n, Z

Z^∗p

χdµ_D,s=Ap(s, χ)D^∗(s,f, χ) Ωf

(under the inclusionip), with elementary factorsAp(s, χ) = Q

q|pAq(s, χ)including a nite Euler product, Satake parameterst_q,i, gaussian sums, the conductor ofχ; the integral is a nite sum.

(ii) iford_p (Q

q|p

Qn

i=1t_q,i)p^−n(n+1)

= 0then the above distributionsµ_D,s are bounded mea- sures, we setµ_D=µ_D,s^∗and the integral is dened for all continuous charactersy∈Hom(Z^∗p,C^∗p) =:

Yp.

Their Mellin transforms Lµ_D(y) = R

Z^∗p

ydµ_D, Lµ_D : Yp → Cp, give bounded p-adic analytic interpolation of the above L-values to on the Cp-analytic group Yp; and these distributions are related by: Z

X

χdµ_D,s= Z

X

χx^s∗−sdµ^∗_D,X=Z^∗p, wheres^∗=`−n,s_∗=n.

Main theorem (continued)

(iii) in the admissible case assume that 0 < h ≤ s^∗−s_∗+ 1

2 = `+ 1−2n

2 , where h = ord_p

(Q

q|p

Qn

i=1t_q,i)p^−n(n+1)

>0, Then there exist hadmissible measures µ_D whose integrals R

Z^∗p

χx^s_pdµ_D are given by i_p

A_p(s, χ)D^∗(s,f, χ) Ωf

∈Cp withA_p(s, χ)as in (i); their Mellin trans- formsLD(y) =R

Z^∗pydµ_D, belong to the type o(logx^h_p). (iv) the functionsLD are determined by (i)-(iii).

Remarks.

(a) Interpretation ofs^∗: the smallest of the "big slopes" ofPH

(b) Interpretation of s∗−1: the biggest of the "small slopes" ofPH.

Eisenstein series and congruences (KEY POINT!)

The (Siegel-Hermite) Eisenstein series E2`,n,K(Z) of weight2`, character det<sup>−`</sup>, is dened in [Ike08] by E<sub>2`,n,K</sub>(Z) = X

g∈Γn,K,∞\Γn,K

(detg)^{`</sup>j(g, Z)<sup>−2`} (converges for ` > n). The normalized Eisenstein series is given byE2`,n,K(Z) = 2⁻ⁿQn

i=1L(i−2`, θ<sup>i−1</sup>)·E2`,n,K(Z). If H ∈Λn(O)⁺, then theH-th Fourier coecient ofE⁽ⁿ⁾2` (Z)ispolynomial overZin variables{p<sup>`−(n/2)</sup>}p, and equals

|γ(H)|^`−(n/2) Y

p|γ(H)

F˜_p(H, p^−`+(n/2)), γ(H) = (−D_K)^[n/2]detH.

Here, F˜p(H, X)is a certain Laurent polynomial in the variables {Xp =p^−s, X_p⁻¹}p over Z. This polynomial is akey point in proving congruencesfor the modular forms in aRankin-Selberg integral.

Also, for a certain congruence subgroup C = Γc, s∈C and a Hecke ideal characterψmodc, the series is dened

E(Z, s, `, ψ) = X

g∈C∞\C

ψ(g)(detg)^{`</sup>j(g, Z)<sup>−2`}|(detg)j(g, Z)|^−s.

An integral representation of Rankin-Selberg type

The integral representation of Rankin-Selberg type in the Hermitian modular case: is stated for the level c moodular forms:

Theorem 4.1 (Shimura, Klosin)

, see [Bou16], p.13. Let 0 6= f ∈ M`(Γ<sub>c</sub>, ψ)) of scalar weight `, ψmodc, such that ∀a,f|T(a) = λ(a)f, and assume that 2`≥n, then there existsT∈S₊∩GL_n(K)andR∈GL_n(K)such that

Γ((s))ψ(det(T))Z(s+ 3n/2,f, χ) =

Λc(s+ 3n/2, θψχ)·C0hf, θ_T(χ)E(¯s+n, `−`θ, χ^ρψ)iC⁰⁰,

whereE(Z, s, `−`θ, ψ)C⁰⁰ is a normalized group theoretic (or adelic) Eisenstein series with compo- nents as above of levelc⁰⁰ divisible byc, and weight`−`θ. Hereh·,·iC⁰⁰ is the normalized Petersson inner product associated to the congruence subgroupC⁰⁰ of level c⁰⁰. Γ((s)) = (4π)^−n(s+h)Γ^ι_n(s+ h),Γ^ι_n(s) =π^n(n−1)2

n−1

Y

j=0

Γ(s−j), whereh= 0or1, C0 the index of a subgroup.

Proof of the Main Theorem (ii): Kummer congruences Let us se the notationD^algp (m,f, χ) =Ap(s, χ)D^∗(m,f, χ)

Ωf The integrality of measures is proven representing D^algp (m, χ) as Rankin-Selberg type integral at critical points s =m. Coecients of modular forms in this integral satisfy Kummer-type congruences and produce bounded measures µ_D whose construction reduces to congruences of Kummer type between the Fourier coecients of modular forms, see also [Bou16]. Suppose that we are given innitely many "critical pairs"

(sj, χj)at which one has an integral representation D^algp (sj,f, χj) =Ap(s, χ)hf, hji

Ω_f with all hj =

P

Tbj,Tq^T∈Min a certain nite-dimensional spaceMcontainingf and dened overQ. We prove¯ the following Kummer-type congruences:

∀x∈Z^∗p, X

j

βjχjx^kj ≡0 modp^N =⇒X

j

βjD^algp (sj,f, χ)≡0 modp^N βj∈Q¯, kj =s^∗−sj, where s^∗=`−nin our case.

Computing the Petersson products

of a given modular form f(Z) = P

HaHq^H ∈ M∗( ¯Q) by another modular form h(Z) = P

Hb_Hq^H ∈ M∗( ¯Q) uses a linear form`_f : h7→ hf, hi hf,fi dened over a subeldk⊂ ¯

Q.

Admissible Hermitian case

Letf ∈S`(C, ψ)be a Hecke eigenform for the congruence subgroupC= Γ<sub>c</sub> of levelc. Letq be a prime ofK over p, which is inertoverQ. Then we say that f ispre-ordinaryat qif there exists an eigenform06=f<sub>0</sub>∈M{p}⊂S`(Cp, ψ)with Satake parameterst_q,i such that

n

Y

i=1

t_q,i

!

N(q)^−n(n+1)2 _p

= 1, wherekkp the normalized absolute value atp.

Theadmissible casecorresponds to



 Y

q|p n

Y

i=1

tq,i



p^−n(n+1) _p

=p^−h for a positiveh >0.

An interpretation of h as the dierenceh =P_N,p(d/2)−P_H(d/2) comes from the above explicit relations.

Existence of h-admissible measures

of Amice-Vélu-type gives an unbounded p-adic analytic interpolation of theL-values of growth log^h_p(·), using the Mellin transform of the constructed measures. This condition says that the productQn

i=1t_p,i is nonzero and divisible by a certain power ofpin O:

ordp



 Y

q|p n

Y

i=1

tq,i

!

p^−n(n+1)



=h.

We use aneasy condition of admissibilityof a sequence of modular distributionsΦj onX=Z^∗p

with values in the semigroup algebraO[[q]] = O[[q^H]]_H∈Λ(O)⁺ as in Theorem 4.8 of [CourPa]. It suces to checkcongruencesof the type (with κ= 4)

U^κv

j

X

j⁰=0

j j⁰

(−a⁰_p)^j−j0Φ_j⁰(a+ (p^v)

∈Cp^vjO[[q]]

for allj= 0,1, . . . ,κh−1. Heres=s^∗−j⁰,Φj⁰(a+ (p^v))a certain convolution of two Hermitian modular forms, i.e.

Φj⁰(χ) =θ(χ)·E(s, χ)

of a Hermitian theta seriesθ(χ)and an Eisenstein seriesE(s, χ)with any Dirichlet characterχmod p^r. We use a general sucient condition of admissibility of a sequence of modular distributionsΦ_j onX=Z^∗p with values inO[[q]]as in Theorem 4.8 of [CourPa].

Proof of the Main Theorem (iii): (admissible case)

Using a Rankin-Selberg integral representation forD^alg(s,f, χ)and an eigenfunctionf0of Atkin's operatorU(p)of eigenvalueαf onf0 the Rankin-Selberg integral ofFs,χ:=θ(χ)·E(s, χ)gives

D^alg(s,f, χ) =hf0, θ(χ)·E(s, χ)i

hf,fi (the Petersson product onG=GU(ηn))

=α^−v_f hf₀, U(p^v)(θ(χ)·E(s, χ))i

hf,fi =α^−v_f hf₀, U(p^v)(Fs,χ)i hf,fi .

Modication in the admissible case: instead of Kummer congruences, to estimate p-adically the integrals of test functions: M =p^v: Z

a+(M)

(x−a)^jdD^alg :=

j

X

j⁰=0

j j⁰

(−a)^j−j0 Z

a+(M)

x^j0dD^alg, using the orthogonality of characters and the sequence of zeta distributions Z

a+(M)

x^jdD^alg = 1

](O/MO)^× X

χmodM

χ⁻¹(a) Z

X

χ(x)x^jdD^alg,R

XχdD^algs^∗−j=D^alg(s^∗−j, f, χ) =:R

Xχ(x)x^jdD^alg.

Congruences between the coecients of the Hermitian modular forms

In order to integrate any locally-analytic function onX, it suces to check the following binomial congruences for the coecients of the Hermitian modular formFs^∗−j,χ=P

ξv(ξ, s^∗−j, χ)q^ξ :for v0, and a constantC

1 ](O/MO)^×

j

X

j⁰=0

j j⁰

(−a)^j−j0 X

χmodM

χ⁻¹(a)v(p^vξ, s^∗−j⁰, χ)q^ξ

∈Cp^vjO[[q]] (This is a quasimodular form ifj⁰6=s^∗)

The resulting measure µ_D allows to integrate all continuous characters inYp = Hom_cont(X,C^∗p), including Hecke characters, as they are always locally analytic.

Itsp-adic Mellin transformLµ_D is an analytic function onYpof the logarithmic growthO(log^h), h= ordp(α).

Proof of the main congruences

Thus the Petersson product in `_f can be expressed through the Fourier coecients of h in the case when there is a nite basis of the dual space consisting of certain Fourier coecients:

`<sub>Ti</sub>:h7→b<sub>Ti</sub>(i= 1, . . . , n). It follows that `f(h) =P

iγib_Ti, whereγi∈k.

Using the expression for`f(hj) =P

iγi,jbj,Ti, the above congruences reduce to X

i,j

γ_i,jβ_jb_j,Ti≡0 modp^N.

The last congruence is done by an elementary check on the Fourier coecientsb_j,Ti. The abstract Kummer congruences are checked for a family of test elements.

In the admissible case it suces to check binomial congruences for the Fourier coecients as above in place of Kummer congruences.

Appendix A. Rewriting the local factor at pwith character θ Notice that ifθ is the quadratic character attached toK/Qthen

(1−αpX)(1−αpθ(p)X) =







(1−αpX)² ifθ(p) = 1, pr=q1q2, N(qi) =p, (1−α²_pX²), ifθ(p) =−1, pr=q, N(q) =p², (1−αpX) ifθ(p) = 0, pr=q², N(q) =p.

Thus, ifX=p^−s,X²=p^−2s,N(q) =p,Zq(X)⁻¹

=





 Q2n

i=1(1−N(q1)²ⁿt⁻¹_q1q2,iX)(1−N(q2)⁻¹tq1q2,iX), ifθ(p) = 1, Qn

i=1(1−N(q)ⁿ⁻¹tq,iX²)(1−N(q)ⁿt⁻¹_q,iX²), ifθ(p) =−1, Qn

i=1(1−N(q)ⁿ⁻¹tq,iX)(1−N(q)ⁿt⁻¹_q,iX), ifθ(p) = 0.

=





 Qn

i=1(1−γ_p,iX)²Qn

i=1(1−δ_p,iX)² ifθ(p) = 1, i.e. pr=q₁q₂, Qn

i=1(1−α²_p,iX²)Qn

i=1(1−β_p,i² X²), ifθ(p) =−1, i.e. pr=q, Qn

i=1(1−α⁰_p,iX)Qn

i=1(1−β_p,i⁰ X) ifθ(p) = 0, i.e. pr=q², whereα⁰_p,i=pⁿ⁻¹tq,i,β_p,i⁰ pⁿt⁻¹_q,i,γp,i=p²ⁿt⁻¹_q1q2,i,p⁻¹tq₁q₂,i. It follows thatQ

q|pZq(N(q)^−n−(1/2)X) = X⁴ⁿ+· · ·

Appendix A (continued).Relations between αi(p) and ti,q

were studied and explained by M.Harris [Ha97] for general Hermitian zeta functionsZ(s,f)of type introduced in [Shi00], using reprsentation theory of unitary groups and Deligne's approach to L-functions, see [De79], in terms of an-dimensional Galois representations ρλ: Gal( ¯K/K)−→

GL(Mf,λ) ∼= GLn(Eλ) over a completion Eλ of a number eld E containing K and the Hecke eigenvalues of a vector-valued Hermitian modular formf:

Z(s−n⁰−1

2,f) =D(s,f) =L(s, Mf,λM(ψ))

for an algebraic Hecke ideal characterψas above of the innity typemψ, see [GH16], p.20. Here the symbolL(s, Mf,λM(ψ))denotes the Rankin-Selberg type convolution (it corresponds to tensor product of Galois representations). Notice thatL(s, M_f,λ)is of degree2n, and L(s, M_f,λM(ψ)) is of degree4nbecauseL(s, ψ) =L(s, R(ψ))is of degree 2.

Moreover, M.Harris suggested a general description of D(s) with given Gamma factors and analytic properties as someD(s,f)some under natural conditions on Gamma factors, giving higher

versions of Shimura-Taniyama-Weil conjecture (i.e. higher Wiles' modularity theorem). This can be stated also over a totally real eldF(instead ofQ), and its quadratic totally imaginary extension K, see [GH16], [Pa94].

Appendix B. Shimura's Theorem: algebraicity of critical values in Cases Sp and UT, p.234 of [Shi00]

Letf ∈V( ¯Q)be a non zero arithmetical automorphic form of type Sp or UT. Letχbe a Hecke character of K such that χ_a(x) = x^{`</sup><sub>a</sub>|x<sub>a</sub>|<sup>−`} with ` ∈ Z<sup>a</sup>, and let σ<sub>0</sub> ∈ 2<sup>−1</sup>Z. Assume, in the notations of Chapter 7 of [Shi00] on the weightskv, µv, `v, that

Case Sp 2n+ 1−kv+µv≤2σ0≤kv−µv, whereµ_v= 0 if[k_v]−l_v∈2Z

andµv= 1if[kv]−lv6∈2Z; σ0−kv+µv

for everyv∈a ifσ₀> nand

σ0−1−kv+µv ∈2Zfor everyv∈aifσ0≤n.

Case UT 4n−(2kvρ+`v)≤2σ0≤mv− |kv−kvρ−`v| and2σ0−`v∈2Zfor everyv∈a.

Appendix B. Shimura's Theorem (continued) Further exclude the following cases

(A) Case Sp σ0=n+ 1, F =Qandχ²= 1;

(B) Case Sp σ₀=n+ (3/2), F =Q;χ²= 1and[k]−`∈2Z (C) Case Sp σ0= 0,c=gandχ= 1;

(D) Case Sp 0< σ0≤n,c=g, χ²= 1 and the conductor ofχisg;

(E) Case UT 2σ₀= 2n+ 1, F =Q, χ₁=θ, andk_v−k_vρ=`_v;

(F) Case UT 0≤2σ0<2n,c=g, χ1=θ^2σ0 and the conductor ofχ isr Then Z(σ₀,f, χ)/hf,fi ∈π^n|m|+dεQ¯,

whered= [F :Q],|m|=P

v∈am_v, and

ε=







(n+ 1)σ₀−n²−n, Case Sp, k∈Z^a, andσ₀> n₀), nσ0−n², Case Sp, k6∈Z^a,orσ0≤n0), 2nσ₀−2n²+n Case UT

Notice that π^n|m|+dε ∈ Z in all cases; if k 6∈ Z^a, the above parity condition on σ₀ shows that σ₀+k_v∈Z, so thatn|m|+dε∈Z.

Appendix C. Examples of Hermitian cusp forms

The Hermitian Ikeda lift, [Ike08]

^{. Assumen= 2n0 even. Letf(τ) = X∞}

N=1

a(N)q^N ∈ S2k+1(Γ0(DK), χ)be a primitive form, whoseL-function is given by

L(f, s) = Y

p6 |DK

(1−a(p)p^−s+θ(p)p^2k−2s)⁻¹ Y

p|DK

(1−a(p)p^−s)⁻¹.

For each primep6 |DK, dene the Satake parameter{αp, βp}={αp, θ(p)α⁻¹_p }by (1−a(p)X+θ(p)p^2kX²) = (1−p^kαpX)(1−p^kβpX) Forp|DK, we putαp=p^−ka(p). Put

A(H) =|γ(H)|^k Y

p|γ(H)

F˜p(H;αp), H∈Λn(O)⁺

f(Z) = X

H∈Λ_n(O)⁺

A(H)q^H, Z∈H2n.

Appendix C (continued).The rst theorem (even case)

Theorem 5.1 (Case E) of [Ike08]

Assume thatn= 2n⁰ is even. Letf(τ),A(H)and f(Z)be as above. Then we havef ∈S2k+2n⁰(Γ⁽ⁿ⁾_K ,det^−k−n0).

In the case whennis odd, consider a similar lifting for a normalized Hecke eigenformn= 2n⁰+ 1 is odd. Letf(τ) =

∞

X

N=1

a(N)q^N ∈S2k(SL2(Z))be a primitive form, whoseL-function is given by

L(f, s) =Y

p

(1−a(p)p^−s+p^2k−1−2s)⁻¹. For each primep, dene the Satake parameter{αp, α⁻¹_p } by

(1−a(p)X+p^2k−1X²) = (1−p^k−(1/2)αpX)(1−p^k−(1/2)α⁻¹X).

Put

A(H) =|γ(H)|^k−(1/2) Y

p|γ(H)

F˜_p(H;α_p), H ∈Λ_n(O)⁺ f(Z) = X

H∈Λn(O)⁺

A(H)q^H, Z∈Hn.

Appendix C (continued). The second theorem (odd case)

Theorem 5.2 (Case O) of [Ike08]

. Assume thatn= 2n⁰+ 1is odd. Let f(τ), A(H) andf(Z)be as above. Then we havef ∈S2k+2n⁰(Γ⁽ⁿ⁾_K ,det^−k−n0).

The lift Lif t⁽ⁿ⁾(f) of f is a common Hecke eigenform of all Hecke operators of the unitary group, if it is not identically zero (Theorem 13.6).

Theorem 18.1 of [Ike08]

^{. Letn,n0, andf} be as in Theorem 5.1 or as in Theorem 5.2.

Assume thatLif t⁽ⁿ⁾(f)6= 0. LetL(s, Lif t⁽ⁿ⁾(f), st)be theL-function ofLif t⁽ⁿ⁾(f)associated to st:^LG→GL4n(C). Then up to bad Euler factors,L(s, Lif t⁽ⁿ⁾(f), st)is equal to

n

Y

i=1

L(s+k+n⁰−i+1

2, f)L(s+k+n⁰−i+1 2, f, θ).

Moreover, the4ncharcteristic roots ofL(s, Lif t⁽ⁿ⁾(f), st)given as follows: fori= 1,· · ·, n α_pp^{−k−n0+i−12}, α⁻¹_p p^{−k−n0+i−12}, θ(p)α_pp^{−k−n0+i−12}, θ(p)α⁻¹_p p^{−k−n0+i−12}

Functional equation of the lift (thanks to Sho Takemori!) There are two cases [Ike08]: the even case (E) and the odd case (O):











f ∈S_2k+1(Γ₀(D), θ),f =Lif t⁽ⁿ⁾(f)∈S2k+2n⁰(Γ_K,n) (E) (of even degreen= 2n⁰ and of weight 2k+ 2n⁰)

f ∈S_2k(SL(Z)),f =Lif t⁽ⁿ⁾(f)∈S2k+2n⁰(Γ_K,n) (O) (of odd degreen= 2n⁰+ 1 and of weight2k+ 2n⁰).

Then, up to bad Euler factors, the

standardL-function off =Lif t⁽ⁿ⁾(f)is given by Z(s,f)=Qn

i=1L(s+k+n⁰−i+¹₂, f)L(s+k+ n⁰−i+¹₂, f, θ)

Let us denotet(s, i) =s+k+n⁰−i+¹₂ then

=





























 Q2n⁰

i=1L(s+k+n⁰−i+¹₂, f)L(s+k+n⁰−i+¹₂, f, θ) (E) Qn⁰

i=1L(t(s, i), f)L(t(s, n+ 1−i), f) L(t(s, i), f, θ)L(t(s, n+ 1−i), f, θ) Q2n⁰+1

i=1 L(s+k+n⁰−i+¹₂, f)

×L(s+k+n⁰−i+¹₂, f, θ) (O)

=L(s+k−¹₂, f)L(s+k−¹₂, f, θ) Qn⁰

i=1L(t(s, i), f)L(t(s, n+ 1−i), f) L(t(s, i), f, θ)L(t(s, n+ 1−i), f, θ).

The Gamma factor Γ_Z(s) of Ikeda's lift

In the even caset(1−s, n+ 1−i) =t(1−s,2n⁰+ 1−i)=(2k+ 1)−t(s, i). The Hecke functional equations7→2k+ 1−sin all symmetric terms of the product, gives the functional equation of the standardL-function of the forms7→1−s, and the gamma factor is then

n

Y

i=1

Γ_C(s+k+n⁰−i+ 1/2)²= Γ_D(s+n⁰+1 2).

In the odd casen= 2n⁰+ 1whenf ∈S_2k(SL₂(Z)), the Lif t(f)∈S2k+2n⁰(Γ_K,n). By2k−t(s, i)

=t(1−s, n+ 1−i), the standardLfunctions has functional equation of the form s7→1−sand the gamma factor is the same.

Hence the Gamma factor of Ikeda's lifting, denoted byf, of an elliptic modular formf andused as a pattern, extends to a general (not necessarily lifted) Hermitian modular formf of even weight

`, which equals in the lifted case to`= 2k+ 2n⁰, wherek= (`−2n<sup>0</sup>)/2 =`/2−n⁰=`/2−n⁰, when the Gamma factor of the standard zeta function with the symmetrys7→1−sbecomes (see p.14) Qn

i=1Γ_C(s+`/2−n⁰+n⁰−i+(1/2))²=Qn

i=1Γ_C(s+`/2−i+(1/2))²=Qn−1

i=0 Γ_C(s+`/2−i−(1/2))².

Acknowledgement

Many thanks to Academy of Sciences and Arts of Bosnia and Herzegovina (ANUBiH) for the invita- tion to International Scientic Conference "Modern Algebra and Analysis and their Applications", to Siegfried Boecherer (Mannheim), Sho Takemori (MPIM) and Emmanuel Royer (University Cler- mont Auvergne) for valuable discussions and observations.

Best wishes to dear Mirjana Vukovic of good health, enthusiasm in life and mathematics, hap- pieness and many new good friends and achievements!

References

[Am-V] Amice, Y. and Vélu, J., Distributionsp-adiques associées aux séries de Hecke, Journées Arithmétiques de Bordeaux (Conf. Univ. Bordeaux, 1974), Astérisque no. 24/25, Soc.

Math. France, Paris 1975, pp. 119-131

[Boe85] Böcherer, S., Über die Funktionalgleichung automorpherLFunktionen zur Siegelscher Modulgruppe. J. reine angew. Math. 362 (1985) 146-168

[BoeNa13] Boecherer, S., Nagaoka, S. , Onp-adic properties of Siegel modular forms, in: Auto- morphic Forms. Research in Number Theory from Oman. Springer Proceedings in Math- ematics and Statistics 115. Springer 2014.

[Boe-Pa11] Böcherer, S., Panchishkin, A.A., Higher Twists and Higher Gauss Sums Vietnam Journal of Mathematics 39:3 (2011) 309-326

[BS00] Böcherer, S., and Schmidt, C.-G.,p-adic measures attached to Siegel modular forms, Ann. Inst. Fourier 50,N^◦5, 1375-1443 (2000).

[Bou14] Bouganis T., Non-abelian p-adic L-functions and Eisenstein series of unitary groups;

the CM method, Ann. Inst. Fourier (Grenoble), 64 no. 2 (2014), p. 793-891.

[Bou16] Bouganis T., p-adic Measures for Hermitian Modular Forms and the Rankin-Selberg Method. in Elliptic Curves, Modular Forms and Iwasawa Theory - Conference in honour of the 70th birthday of John Coates, pp 33-86

[Hur899] Carlitz, L. , The coecients of the lemniscate function, Math. Comp., 16 (1962), 475-478.

[CourPa] Courtieu, M., Panchishkin ,A.A., Non-Archimedean L-Functions and Arithmetical Siegel Modular Forms, Lecture Notes in Mathematics 1471, Springer-Verlag, 2004 (2nd augmented ed.)

[CoWi77] Coates, J. and Wiles, A., On the conjecture of Birch and Swinnerton-Dyer, Inventiones math. 39, 223-251

[Coh17] Cohen, H. Computing L -Functions: A Survey. Journal de théorie des nombres de Bor- deaux, Tome 27 (2015) no. 3 , p. 699-726

[De79] Deligne P., Valeurs de fonctionsLet périodes d'intégrales, Proc.Sympos.Pure Math. vol.

55. Amer. Math. Soc., Providence, RI, 1979 , 313-346.

[EE] Eischen, Ellen E., p-adic Dierential Operators on Automorphic Forms on Unitary Groups. Annales de l'Institut Fourier 62, No.1 (2012) 177-243.

[EHLS] Eischen Ellen E., Harris, Michael, Li, Jian-Shu, Skinner, Christopher M., p-adic L- functions for unitary groups, arXiv:1602.01776v3 [math.NT]

[EZ85] Eichler, M., Zagier, D., The theory of Jacobi forms, Progress in Mathematics, vol. 55 (Birkhéuser, Boston, MA, 1985).

[Ike01] Ikeda, T., On the lifting of elliptic cusp forms to Siegel cusp forms of degree2n, Ann. of Math. (2) 154 (2001), 641-681.

[Ike08] Ikeda, T., On the lifting of Hermitian modular forms, Compositio Math. 144, 1107-1154, (2008)

[Iw] K. Iwasawa, Lectures on p-Adic L-Functions, Ann. of Math. Studies, N^◦ 74. Princeton Univ. Press (1972).

[GMPS14] Gelbart, S., Miller, S.D, Panchishkin, S., and Shahidi, F., Ap-adic integral for the reciprocal ofL-functions. Travaux du Colloque "Automorphic Forms and Related Geome- try, Assessing the Legacy of I.I. Piatetski-Shapiro" (23 - 27 April, 2012, Yale University in New Haven, CT), Contemporary Mathematics, 345-374 (avec Stephen Gelbart, Stephen D. Miller, and Freydoon Shahidi), 53-68, 2014.

[GeSha] Gelbart, S., and Shahidi, F., Analytic Properties of Automorphic L-functions, Aca- demic Press, New York, 1988.

[GPSR] Gelbart S.,Piatetski-Shapiro I.I., Rallis S. Explicit constructions of automorphic L-functions. Springer-Verlag, Lect. Notes in Math. N 1254 (1987) 152p.

[Ge16] Guerberoff, L., Period relations for automorphic forms on unitary groups and critical values ofL-functons, Preprint, 2016.

[GH16] Grobner, H. and Harris, M. Whittaker periods, motivic periods, and special values of tensor productl-functions, Journal of the Institute of Mathematics of Jussieu Volume 15, Issue 4, October 2016, pp. 711-769

[Ha81] Harris, M., Special values of zeta functions attached to Siegel modular forms. Ann. Sci.

Ecole Norm Sup. 14 (1981), 77-120.

[Ha97] Harris, M., L-functions and periods of polarized regular motives. J. Reine Angew. Math, (483):75-161, 1997.

[Ha14] Harris, M., Automorphic Galois representations and the cohomology of Shimura vari- eties. Proceedings of the International Congress of Mathematicians, Seoul, 2014

[Hur1899] Hurwitz, A., Ueber die Entwicklungskoezienten der lemniskatischen Funktionen, Math. Ann., 51 (1899), 196-226; Mathematische Werke. Vols. 1 and 2, Birkhaeuser, Basel, 1962-1963, see Vol. 2, No. LXVII.

[Ich12] Ichikawa, T., Vector-valued p-adic Siegel modular forms, J. reine angew. Math., DOI 10.1515/ crelle-2012-0066.

[Ka76] Katz, N.M., p-adic interpolation of real analytic Eisenstein series. Ann. of Math. 104 (1976) 459571

[Ka78] Katz, N.M.,p- adic L-functions for CM-elds. Invent. Math. 48 (1978) 199-297

[KiNa16] Kikuta, Toshiyuki, Nagaoka, Shoyu, Note on mod p property of Hermitian modular forms arXiv:1601.03506 [math.NT]

[Kl] Klosin ,K., Maass spaces on U(2,2) and the Bloch-Kato conjecture for the symmetric square motive of a modular form, Journal of the Mathematical Society of Japan, Vol. 67, No. 2 (2015) pp. 797-860.

[Ko80] Koblitz, Neal, p-adic Analysis. A Short Course on Recent Work, Cambridge Univ. Press, 1980

[KuLe64] Kubota, T., Leopoldt, H.W. (1964): Einep−adische Theorie der Zetawerte. I. J. reine u. angew. Math., 214/215, 328-339 (1964).

[Kr44] Krasner, M., Une généralisation de la notion de corps-corpoéde. Un corpoéde remar- quable de la théorie des corps valués, C. R. Acad. Sci. Paris Sér. IMath. 219 (1944), 345-347.

[Kr50] Krasner, M., Quelques méthodes nouvelles dans la théorie des corps valués complets.

Algébre et Théorie des Nombres, 29-39. Colloques Internationaux du Centre National de la Recherche Scientique, no. 24, Centre National de la Recherche Scientique, Paris, 1950.

[Kr66] Krasner, M., Prolongement analytique uniforme et multiforme, Collogue C.N.R.,S. né 143, Clermont-Ferrand, 1963, Paris, Ed. C.N.R.S., 1966, p. 97-141

[Kr74] Krasner, M., Rapport sur le prologement analytique dans les corps values complete par la methode des e1ements analytiques quasiconnexes, Bull. Soc. Math. France, Mem. 39-40 (1974) 131-254.

[Kr80] Krasner, M., Anneaux gradués généraux, Colloque d'Algébre Rennes (1980), 209-308.

[KrKa51] Krasner, M., Kaloujnine, L. Produit complet des groupes de permutations et probléme d'extension de groupes II , Acta Sci. Math. Szeged , 14 (1951) p. 39-66 et 69-82.

[KrVu87] Krasner, M., Vukovi¢, M. Structures paragraduées (groupes, anneaux, modules), Queen's Papers in Pure and Applied Mathematics, 77, Queen's University, Kingston, Ontario, Canada, 1987.

[LangMF] Lang, Serge. Introduction to modular forms. With appendixes by D. Zagier and Walter Feit. Springer-Verlag, Berlin, 1995

[Ma73] Manin, Yu. I., Periods of cusp forms and p-adic Hecke series, Mat. Sbornik, 92 , 1973, pp. 378-401

[Ma76] Manin, Yu. I., Non-Archimedean integration and Jacquet-Langlands p-adic L-functions, Uspekhi Mat. Nauk, 1976, Volume 31, Issue 1(187), 5-54

[MaPa] Manin, Yu. I.,Panchishkin, A.A., Introduction to Modern Number Theory: Funda- mental Problems, Ideas and Theories (Encyclopaedia of Mathematical Sciences), Second Edition, 504 p., Springer (2005)

[Ma-Vi] Manin, Yu.I., Vishik, M. M.,p-adic Hecke series of imaginary quadratic elds, (Russian) Mat. Sb. (N.S.) 95(137) (1974), 357-383.

[MTT] Mazur, B., Tate J., Teitelbaum, J., On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer. Invent. Math. 84, 1-48 (1986).

[Mi-St] J. Milnor, J. Stasheff, Characteristic Classes, Ann. of Math. Studies N^◦76, Princeton Univ. Press. (1974), p 231-264.

[Pa88] Panchishkin, A.A., Non-Archimedean automorphic zeta functions, Moscow University Press (1988).

[Pa91] Panchishkin, A.A., Non-ArchimedeanL-Functions of Siegel and Hilbert Modular Forms.

Volume 1471 (1991)

[Pa94] Panchishkin, A., Motives over totally real elds and padic Lfunctions. Annales de l'Institut Fourier, Grenoble, 44, 4 (1994), 9891023

[PaMMJ] Panchishkin, A.A., A new method of constructing p-adic L-functions associated with modular forms, Moscow Mathematical Journal, 2 (2002), Number 2, 1-16

[PaTV] Panchishkin, A. A., Two variablep-adicLfunctions attached to eigenfamilies of positive slope, Invent. Math. v. 154, N3 (2003), pp. 551 - 615

[Pa14] Panchishkin, A.A. Analytic constructions of p-adic L-functions and Eisenstein series.

Travaux du Colloque "Automorphic Forms and Related Geometry, Assessing the Legacy of I.I.Piatetski-Shapiro (23-27 April, 2012, Yale University in New Haven, CT)", 345-374, 2014

[Pa16] Panchishkin, A.A., Graded structures and dérential oérators on nearly holomorphic and quasimodular forms on classical groups. Sarajevo Journal of Mathematics Vol.1 (25), No.2, Suppl., 2016, 1-17

[Sha69] Shafarevich, I.R. Zeta Function, Moscow University Press (1969).

[Sl] Sloane N.J.A,, A047817. Denominators of Hurwitz numbers H_n The On-Line Encyclo- pedia of Integer Sequences https://oeis.org/A047817.

[Se73] Serre, J.P., Cours d'arithmétique. Paris, 1970.

[Se70] Serre, J.P., Facteurs locaux des fonctions zêta des variétés algébriques (dénitions et conjectures). Sém. Delange - Pisot - Poitou, exp. 19, 1969/70.

[Se73] Serre, J.P., Formes modulaires et fonctions zêta p-adiques, Lect Notes in Math. 350 (1973) 191268 (Springer Verlag)

[Shi97a] Shimura G., Euler Products and Eisenstein series, CBMS Regional Conference Series in Mathematics, No.93, Amer. Math. Soc, 1997.

[Shi97b] Shimura G., Colloquium Paper: Zeta functions and Eisenstein series on classical groups, Proc Nat. Acad Sci U S A. 1997 Oct 14; 94(21): 11133-11137

[Shi00] Shimura G., Arithmeticity in the theory of automorphic forms, Mathematical Surveys and Monographs, vol. 82 (Amer. Math. Soc., Providence, 2000).

[MC] Skinner, C. and Urban, E. The Iwasawa Main Cconjecture for GL(2). Invent. Math.

195 (2014), no. 1, 1-277. MR 3148103

[U14] Urban, E., Nearly Overconvergent Modular Forms, in: Iwasawa Theory 2012. State of the Art and Recent Advances, Contributions in Mathematical and Computational Sciences book series (CMCS, Vol. 7), pp. 401-441

[VoKar] Voronin, S.M., Karatsuba, A.A., The Riemann zeta-function, Moscow, Fizmatlit, 1994.

[Vu] Vukovi¢, M., Structures graduéees et paragradués. Prpublication de l'Institut Fourier no 531(2006).

[Wa] Washington,L., Introduction to Cyclotomic Fields, Springer (1982).

[YS] Yoshida, H., Review on Goro Shimura, Arithmeticity in the theory of automorphic forms [Shi00], Bulletin (New Series) of the AMS, vol. 39, N3 (2002), 441-448.