Constructing vector-valued automorphic forms on unitary groups

This paper has two main parts. First, we construct certain differential operators, which generalize operators studied by G. Shimura. Then, as an application of some of these differential operators, we construct certain p-adic families of automorphic forms. Building on the author's earlier work, these differential operators map automorphic forms on a unitary group of signature (n,n) to (vector-valued) automorphic forms on the product $U^\varphi\times U^{-\varphi}$ of two unitary groups, where $U^\varphi$ denotes the unitary group associated to a Hermitian form $\varphi$ of arbitrary signature on an n-dimensional vector space. These differential operators have both a p-adic and a C-infinity incarnation. In the scalar-weight, C-infinity case, these operators agree with ones studied by Shimura. In the final section of the paper, we also discuss some generalizations to other groups and settings. The results from this paper apply to the author's paper-in-preparation with J. Fintzen, E. Mantovan, and I. Varma and to her ongoing joint project with M. Harris, J. -S. Li, and C. Skinner; they also relate to her recent paper with X. Wan.

In the 1970's, Serre exploited congruences between $q$-expansion coefficients of Eisenstein series to produce $p$-adic families of Eisenstein series and, in turn, $p$-adic zeta functions. Partly through integration with more recent machinery, including Katz's approach to $p$-adic differential operators, his strategy has influenced four decades of developments. Prior papers employing Katz's and Serre's ideas exploiting differential operators and congruences to produce families of automorphic forms rely crucially on $q$-expansions of automorphic forms. The overarching goal of the present paper is to adapt the strategy to automorphic forms on unitary groups, which lack $q$-expansions when the signature is of the form $(a, b)$, $a\neq b$. In particular, this paper completely removes the restrictions on the signature present in prior work. As intermediate steps, we achieve two key objectives. First, partly by carefully analyzing the action of the Young symmetrizer on Serre-Tate expansions, we explicitly describe the action of differential operators on the Serre-Tate expansions of automorphic forms on unitary groups of arbitrary signature. As a direct consequence, for each unitary group, we obtain congruences and families analogous to those studied by Katz and Serre. Second, via a novel lifting argument, we construct a $p$-adic measure taking values in the space of $p$-adic automorphic forms on unitary groups of any prescribed signature. We relate the values of this measure to an explicit $p$-adic family of Eisenstein series. One application of our results is to the recently completed construction of $p$-adic $L$-functions for unitary groups by the first named author, Harris, Li, and Skinner.

In the 1970's, Serre exploited congruences between q -expansion coefficients of Eisenstein series to produce p -adic families of Eisenstein series and, in turn, p -adic zeta functions. Partly through integration with more recent machinery, including Katz's approach to p -adic differential operators, his strategy has influenced four decades of developments. Prior papers employing Katz's and Serre's ideas exploiting differential operators and congruences to produce families of automorphic forms rely crucially on q -expansions of automorphic forms. The overarching goal of the present paper is to adapt the strategy to automorphic forms on unitary groups, which lack q -expansions when the signature is of the form (a, b) , a\ne b . In particular, this paper completely removes the restrictions on the signature present in prior work. As intermediate steps, we achieve two key objectives. First, partly by carefully analyzing the action of the Young symmetrizer on Serre-Tate expansions, we explicitly describe the action of differential operators on the Serre-Tate expansions of automorphic forms on unitary groups of arbitrary signature. As a direct consequence, for each unitary group, we obtain congruences and families analogous to those studied by Katz and Serre. Second, via a novel lifting argument, we construct a p -adic measure taking values in the space of p -adic automorphic forms on unitary groups of any prescribed signature. We relate the values of this measure to an explicit p -adic family of Eisenstein series. One application of our results is to the recently completed construction of p -adic L -functions for unitary groups by the first named author, Harris, Li, and Skinner.

Unitary groups generated by differential operators have special properties that can be used to study completeness of the set of eigenvectors of the infinitesimal generator. Unitary groups also occur in differential operator theory in another manner, associated with unitary equivalence of differential operators. We discuss what happens to A t = U t − 1 A U t {A_t} = U_t^{ - 1}A{U_t} as t approaches infinity provided that A t {A_t} is a differential operator for all t and A has certain properties.

The goal of this paper is to study certain <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math>-adic differential operators on automorphic forms on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>U</mml:mi><mml:mo>(</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>. These operators are a generalization to the higher-dimensional, vector-valued situation of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math>-adic differential operators constructed for Hilbert modular forms by N. Katz. They are a generalization to the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math>-adic case of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>C</mml:mi> <mml:mi>∞</mml:mi> </mml:msup></mml:math>-differential operators first studied by H. Maass and later studied extensively by M. Harris and G. Shimura. The operators should be useful in the construction of certain <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math>-adic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>L</mml:mi></mml:math>-functions attached to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math>-adic families of automorphic forms on the unitary groups <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>U</mml:mi><mml:mo>(</mml:mo><mml:mi>n</mml:mi><mml:mo>)</mml:mo><mml:mo>×</mml:mo><mml:mi>U</mml:mi><mml:mo>(</mml:mo><mml:mi>n</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>.

Unitary groups generated by differential operators have special properties that can be used to study completeness of the set of eigenvectors of the infinitesimal generator.Unitary groups also occur in differential operator theory in another manner, associated with unitary equivalence of differential operators.We discuss what happens to At=Ut~ AUt as t approaches infinity provided that A, is a differential operator for all t and A has certain properties.

Holomorphic vector valued differential operators acting on Siegel modular forms and preserving automorphy under the restriction to diagonal blocks are important in many respects, including application to critical values of L functions. Such differential operators are associated with vectors of new special polynomials of several variables defined by certain harmonic conditions. They include the classical Gegenbauer polynomial as a prototype, and are interesting as themselves independently of Siegel modular forms. We will give formulas for all such polynomials in two different ways. One is to describe them using polynomials characterized by monomials in off-diagonal block variables. We will give an explicit and practical algorithm to give the vectors of polynomials through these. The other one is rather theoretical but seems much deeper. We construct an explicit generating series of polynomials mutually related under certain mixed Laplacians. Here substituting the variables of the polynomials to partial derivatives, we obtain the generic differential operator from which any other differential operators of this sort are obtained by certain projections. This process exhausts all the differential operators in question. This is also generic in the sense that for any number of variables and block partitions, it is given by a recursive unified expression. As an application, we prove that the Taylor coefficients of Siegel modular forms with respect to off-diagonal block variables, or of corresponding expansion of Jacobi forms, are essentially vector valued Siegel modular forms of lower degrees, which are obtained as images of the differential operators given above. We also show that the original forms are recovered by the images of our operators. This is an ultimate generalization of Eichler–Zagier’s results on Jacobi forms of degree one. Several more explicit results and practical construction are also given.

We wish to use Krasner graded and Krasner-Vukovi´paragraded structures on differential operators and quasimod-ular forms on classical groups and show that these structures provide a tool to construct p-adic measures and p-adic L-functions on the corre-sponding non-archimedean weight spaces. An approach to constructions of automorphic L-functions on unitary groups and their p-adic analogues is presented. For an algebraic group G over a number field K these L functions are certain Euler products L(s, π, r, χ). We present a method using arithmetic nearly-holomorphic forms and general quasi-modular forms, related to algebraic automorphic forms. It gives a technique of constructing p-adic zeta-functions via quasi-modular forms and their Fourier coefficients. * This paper was presented at the International Scientific Conference Graded structures in algebra and their applications, dedicated to the memory of Prof. Marc Krasner, IUCDubrovnik, Cratia, September, 22-24, 2016.

This paper is a joint project with Siegfried Bocherer (Mannheim), developing a recent preprint of Yubo Jin (Durham UK) previous works of Anh Tuan Do (Vietnam) and Dubrovnik, IUC-2016 papers from \textit{Sarajevo Journal of Mathematics} (Vol.12, No.2-Suppl., 2016). We wish to use paragraded structures on {differential operators and arithmetical automorphic forms on classical groups and show that these structures provide a tool to construct $p$-adic measures and $p$-adic $L$-functions on the corresponding non-archimedean weight spaces.} An approach to constructions of automorphic $L$-functions on unitary groups and their $p$-adic analogues is presented. For an algebraic group $G$ over a number field $K$ these $L$ functions are certain Euler products $L(s,\pi, r, \chi)$. In particular, our constructions cover the $L$-functions in \cite{Shi00} via the doubling method of Piatetski-Shapiro and Rallis. A $p$-adic analogue of $L(s,\pi, r, \chi)$ is a $p$-adic analytic function $L_p(s,\pi, r, \chi)$ of $p$-adic arguments $s \in \Z_p$, $\chi \bmod p^r$ which interpolates algebraic numbers defined through the normalized critical values $L^*(s,\pi,r, \chi)$ of the corresponding complex analytic $L$-function. We present a method using arithmetic nearly-holomorphic forms and general quasi-modular forms, related to algebraic automorphic forms. It gives a technique of constructing $p$-adic zeta-functions via general quasi-modular forms and their Fourier coefficients.

Multilinear Operators on Siegel Modular Forms of Genus 1 and 2

Introduction. In his papers [5; 6] Eichler demonstrated the significance for the study of automorphic forms of Bol 's discovery [3 ] of some remarkably simple differential operators taking automorphic forms into automorphic forms. In [6] in particular Eichler discussed a relation between the automorphic forms associated to a transformation group 9 on a Riemann surface D and some purely algebraic constructions involving the group 9, the first cohomology groups of 9 with certain modules of polynomials as coefficients; the cocycles appeared as the periods of the automorphic forms under iterated indefinite integration, generalizing the classical interpretation of the periods of the abelian integrals on D/g (which can of course be considered as automorphic forms on D) as cocycles of the group 9 or alternatively of the space ID/9. The object of studying such a relation is the development of tools for calculating the dimensions of spaces of automorphic forms and the traces of the Hecke operators on automorphic forms. The aim of the present paper is the study of a more general form of this relation in somewhat greater detail for one complex variable, but in such a manner that the results can be extended to several complex variables; the actual extension to several complex variables, as well as the application to the study of the Hecke operators, will be discussed elsewhere. As for the contents of this paper, ? 1 is devoted to an exposition of Bol's differential operators in a form more useful in the present context than that of [3]. In ?2 these differential operators are applied to give an exact cohomology sequence containing, in a rather more transparent form, the relation of Eichler discussed above. The interpretations of the terms appearing in this exact sequence are discussed in ??3 through 5; the only point of difficulty arises in ?4, Theorem 3 of that section really being a form of the Serre duality theorem [12] appropriate to the occasion. These results are combined in ?6 to give a formal statement of the fundamental result of the paper. 1. Differential operators preserving automorphic forms. Let SC be the group of 2 X 2 real matrices

Introduction. In the investigation of a non-degenerate quadratic form f (x) with coefficients in a local or global field K, one associates two groups with f (x), the orthogonal and metaplectic groups. If K is, e.g., the field R of real numbers, the second group is the unitary group generated by Ut = exp(it -f(x)) and its conjugate V, under the Fourier transformation for all t in R; and it is locally isomorphic to SL2(R). By using a similar definition one can associate with any polynomial f (x) of higher degree a group. Such groups have appeared in connection with the Siegel-Weil formulas for the norm forms of certain simple Jordan algebras. On the other hand Kubota [4], [5] and Yamazaki [8] have associated for K = C and R metaplectic groups with where n need not be 2, in fact n 2 and n = 2k 2, respectively. In both cases they have used certain Bessel transformations instead of the Fourier transformation to get the metaplectic groups. In a recent paper [3] we have shown that the Lie algebra of a hypermetaplectic group over R is infinite dimensional (at least if the Fourier transformation is defined relative to the bilinear form associated with a definite quadratic form). In this paper we have restricted ourselves to the one-variable case and tried to obtain all metaplectic groups over K = R and C. More precisely we have started from an arbitrary polynomial f (x) of one variable x with coefficients in K = R or C, put U, = exp (it .f (x)) or exp(2iRe(tf(x)) and determined all cases where U, and another oneparameter subgroup V, of the unitary group of a certain L2-space generate a Lie group G locally isomorphic to SL2(K). In doing so we have assumed that the tangent vector or vectors of V, at t = 0 is a linear differential operator with polynomial coefficients, a condition satisfied in the known cases. The result is that f (x) simply becomes x, x2 under a suitable normalization and, accordingly, the differential operator becomes i-times

This paper completes the construction of$p$-adic$L$-functions for unitary groups. More precisely, in Harris, Li and Skinner [‘$p$-adic$L$-functions for unitary Shimura varieties. I. Construction of the Eisenstein measure’,Doc. Math.Extra Vol.(2006), 393–464 (electronic)], three of the authors proposed an approach to constructing such$p$-adic$L$-functions (Part I). Building on more recent results, including the first named author’s construction of Eisenstein measures and$p$-adic differential operators [Eischen, ‘A$p$-adic Eisenstein measure for unitary groups’,J. Reine Angew. Math.699(2015), 111–142; ‘$p$-adic differential operators on automorphic forms on unitary groups’,Ann. Inst. Fourier (Grenoble)62(1) (2012), 177–243], Part II of the present paper provides the calculations of local$\unicode[STIX]{x1D701}$-integrals occurring in the Euler product (including at$p$). Part III of the present paper develops the formalism needed to pair Eisenstein measures with Hida families in the setting of the doubling method.

The purpose of this course is to give an introduction to the theory of p-adic integration with values in spaces of modular forms (elliptic modular forms, Siegel modular forms, . . .). We show that very general p-adic families of modular forms can be constructed as moments of certain p-adic measures on a profinite group Y = lim ←− Yi with values in a formal q-expansion ring like Zp[[q ]] where B is an additive semi-group, and q = {q |ξ ∈ B} the corresponding formally written multiplicative semi-group (for example B = Bn = {ξ = ξ ∈ Mn(Q)|ξ ≥ 0, ξ half-integral} is the semi-group, important for the theory of Siegel modular forms). We discuss some applications of this theory to the construction of certain new p-adic families of modular forms (families of Klingen-Eisenstein series, families of theta-series with spherical polynomials. . .). Main sources of this theory are: • Serre’s theory of p-adic forms as certain formal q-expansions (J.-P. Serre, Formes modulaires et fonctions zeta p-adiques, LNM 350 (1973) 191-268) [Se73]. • Hida’s theory of p-adic modular forms and p-adic Hecke algebras (H. Hida, Elementary theory of L-functions and Eisenstein series, Cambridge University Press, 1993 [Hi93]). • Construction of p-adic Siegel-Eisenstein series by the author, see [PaSE]. As an application, we describe a solution of a problem of Coleman-Mazur in [PaTV], using the RankinSelberg method and the p-adic integration in a Banach algebra A. An introductory cours given on November 29 in POSTECH (Pohang, Korea) 0 Introduction Let p be a prime number (we often assume p≥ 5). There are two different ways of introducing p-adic modular forms: the first approach uses formal q-expansions with coefficients in a p-adic ring [Se73], and the second approach is the p-adic interpolation of Galois representations attached to classical automorphic forms. The first approach was extensively developped by Katz [Ka78] for the group G = GL2 over a totally real number field, in order to construct p-adic L-functions for CM-fields using p-adic Hilbert-Eisenstein series. In general, in this q-expansion method a typical p-adic family φ of modular (automorphic) forms is an element of the Serre ring: φ ∈ Λ[[q]] where Λ = Zp[[T ]] is the Iwasawa algebra. In the second approach one considers Λ-adic Galois representations of type ρ : Gal(Q/Q) → GLm(Λ) (“Big Galois representations”, see [Hi86], [Til-U]). These two theories are essentially equivalent if we start from holomorphic automorphic forms on the group G = GL2 over a totally real field, but in other cases there is no direct link between φ and ρ. On the other hand there exist interesting examples of p-adic L-functions Lφ,p and Lρ,p attached to φ and to ρ. In general Lφ,p and Lρ,p should belong to the quotient field L = QuotΛ or to its finite extensions. If ρ interpolates a p-adic family of motives then there are conjectural general definitions of Lρ,p (see [Co-PeRi], [Colm98], [PaAdm]). It would be very interesting to formulate a general Langlands-type conjecture relating Λ-adic automorphic forms and Λ-adic Galois representations. As an application, we describe a solution of a problem of Coleman-Mazur, using the Rankin-Selberg method and the theory of p-adic integration with values in a p-adic algebra A. This problem was stated in "The Eigencurve" (1998), R.Coleman and B.Mazur stated the following as follows: Given a prime p and Coleman’s family {fk′} of cusp eigenforms of a fixed positive slope σ = ordp(αp(k )) > 0, to construct a two variable p-adic L-function interpolating on k the Amice-Velu p-adic L-functions Lp(fk′ ). Our p-adic L-functions are p-adic Mellin transforms of certain A-valued measures. Such measures come from Eisenstein distributions with values in certain Banach A-modules M = M(N ;A) of families of overconvergent forms over A.

We construct a [math] -adic Eisenstein measure with values in the space of vector-weight [math] -adic automorphic forms on certain unitary groups. This measure allows us to [math] -adically interpolate special values of certain vector-weight [math] automorphic forms, including Eisenstein series, as their weights vary. This completes a key step toward the construction of certain [math] -adic [math] -functions.\n¶ We also explain how to extend our methods to the case of Siegel modular forms and how to recover Nicholas Katz’s [math] -adic families of Eisenstein series for Hilbert modular forms.