Global Eisenstein Measure in p-adic Automorphic Forms

• Global Eisenstein Measure is a p-adic measure valued in Banach spaces of automorphic forms that interpolates classical Eisenstein series across varying weights.
• It employs adelic constructions, particularly on higher-rank groups and unitary settings, to capture holomorphic, nearly holomorphic, and vector-weight forms.
• The framework links special values, p-adic L-functions, and analytic spectral measures, while addressing challenges like non-ordinary and singular weight interpolation.

Searching arXiv for recent and foundational papers on Eisenstein measures, especially global and p-adic constructions. Searching for "Eisenstein measure p-adic unitary groups". Searching for "An introduction to Eisenstein measures". Searching for "Eisenstein-Kronecker Katz p-adic measure Poincare bundle". A global Eisenstein measure is, in the arithmetic sense, a $p$-adic measure with values in a Banach space of $p$-adic modular or automorphic forms such that specialization at a weight character recovers the $q$-expansion of a classical Eisenstein series. In the most general form used in modern automorphic theory, it is an adelic construction on higher-rank groups—especially unitary groups—whose values interpolate holomorphic, nearly holomorphic, and vector-weight Eisenstein series across rigid-analytic weight spaces. Related literature also uses closely connected notions: global pseudo-measures whose Mellin transforms are reciprocals of $p$-adic $L$-functions, and global analytic measures on arithmetic quotients obtained from the squared absolute values or residues of Eisenstein series (Eischen, 2021, Eischen, 2011, Gelbart et al., 2016, Laaksonen, 2015).

1. Conceptual framework

The basic abstract formulation fixes an odd prime $p$, a $p$-adically complete ring $O$, a profinite abelian group $G$, and a Banach $O$-module $p$0 carrying a space of $p$1-adic modular or automorphic forms. An Eisenstein measure is then a measure

$p$2

characterized by the condition that for each weight or point $p$3 in a suitable rigid-analytic weight space,

$p$4

is the $p$5-expansion of the classical Eisenstein series of weight $p$6. Equivalently, the entire family is encoded by a single element $p$7 whose evaluation at $p$8 recovers the corresponding Eisenstein $p$9-expansion (Eischen, 2021).

This formulation organizes a historical sequence. Serre introduced the space of $q$0-adic modular forms and constructed $q$1-adic families of Eisenstein series for totally real fields. Deligne–Ribet produced Hilbert–Eisenstein measures, and Katz constructed Eisenstein measures for CM fields using partial Fourier transforms and $q$2-adic differential operators. Eischen’s work extends this picture to adelic, higher-rank unitary groups, first for scalar weights and then for vector weights (Eischen, 2021, Eischen, 2011, Eischen, 2013).

The qualifier global has several precise uses in the literature. In higher-rank arithmetic geometry it refers to an adelic construction on a global reductive group, such as a unitary similitude group, with interpolation across a global weight space (Eischen, 2011). In Iwasawa-theoretic settings it can refer to the packaging of local Eisenstein distributions into an object on $q$3, possibly only as a pseudo-measure rather than a bounded measure (Gelbart et al., 2016). In analytic spectral theory, weak-limit points of measures of the form $q$4 on an arithmetic quotient have also been regarded as a “global Eisenstein measure” capturing the continuous spectrum (Laaksonen, 2015).

2. Interpolation, congruences, and uniqueness

The core property of an Eisenstein measure is interpolation of classical values by a single $q$5-adic distribution. For $q$6, continuous characters factor as $q$7 with $q$8, and one basic identity is

$q$9

for even $p$0. In this form, a single measure encodes the Kummer congruences among the values $p$1 as $p$2 and $p$3 vary (Eischen, 2021).

The measure-theoretic structure is governed by additivity and equivariance. On $p$4, a measure is determined by its values on compact opens, and the relation

$p$5

expresses the refinement of a disk into its $p$6 subdisks. Together with twists by characters and Kummer congruences, such relations determine the Eisenstein measure (Eischen, 2021).

Existence and uniqueness are controlled by abstract Kummer congruences and density of character values. A collection of values on a dense family of test functions arises from a measure precisely when the corresponding congruences are satisfied, and, when $p$7 is flat over $p$8 and enough roots of unity are present, a measure on a profinite abelian group is uniquely determined by its values on finite-order characters. In Serre’s setting, boundedness of the nonzero Fourier coefficients in the Iwasawa algebra forces the constant term into the same algebra, except for the tame-trivial case, where a standard renormalization is required (Eischen, 2021).

These results explain why a global Eisenstein measure is not merely a collection of compatible classical forms but a single object in $p$9 whose coefficient functions already satisfy the relevant $L$0-adic congruence system.

3. Adelic construction on unitary groups

For higher-rank unitary groups, the global set-up begins with a CM field $L$1 with totally real subfield $L$2, an $L$3-dimensional $L$4-vector space $L$5 with non-degenerate hermitian form $L$6 of signature $L$7 at the infinite places, and

$L$8

which has induced signature $L$9. The unitary similitude groups are

$p$0

viewed adelically. Fixing a prime $p$1 split in $p$2, a tame level with hyperspecial factor at $p$3, and the space $p$4 of $p$5-adic automorphic forms on $p$6 at that tame level, one constructs an Eisenstein family by starting from adelic Siegel Eisenstein series (Eischen, 2011).

The input data consist of a unitary algebraic Hecke character $p$7 of conductor dividing $p$8, the unnormalized induced representation

$p$9

for the Siegel parabolic $p$0, and a factorizable section $p$1. The local choices are essential. At archimedean places, $p$2 is Shimura’s Gaussian-type holomorphic or nearly holomorphic section of weight $p$3. At $p$4-adic places, $p$5 is built from a Schwartz function $p$6 so that the local Whittaker integral $p$7 is exactly the partial Fourier transform $p$8 for $p$9 Hermitian. Away from $O$0 and $O$1, the local Fourier coefficient is a polynomial $O$2 in the Hecke eigenvalue. The global Eisenstein series is

$O$3

initially convergent for $O$4 large and then meromorphically continued (Eischen, 2011).

Shimura’s computations give a factorization of the nonzero Fourier coefficient at $O$5: $O$6 When $O$7 specializes to $O$8 with $O$9, the archimedean integrals pick out the holomorphic form; other integer shifts yield nearly holomorphic Eisenstein series (Eischen, 2011).

The $G$0-adic interpolation step identifies a rigid-analytic weight space $G$1 parametrizing tuples $G$2 of $G$3-adic continuous characters, and produces a $G$4-valued measure

$G$5

such that for each algebraic point $G$6,

$G$7

Equivalently, for any locally constant test function $G$8 on weight space, the Mellin transform $G$9 is the $O$0-expansion of the Eisenstein series attached to $O$1. In rank one, this recovers

$O$2

The resulting measure is continuous, $O$3-adically analytic in $O$4, and satisfies distribution relations under an appropriate Hecke algebra action (Eischen, 2011).

4. Vector weights and geometric realizations

Eischen’s vector-weight extension replaces scalar-weight interpolation by measures valued in spaces of vector-weight $O$5-adic automorphic forms on $O$6. The $O$7-adic weight space $O$8 has $O$9-points given by locally analytic characters

$p$00

with $p$01, so that $p$02 is a union of open unit polydiscs. The module of vector-weight forms is

$p$03

realized either as global sections on the Igusa tower over the ordinary locus or, via Hida–Hida theory, as continuous functions on $p$04 valued in the Iwasawa algebra and satisfying the usual $p$05- and level-compatibilities (Eischen, 2013).

The measure is defined on

$p$06

by first sending a continuous function $p$07 on $p$08 to a function

$p$09

extended by zero outside $p$10. Theorem 10 then produces a form $p$11 with Fourier coefficients given by finite $p$12-linear combinations of values $p$13, and the assignment

$p$14

defines the Eisenstein measure $p$15 (Eischen, 2013).

At an arithmetic weight $p$16 and for a Hecke character $p$17 of conductor dividing $p$18, the measure specializes to the classical $p$19 Eisenstein series $p$20. In particular,

$p$21

in the formal interpolation sense. The same construction extends to Siegel modular forms by replacing $p$22 with $p$23, and in the symplectic case $p$24 recovers Katz’s Eisenstein families for Hilbert modular forms (Eischen, 2013).

A complementary geometric perspective comes from Eisenstein–Kronecker classes and the Poincaré bundle. For elliptic curves, the Poincaré bundle on $p$25 yields a $p$26-adic theta function whose Amice transform is a two-variable measure

$p$27

with moments

$p$28

In this approach, Katz’s two-variable $p$29-adic Eisenstein measure becomes the Amice transform of a single $p$30-adic theta function (Sprang, 2018). Gamarra extends the Eisenstein–Kronecker construction to Hilbert moduli spaces with $p$31-level, producing a universal measure

$p$32

whose $p$33th moment is the $p$34-adic Hilbert modular form $p$35 and which recovers essentially Katz’s measure after comparison (Segovia, 9 Jun 2026).

5. Relation to special values and $p$36-adic $p$37-functions

The principal arithmetic role of a global Eisenstein measure is as an input to the construction of $p$38-adic $p$39-functions. On unitary groups, pairing the Eisenstein family with suitable $p$40-adic families of cusp forms via doubling integrals yields $p$41-adic $p$42-functions that, at classical arithmetic points, recover the algebraic part of special complex $p$43-values up to explicit period factors. In the informal formulation of Eischen’s main corollary, if $p$44 is a cohomological cuspidal automorphic representation of a unitary group $p$45, then pairing $p$46 against a locally analytic family of $p$47-stabilized forms in $p$48 gives a $p$49-adic $p$50-function $p$51 satisfying

$p$52

at arithmetic weights $p$53 (Eischen, 2011).

This role explains why the higher-rank theory differs from the classical $p$54 situation. In rank one, the constant term of an Eisenstein family can often be read directly as the desired $p$55-adic $p$56-function. For higher-rank groups such as unitary and symplectic groups, the constant-term approach fails, and integral representations of $p$57-functions involve pairings of Eisenstein series with cusp forms, for example through the doubling method. Current approaches combine Hida’s pairing and Hecke algebras or Pilloni’s higher Hida theory, but the general theory remains incomplete (Eischen, 2021).

The geometric constructions via Poincaré bundles and Eisenstein–Kronecker classes are compatible with this viewpoint. Their moment maps produce modular forms whose values at CM points recover classical Eisenstein or Eisenstein–Kronecker series, and these series are precisely the objects entering formulas for Hecke $p$58-values in the CM and Hilbert settings (Sprang, 2018, Segovia, 9 Jun 2026).

6. Pseudo-measures, analytic measures, and open problems

A central caution is that not every Eisenstein distribution is a bounded $p$59-adic measure. Gelbart–Greenberg–Miller–Shahidi study the distribution $p$60 obtained from Fourier coefficients of normalized Eisenstein series on $p$61, characterized by

$p$62

For regular primes $p$63, this distribution is bounded on compact opens and hence is a genuine measure. For irregular primes, however, there exist constants $p$64 such that

$p$65

so the distribution fails boundedness and becomes only a pseudo-measure in the sense of Iwasawa. Packaging the local factors produces a global Eisenstein object on $p$66, but at irregular primes it lies in the total ring of fractions rather than in the subring of bounded measures (Gelbart et al., 2016).

A second, analytically distinct usage appears on arithmetic hyperbolic manifolds. For $p$67 with $p$68, the measures

$p$69

attached to the global Eisenstein series satisfy precise quantum-limit statements. If $p$70, then for compact Jordan-measurable $p$71,

$p$72

If $p$73 and $p$74, then the normalized measures become equidistributed with respect to hyperbolic volume. Residues at poles of the scattering matrix define scattering-state measures, and under the Generalized Riemann Hypothesis these converge weakly to $p$75. In that setting, weak-limit points of $p$76 may be regarded as a global Eisenstein measure on the arithmetic quotient, capturing the continuous spectrum (Laaksonen, 2015).

Several open problems remain standard reference points. If $p$77 is inert in a CM field, the ordinary locus disappears; this has been settled for quadratic imaginary fields by Andreatta–Iovita and Kriz via overconvergent differential operators but remains open in general PEL settings. For higher-rank groups, the challenge is to construct Eisenstein measures where the constant-term method fails and then interpolate the relevant pairings $p$78-adically. Extending the theory to non-ordinary or singular weights via eigenvarieties and overconvergent methods remains largely conjectural (Eischen, 2021).

Taken together, these constructions show that the notion of global Eisenstein measure is structurally stable across several domains: as a $p$79-adic distribution in $p$80, as a higher-rank adelic interpolating family on unitary and symplectic groups, as a geometric measure arising from Poincaré bundles and Eisenstein–Kronecker classes, and, in analytic spectral theory, as a weak-limit measure encoding the continuous Eisenstein spectrum. The precise category—measure, pseudo-measure, or analytic measure on a quotient—is determined by the interpolation problem and the ambient automorphic theory.