Parabolic Eigenvariety: p-adic Hecke Families

• Parabolic eigenvariety is a rigid analytic parameter space that organizes p-adic Hecke eigensystems through parabolic and Levi data while imposing finite slope conditions.
• It employs parahoric overconvergent cohomology to interpolate analytic weights along the Levi factor and permits infinite slope away from designated parabolic directions.
• These constructions reveal deep intersections between automorphic forms, p-adic L-functions, and eigenvariety geometry, offering fresh insights in higher rank settings.

A parabolic eigenvariety is a rigid analytic parameter space for finite-slope Hecke eigensystems in which the $p$-adic variation is organized by a parabolic datum. In the direct overconvergent-cohomological construction, one fixes a standard parabolic $Q\subset G$, introduces parahoric overconvergent cohomology with respect to $Q$, and constructs $Q$-parabolic eigenvarieties that parametrise $p$-adic families of systems of Hecke eigenvalues that are finite slope at $Q$, but that allow infinite slope away from $Q$ (Salazar et al., 2020). Closely related literatures use the same adjective for eigenvarieties built from parabolic cohomology, for eigenvarieties organized by degree of cuspidality or corank, and for Levi-based interpolation on Shimura varieties with non-ordinary or partially classical structures (Wu, 2020).

1. Core definition and terminological range

For $G=\mathcal G_{/\mathbf Q_p}$ quasi-split, with fixed Borel $B\supset T$, and a standard parabolic $Q=L_QN_Q$, the associated parahoric subgroup is

$Q\subset G$0

The resulting eigenvariety is built over a weight space in which the weights vary only through the Levi: $Q\subset G$1 Its points correspond to Hecke eigensystems occurring in parahoric overconvergent cohomology and satisfying a finite-slope condition only for $Q\subset G$2-controlling operators (Salazar et al., 2020).

Other constructions use “parabolic” in different but related senses. For $Q\subset G$3, the cuspidal eigenvariety is defined by replacing ordinary cohomology with the image of compactly supported cohomology in ordinary cohomology,

$Q\subset G$4

so that the eigenvariety is the parabolic or cuspidal part of the full eigenvariety (Wu, 2020). For PEL Shimura varieties, Brasca–Rosso construct eigenvarieties for non-cuspidal forms by a filtration by degree of cuspidality, producing eigenvarieties of intermediate dimension indexed by corank (Brasca et al., 2016). In a derived setting, a parabolic subgroup enters through Jacquet modules and parabolic induction, but the construction does not define a separate canonical parabolic eigenvariety; instead it produces a subeigenvariety associated with induction data (Fu, 2021).

A plausible synthesis is that “parabolic eigenvariety” denotes a class of $Q\subset G$5-adic eigenvariety constructions in which a parabolic subgroup, a Levi factor, boundary depth, or parabolic cohomology determines which analytic directions vary and which finite-slope conditions are imposed.

2. Parahoric overconvergent cohomology and the $Q\subset G$6-parabolic construction

The basic coefficient module is a hybrid object that is locally analytic transverse to $Q\subset G$7 and algebraic along $Q\subset G$8. For a classical dominant weight $Q\subset G$9, one fixes the algebraic representation $Q$0 of $Q$1 and defines

$Q$2

Letting $Q$3 gives

$Q$4

When $Q$5, this recovers the usual overconvergent distributions; when $Q$6, one recovers classical algebraic coefficients: $Q$7 Thus the theory interpolates between fully analytic Iwahori coefficients and fully algebraic coefficients (Salazar et al., 2020).

The finite-slope condition is encoded by a monoid

$Q$8

with criterion

$Q$9

A $Q$0-controlling operator therefore contracts only the root directions outside $Q$1. This is the source of the central flexibility: a Hecke eigensystem may be finite slope with respect to $Q$2 even when it has infinite slope for the full Iwahori/Borel collection of operators. The local eigenvariety pieces are

$Q$3

$Q$4

and the global construction is obtained from a Fredholm series

$Q$5

and the associated Fredholm hypersurface $Q$6. The resulting rigid analytic space $Q$7 carries a weight map

$Q$8

and its $Q$9-points above $p$0 correspond to systems of Hecke eigenvalues $p$1 with $p$2 occurring in $p$3 (Salazar et al., 2020).

The classicality theorem is formulated along a maximal chain of parabolics

$p$4

with $p$5 obtained from $p$6 by adjoining one simple root $p$7, and with operators $p$8. For

$p$9

the theorem states that if a system of Hecke eigenvalues $Q$0 occurring in classical cohomology satisfies

$Q$1

then the $Q$2-generalized eigenspaces in parahoric overconvergent cohomology and in classical cohomology coincide. When $Q$3, the theory recovers the usual overconvergent cohomology, and the slope bound is stronger than the standard one in the literature because it gives separate inequalities in each simple-root direction rather than a single combined inequality (Salazar et al., 2020).

A further structural invariant is the overconvergent defect

$Q$4

Any irreducible component through $Q$5 has dimension at least $Q$6. If $Q$7 admits discrete series, then cuspidal $Q$8-non-critical points lie on components of maximal possible dimension $Q$9 (Salazar et al., 2020).

3. The $Q$0 parabolic eigenvariety and Shalika families

For

$Q$1

the paper on $Q$2-adic $Q$3-functions in finite slope Shalika families studies the maximal parabolic $Q$4 with Levi

$Q$5

The relevant weight space is the parabolic subspace

$Q$6

which has dimension $Q$7 when $Q$8. The local eigenvariety chart is

$Q$9

where $G=\mathcal G_{/\mathbf Q_p}$0 is the image of the Hecke algebra acting on the slope-$G=\mathcal G_{/\mathbf Q_p}$1 part of parahoric overconvergent cohomology

$G=\mathcal G_{/\mathbf Q_p}$2

A point is classical if it comes from a cohomological automorphic representation, and a Shalika point if that representation is cuspidal and admits a Shalika model (Salazar et al., 2021).

The automorphic input is a regular algebraic cuspidal automorphic representation $G=\mathcal G_{/\mathbf Q_p}$3 of $G=\mathcal G_{/\mathbf Q_p}$4 that is of symplectic type and admits an $G=\mathcal G_{/\mathbf Q_p}$5-Shalika model. The paper fixes a $G=\mathcal G_{/\mathbf Q_p}$6-refinement

$G=\mathcal G_{/\mathbf Q_p}$7

with $G=\mathcal G_{/\mathbf Q_p}$8 spherical at $G=\mathcal G_{/\mathbf Q_p}$9, and studies parahoric overconvergent cohomology together with distribution-valued evaluation maps

$B\supset T$0

These maps are constructed by $B\supset T$1-adic interpolation of branching laws for

$B\supset T$2

They yield a parabolic $B\supset T$3-adic $B\supset T$4-function

$B\supset T$5

interpolating critical complex $B\supset T$6-values. Its growth is controlled by

$B\supset T$7

and the noncritical slope condition is

$B\supset T$8

Under this hypothesis, the refinement is strongly non-$B\supset T$9-critical, so the classical class lifts uniquely to overconvergent cohomology (Salazar et al., 2021).

The main geometric theorem states that if $Q=L_QN_Q$0 is regular, $Q=L_QN_Q$1 is irreducible, and the refinement is non-$Q=L_QN_Q$2-critical, then the parabolic eigenvariety $Q=L_QN_Q$3 is étale over $Q=L_QN_Q$4 at the point $Q=L_QN_Q$5. After shrinking $Q=L_QN_Q$6, the connected component through that point maps isomorphically onto $Q=L_QN_Q$7. Equivalently, the local ring is finite étale over the weight space local ring, so the eigenvariety is smooth of the expected dimension $Q=L_QN_Q$8. The mechanism runs in the reverse of many earlier arguments: non-vanishing of the standard $Q=L_QN_Q$9-adic $Q\subset G$00-function implies non-vanishing of the family evaluation map; because the target is torsion-free over $Q\subset G$01, the relevant overconvergent cohomology module is faithful over $Q\subset G$02, and this forces full support over $Q\subset G$03. Under $Q\subset G$04-regularity and the existence of a nonzero critical $Q\subset G$05-value, the component through $Q\subset G$06 contains a Zariski-dense set of classical points admitting Shalika models, so it becomes a Shalika family (Salazar et al., 2021).

This is one of the few higher-rank results showing that an eigenvariety for $Q\subset G$07 is smooth or étale at a genuinely nonordinary finite-slope classical point. It also exhibits an explicit interaction between automorphic periods, branching laws, and local eigenvariety geometry.

4. Parabolic cohomology and cuspidal eigenvarieties

For $Q\subset G$08, the parabolic terminology refers to cohomology rather than to a separate parabolic weight space. Starting from Johansson–Newton’s full eigenvariety, one defines a cuspidal eigenvariety

$Q\subset G$09

by replacing ordinary cohomology with parabolic cohomology,

$Q\subset G$10

Equivalently, this is the image of compactly supported cohomology inside ordinary cohomology, or the kernel of the boundary map in the long exact sequence. In this sense, the eigenvariety is the parabolic or cuspidal part of the full overconvergent-cohomological eigenvariety (Wu, 2020).

The coefficient modules are analytic distributions $Q\subset G$11, and a central ingredient is a pairing on these distributions. For $Q\subset G$12, the pairing is

$Q\subset G$13

equivalently with integrand

$Q\subset G$14

It satisfies the equivariance relation

$Q\subset G$15

Via cup product, this induces a pairing on parabolic cohomology and then on the coherent sheaves living over the Fredholm hypersurface $Q\subset G$16 and over the cuspidal eigenvariety itself (Wu, 2020).

The geometric application is a ramification criterion for the weight map

$Q\subset G$17

At a good point, the pairing defines an adjoint $Q\subset G$18-ideal $Q\subset G$19. Under nondegeneracy of the pairing at a good point $Q\subset G$20,

$Q\subset G$21

If $Q\subset G$22 is also smooth, then

$Q\subset G$23

where $Q\subset G$24 is defined via the $Q\subset G$25-th Fitting ideal of relative differentials. The parabolic eigenvariety in this setting is therefore a higher-rank analogue of the cuspidal eigencurve: it is cut out by excluding boundary contributions and then studied through a Hecke-equivariant pairing on the resulting coherent sheaves (Wu, 2020).

5. Coherent-geometric, Levi-based, and partially classical variants

In the coherent-geometric theory of non-cuspidal forms on certain PEL Shimura varieties, the obstruction to a direct Buzzard construction is that the full space of forms is not projective over weight space. Brasca–Rosso introduce a filtration

$Q\subset G$26

by corank, where $Q\subset G$27 is the space of families of forms of corank at most $Q\subset G$28. For each fixed $Q\subset G$29, the corresponding module is projective, so Buzzard’s machine applies and produces eigenvarieties $Q\subset G$30 of explicit dimension $Q\subset G$31 in the Siegel notation, maximal for cuspidal forms and equal to $Q\subset G$32 for forms that are not cuspidal at all. The reduced eigenvarieties glue into a single non-equidimensional eigenvariety over the full weight space, and the construction is embedded into Hansen’s cohomological eigenvariety. The paper identifies this as exactly the kind of structure one expects for parabolic eigenvarieties (Brasca et al., 2016).

For Picard modular forms on $Q\subset G$33 with $Q\subset G$34 inert, the ordinary locus is empty, so the construction uses the $Q\subset G$35-ordinary locus and the canonical filtration

$Q\subset G$36

The weight space is

$Q\subset G$37

a disjoint union of open $Q\subset G$38-balls, and it matches the Levi factor $Q\subset G$39. The resulting eigenvariety is $Q\subset G$40-dimensional, parametrizes Hecke eigensystems on overconvergent, locally analytic, cuspidal Picard modular forms of finite slope, and is described in the paper as a genuine example of a parabolic eigenvariety because the analytic variation is controlled by a parabolic with Levi $Q\subset G$41 (Hernandez, 2017).

For partially classical Hilbert modular forms, one fixes a subset $Q\subset G$42, where $Q\subset G$43 recovers overconvergent forms and $Q\subset G$44 recovers classical forms. The paper constructs families of partially classical forms over adapted weight spaces and proves the existence of a $Q\subset G$45-classical eigenvariety

$Q\subset G$46

which is equidimensional of dimension

$Q\subset G$47

Its $Q\subset G$48-points above a weight correspond to systems of Hecke eigenvalues occurring in the finite-slope part of the fiber, and it carries a Galois pseudocharacter interpolating Frobenius traces away from $Q\subset G$49. The paper explicitly presents this as a coherent-cohomological analogue of parabolic eigenvarieties, since only selected $Q\subset G$50-adic directions are interpolated and the resulting eigenvariety has smaller dimension than the full Hilbert eigenvariety (Dimitrov et al., 2024).

An earlier PEL-Shimura construction for cuspforms with dense ordinary locus is also described as “naturally ‘parabolic’” because the eigenvalue systems come from sections vanishing on the boundary divisor $Q\subset G$51, so the parabolic condition is encoded geometrically by the boundary twist and analytically by cuspidal growth or vanishing (Brasca, 2014).

6. Functoriality, symplectic loci, and conceptual limits

The study of the classical symplectic locus in the $Q\subset G$52 eigenvariety provides a further parabolic refinement of eigenvariety geometry. At Iwahori level $Q\subset G$53, a standard parabolic $Q\subset G$54 determines a parahoric subgroup

$Q\subset G$55

a local Hecke algebra $Q\subset G$56, and a $Q\subset G$57-parahoric $Q\subset G$58-refinement $Q\subset G$59. The non-$Q\subset G$60-critical slope condition is

$Q\subset G$61

For a spin parabolic $Q\subset G$62, the relevant pure weight subspace

$Q\subset G$63

has dimension $Q\subset G$64. If $Q\subset G$65 is an optimally $Q\subset G$66-spin Iwahori refinement, then any symplectic family $Q\subset G$67 through $Q\subset G$68 satisfies

$Q\subset G$69

Under non-critical slope and regular weight, there is a unique symplectic family through $Q\subset G$70, of dimension exactly $Q\subset G$71, and it is étale over $Q\subset G$72 at $Q\subset G$73. The paper ties this geometry to a $Q\subset G$74-refined Friedberg–Jacquet criterion and formulates the conjecture that every symplectic family is a transfer from $Q\subset G$75 with dimension dictated by the minimal spin parabolic (Salazar et al., 2023).

The derived construction of eigenvarieties gives a different use of parabolic data. It constructs a derived variant of Emerton’s eigenvarieties using the derived Jacquet module

$Q\subset G$76

proves exactness of the finite slope part functor, and applies the standard eigenvariety machine to the resulting essentially admissible $Q\subset G$77-representations. In the global unitary-group application, a parabolic subgroup $Q\subset G$78 gives rise to an induction eigenvariety

$Q\subset G$79

and Theorem 7.3 states that this embeds as a closed subvariety of the unitary-group eigenvariety. The paper explicitly states, however, that it does not construct a separate “parabolic eigenvariety” via a parabolic weight space or a parabolic finite-slope condition distinct from the standard $Q\subset G$80-Jacquet setup (Fu, 2021).

A recurring misconception is therefore that “parabolic eigenvariety” names a single canonical object. The available constructions point in a different direction. Some are attached directly to a chosen parabolic subgroup $Q\subset G$81 and a parahoric finite-slope condition; some isolate the cuspidal or parabolic-cohomological part of a full eigenvariety; some interpolate only Levi directions or only selected $Q\subset G$82-adic places; and some use parabolic induction merely to produce a subeigenvariety. This suggests that the unifying content is not a unique definition but a common principle: the geometry of $Q\subset G$83-adic Hecke families is constrained by a parabolic structure, and the resulting eigenvariety records precisely those automorphic directions that remain analytic under that constraint.