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$p$-adic Fourier theory in families

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Pith's one-line read A p-adic Fourier duality is proven over arbitrary families

desk verdict A strong Fourier-theoretic paper whose proved core is for character v-groups; the advertised full generality awaits an unproven conjecture, but the Eisenstein applications are genuinely new and rest on the proved part. read the letter →

arxiv 2507.05374 v1 pith:FRMCHHGZ submitted 2025-07-07 math.NT math.AGmath.RT

Andrew Graham , Pol van Hoften , Sean Howe This is my paper
classification math.NTmath.AGmath.RT MSC 11F8514L0514G2211S80
keywords p-adicFouriertransformsolidcondensedmathematicssmallv-stacksp-divisiblerigidanalyticgroupslocallycharactersEisensteinmeasureHodge–Tatetriplesquaternionicmodularforms
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proves a p-adic analogue of the classical duality between a locally compact abelian group and its character group, and it works in families: for any locally analytic character datum $(\Lambda,V,\gamma)$ over a small v-stack, integration against the universal character $\kappa_X$ gives mutually dual isomorphisms of solid Hopf $O_Y$-algebras between distributions on the character group $H_X$ and $\gamma$-locally analytic functions on $\Lambda$. This unifies and extends earlier p-adic Fourier constructions and provides a compatible integral version built from Cartier duality for finite flat group schemes. The application is a global Eisenstein measure on the integral p-adic modular curve whose Hodge–Tate moments are the classical Eisenstein series, and whose rigid generic fiber yields new families of quaternionic modular forms that overconverge into the supersingular locus. The advertised scope over arbitrary small v-stacks rests on a classification conjecture that the paper explicitly leaves to future work.

What carries the argument

The load-bearing object is the universal character $\kappa_X$, the $\gamma$-locally analytic pairing $\Lambda\times_Y H_X\to \widehat{\mathbb{G}}_{m,\eta}^\diamondsuit$. The character v-group $H_X$ is the fiber product cutting out characters of $\Lambda$ that factor through $\gamma$, so its points are exactly the analytic characters whose Fourier theory is wanted. The proof strategy is to reduce the isomorphism to the one-dimensional classical p-adic Fourier transform on $\mathbb{Z}_p$: first treat trivial Galois action, then realize $\gamma$-locally analytic functions as the closed subspace killed by certain invariant vector fields, match these on the dual side with multiplication by coordinate functions composed with the logarithm, and finally descend along the Galois action. Around this core the paper builds a sheaf-theoretic functional analysis: solid condensed $O_Y$-modules, reflexive strongly Fréchet sheaves with Künneth isomorphisms for the solid tensor product, and, for the integral theory, Cartier duality for finite flat group schemes.

What would settle it

Exhibit a finite-height p-divisible v-group over a small v-stack that is not isomorphic to any $H_X$; this would disprove the classification bridge on which the advertised full scope rests.

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Extended reading notes

Core claim

The central claim, Theorem 6.3.1, is that for a small v-stack $Y$ and a locally analytic character datum $X=(\Lambda,V,\gamma)$—with $\Lambda$ a $\mathbb{Z}_p$-local system, $V$ a vector bundle, and $\gamma: \Lambda\otimes_{\mathbb{Z}_p}O_Y\to V$ a surjection—the universal $\gamma$-locally analytic character $\kappa_X: \Lambda\times_Y H_X\to \widehat{\mathbb{G}}_{m,\eta}^\diamondsuit$ induces natural isomorphisms of solid Hopf $O_Y$-algebras $D_{H_X/Y} \cong O^{\gamma\text{-la}}_{\Lambda/Y}$ and $D^{\gamma\text{-la}}_{\Lambda/Y} \cong O_{H_X/Y}$, dual to each other, functorial in $X$, and compatible with base change. The transforms exchange invariant derivations with multiplication by coordinate functions. Over a non-archimedean field the construction specializes to the classical p-adic Fourier theory for locally analytic functions on $\mathbb{Z}_p$ and on $O_L$ for finite extensions $L/\mathbb{Q}_p$. An integral version for p-divisible groups over $p$-adically complete rings is proved from Cartier duality and shown compatible with the rigid analytic transform on the generic fiber. In the modular-curve application, the modified Weierstrass function $x(n)=x-n^2[n]^*x$ is turned by the integral Fourier transform into a weight-two measure-valued Eisenstein series $\mathrm{Eis}(n)$ satisfying $\int HT^{k-2}\,d\mathrm{Eis}(n)=2(1-n^k)G_k$ for $k\geq 3$.

Load-bearing premise

The argument assumes that every finite-height p-divisible v-group over a small v-stack is the character group of some locally analytic character datum; the paper's Lemma 4.3.7 relies on a classification conjecture whose proof is deferred to future work, so a failure of that conjecture would narrow the scope of the main theorem to character v-groups.

Editorial extensions

If this is right

  • For any locally analytic character datum over a small v-stack, distributions on the character group and $\gamma$-locally analytic functions on the lattice are canonically dual solid Hopf algebras, so multiplication and convolution are exchanged in families.
  • Over a non-archimedean field, the construction recovers the classical p-adic Fourier theory for locally analytic functions on $\mathbb{Z}_p$ and on $O_L$ for finite extensions $L/\mathbb{Q}_p$.
  • The integral Fourier transforms for p-divisible groups over $p$-adically complete rings are compatible with the rigid analytic transforms on the generic fiber, giving a canonical section from Hodge–Tate locally analytic functions to integral functions after inverting $p$.
  • The global Eisenstein measure $\mathrm{Eis}(n)$ on the Tate module of the universal elliptic curve satisfies $\int HT^{k-2}d\mathrm{Eis}(n)=2(1-n^k)G_k$ for $k\geq 3$ and specializes to previous Eisenstein measures over the ordinary locus and at CM points.
  • The rigid generic fiber of $\mathrm{Eis}(n)$ is a global Eisenstein distribution whose specializations are new families of quaternionic Eisenstein series, overconvergent from profinite formal-CM loci into open subsets of the supersingular locus.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the deferred classification conjecture is proved, the same Fourier isomorphisms apply to every finite-height p-divisible v-group, not just those presented as character groups; otherwise the results still cover the character-group category.
  • The mechanism of cutting out analyticity classes by invariant vector fields suggests testable variants: replacing $\gamma$-local analyticity by higher-order differential conditions should produce matching closed sub- and quotient Hopf algebras on the dual side.
  • The canonical section of the integral-to-rigid comparison is a concrete convergence criterion; comparing it with known explicit descriptions of locally analytic functions on one-dimensional Lubin–Tate groups could yield orthonormal bases and new congruence results.
  • The quaternionic Eisenstein families invite a computational check: if they are Hecke eigenvectors, their eigenvalues should match the Hecke action predicted for the associated Galois representations, giving a route to p-adic L-functions on the supersingular locus.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 4 minor

Summary. The paper develops a p-adic Fourier theory for p-divisible rigid analytic groups over small v-stacks. For a locally analytic character datum X=(Λ,V,γ) over a small v-stack Y, the authors construct the character v-group H_X and the sheaf O^{γ-la}_{Λ/Y} of γ-locally analytic functions on Λ, and prove (Theorem 6.3.1) that integration against the universal character κ_X defines mutually dual isomorphisms of solid Hopf O_Y-algebras D_{H_X/Y} ≅ O^{γ-la}_{Λ/Y} and D^{γ-la}_{Λ/Y} ≅ O_{H_X/Y}, functorial in X and compatible with base change. The proof reduces the general case to the classical Amice transform for Z_p. An integral Fourier transform for p-divisible groups over p-adically complete rings (Theorem 7.0.2) is deduced from Cartier duality, and a compatibility with the rigid analytic theory on the generic fiber is established (Theorem 7.2.6). As an application, the authors construct a global Eisenstein measure Eis(n) on the Tate module of the universal elliptic curve over the integral p-adic modular curve, recover Katz's Eisenstein measure over the ordinary locus, and obtain new families of quaternionic Eisenstein series overconverging from the supersingular locus.

Significance. If correct, this is a major advance in p-adic analysis and arithmetic geometry. It places the classical theories of Amice and Schneider–Teitelbaum into a relative, functorial framework over arbitrary small v-stacks, with proofs carried out in the language of condensed mathematics and solid modules. The main theorems are not fitted to the conclusions: the one-dimensional case recovers the Amice transform and the Schneider–Teitelbaum theory, the Eisenstein measure recovers Katz's construction over the ordinary locus, and the global Eisenstein distribution yields new quaternionic Eisenstein series. The paper is honest about its main unproven input, Conjecture 1, which is deferred to a forthcoming thesis; however, the proven theorems cover character v-groups and, via Proposition 4.3.4, the rigid generic fibers of honest p-divisible groups.

major comments (1)
  1. [§4.3 (Definition 4.3.1, Lemma 4.3.7); §1 (abstract and introduction)] The main theorem (Theorem 6.3.1) is proved for character v-groups H_X associated to locally analytic character data X=(Λ,V,γ), but the abstract and introduction advertise Fourier transforms for 'finite height p-divisible rigid analytic groups' over arbitrary small v-stacks. The bridge is Lemma 4.3.7, whose second part asserts that every finite height p-divisible v-group is of the form H_X; its proof cites Lemma 4.2.10, yet Lemma 4.2.10 only shows that H_L is a p-divisible v-group for L∈HT(Y) and does not establish essential surjectivity. If 'finite height' is defined as H=H_L for L∈HT_fh(Y), as in Definition 4.3.1, then the second part of Lemma 4.3.7 is a tautology and the proof should state this; if it is meant to cover all p-divisible v-groups with finite-rank Tate module, the assertion depends on Conjecture 1, which is deferred to [19]. This is a scope gap between the advertised results and the proven theorems, and it should be resolved by clarifying the definitions and adjusting the abstract and introduction, while noting that Proposition 4.3.4 still covers rigid generic fibers of honest p-divisible groups.
minor comments (4)
  1. [§6.2] The two Fourier transforms are both denoted F_X in the running text, which makes it hard to distinguish them; please use distinct notations such as F_X and F^X (or F and F^∨) throughout.
  2. [Title page] The title and opening page contain typographical spacing errors (e.g., 'F OURIER THEOR Y'); please correct these before publication.
  3. [Proof of Theorem 8.2.1] The phrase '∂^j x(n)|_{x=1}' is unclear; the evaluation should be at the identity of the elliptic curve (or the origin in the local parameter), not at 'x=1'.
  4. [Definition 3.1.9] The definition of 'fiercely v-complete' as the 'largest full subcategory' with the listed closure properties could be phrased more rigorously, for instance as the intersection of all full subcategories satisfying those closure conditions.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Fourier isomorphisms reduce to external Amice and Cartier duality results; the unproven Conjecture 1 is a non-circular scope caveat.

full rationale

The derivation chain is self-contained against external benchmarks rather than circular. Theorem 6.3.1 does not define its conclusion into its premises: the Fourier transforms are defined by integration against the universal character, and the proof reduces to the one-dimensional Amice transform in Lemma 6.2.6, where the condensed transform is identified with the classical bijection of Corollary A.2.6. The general locally analytic case is obtained by passing to W-invariants and quotients (Proposition 6.3.2 and Lemma 5.2.7), and nontrivial Galois actions are handled by descent; none of these steps fits a parameter or renames the desired isomorphism as an input. The integral Fourier theory (Theorem 7.0.2) is deduced from Cartier duality for finite flat group schemes via Lemma 7.1.2, an external result. The Eisenstein measure in Theorem 8.2.1 is constructed by applying the inverse integral Fourier transform to the modified Weierstrass function, and its moments are computed from the classical Laurent expansion of ℘, giving 2(1−n^k)G_k by direct calculation rather than by assuming the conclusion. The comparisons with Schneider–Teitelbaum (Proposition 6.6.6) and with Katz (Theorem 8.2.3 and Example 8.2.6) provide independent confirmation. The one caveat is a scope gap, not a circularity: Lemma 4.3.7's claim that 'every finite height p-divisible v-group is of the form H_X' relies on Conjecture 1, which the paper explicitly leaves to Gerth's forthcoming thesis [19]. If Conjecture 1 fails, Theorem 6.3.1 still holds for all character v-groups H_X, and Proposition 4.3.4 still covers rigid generic fibers of honest p-divisible groups, but the advertised generality over all finite height p-divisible v-groups would be unsupported. This is a deferred conjecture and missing proof, not a self-citation chain or a definitional reduction, so it does not raise the circularity score.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim relies on a heavy apparatus of condensed/solid mathematics and on Fargues' classification. The main unproven input is Conjecture 1, which is used for the advertised scope over arbitrary v-stacks; without it, the main theorems still hold for character v-groups H_X, but not necessarily for all finite height p-divisible v-groups.

assumptions (6)
  • domain assumption Fargues' classification of p-divisible rigid analytic groups over a field by Hodge-Tate triples
    Used to identify every p-divisible rigid analytic group over K with some H_gamma, as in Section 4.1.12 and the introduction.
  • ad hoc to paper Conjecture 1: equivalence between Hodge-Tate triples and p-divisible v-groups over any small v-stack
    Stated as unproven and deferred to the forthcoming thesis of Gerth; used in Lemma 4.3.7 to claim every finite height p-divisible v-group is H_X.
  • standard math Condensed mathematics framework of Clausen-Scholze: solid abelian groups, internal homs, solid tensor product
    Foundational background for all functional analysis in the paper, including the categories Mod_solid_R and the properties used in Sections 2 and 3.
  • standard math Flatness of strongly (countably) Frechet modules for the solid tensor product (Proposition 2.4.3)
    Proved in the paper using Bosco's theory, but used as a key technical backdrop for Kunneth isomorphisms and reflexivity.
  • standard math Classification of finite projective modules over perfectoid affinoid algebras (Bhatt-Scholze, Kedlaya-Liu)
    Invoked in Lemma 4.3.2 and elsewhere to trivialize vector bundles locally on perfectoid spaces.
  • standard math Cartier duality for finite flat group schemes
    Basis for the integral Fourier theory in Section 7, starting from the finite flat group scheme case.

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Pith. "Pith review of $p$-adic Fourier theory in families." pith.science (2026). https://pith.science/paper/FRMCHHGZ

@misc{pith2026250705374,
  author       = {Pith},
  title        = {Pith review of: $p$-adic Fourier theory in families},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FRMCHHGZ}},
  note         = {Machine review of arXiv:2507.05374}
}
abstract

We construct Fourier transforms relating functions and distributions on finite height $p$-divisible rigid analytic groups and objects in a dual category of $\mathbb{Z}_p$-local systems with analyticity conditions. Our Fourier transforms are formulated as isomorphisms of solid Hopf algebras over arbitrary small v-stacks, and generalize earlier constructions of Amice and Schneider--Teitelbaum. We also construct compatible integral Fourier transforms for $p$-divisible groups and their dual Tate modules. As an application, we use the Weierstrass $\wp$-function to construct a global Eisenstein measure over the $p$-adic modular curve, extending previous constructions of Katz over the ordinary locus and at CM points, and show its generic fiber, the global Eisenstein distribution, gives rise to new families of quaternionic modular forms that overconverge from profinite sets in the rigid analytic supersingular locus.

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