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Construction of eigenvarieties

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Pith's one-line read These lecture notes show that the Coleman–Mazur eigencurve and its quaternionic and cohomological analogues are all produced by one construction: a compact Hecke operator, a characteristic power series, and a Fredholm hypersurface.

desk verdict Honest, well-organized lecture notes with no new results; the only real flaw is a typo in the pseudocharacter identity. read the letter →

arxiv 2411.16880 v1 pith:T5B6Q7CW submitted 2024-11-25 math.NT

James Newton This is my paper · ORCID
classification math.NT MSC 11F3311F8514G22
keywords eigenvarietiesColeman–Mazureigencurveoverconvergentmodularformsp-adicfamiliesofHeckeoperatorFredholmhypersurfaceadicspacescohomology
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

These lecture notes explain how eigenvarieties—p-adic analytic spaces that interpolate systems of Hecke eigenvalues attached to automorphic forms—are constructed. Their central example is the Coleman–Mazur eigencurve, built from spaces of overconvergent modular forms and the spectral theory of a single compact Hecke operator $U_p$. The notes show that the same 'eigenvariety machine' produces eigencurves for definite quaternion algebras and, more generally, eigenvarieties from overconvergent cohomology for $GL_n$. Along the way they spell out the common geometric output: the eigenvariety is finite over a Fredholm spectral curve, locally quasi-finite and flat over weight space, one-dimensional, reduced, and has a Zariski-dense, self-accumulating set of classical points. The contents are expository and no new theorems are proved.

What carries the argument

The eigenvariety machine of [Lud24]: a compact operator $U_p$ on a Banach module of overconvergent forms, its entire characteristic power series $F^\dagger$, the Fredholm hypersurface $Z = V(F^\dagger)$ over weight space, and Riesz theory turning slope decompositions into a Hecke-module coherent sheaf. Two further ingredients do specific work: the Katz–Lubin canonical-subgroup theorem (Theorem 2.2.9) makes $U_p$ act on overconvergent affinoids by moving the radius from $v$ to $v/p$, and Coleman's classicality theorem identifies small-slope points as classical points.

What would settle it

Compute the characteristic power series $\det(1 - X U_p)$ on $M^{\dagger,v}_k(N)$ for two different radii $v$ and $v/p$ inside $(0, p/(p+1))$: if the two series differ, the claimed independence of $v$ fails and the spectral-curve gluing cannot proceed; similarly, exhibit an elliptic curve over a $p$-adic field with $v_p(A) < p/(p+1)$ that has no canonical subgroup, which would contradict the quoted theorem on which the construction rests.

Watch

Extended reading notes

Core claim

The central claim of these notes is that the standard eigenvarieties—the Coleman–Mazur eigencurve, eigencurves for definite quaternion algebras, and eigenvarieties from overconvergent cohomology for $GL_n$—are instances of one mechanical construction. Starting with a Banach space of overconvergent forms carrying a compact Hecke operator $U_p$, one forms the characteristic power series $F^\dagger = \det(1 - X U_p)$, defines the Fredholm hypersurface $Z = V(F^\dagger)$ over weight space, and uses Riesz theory to glue the slope decompositions into a coherent sheaf with Hecke action. The output $E$ is an adic space finite over $Z$, locally quasi-finite and flat over weight space, equidimensional of dimension one, reduced, with classical points Zariski dense and self-accumulating. In the prototypical case this $E$ is exactly the Coleman–Mazur eigencurve.

Load-bearing premise

The entire construction rests on a quoted theorem without proof: that a specific, continuously varying subgroup of the $p$-torsion of an elliptic curve—the 'canonical subgroup'—exists throughout the region of overconvergence $v < p/(p+1)$; if that theorem failed, the Hecke operator $U_p$ would not be defined and the eigencurve would not exist.

Editorial extensions

If this is right

  • If the construction is correct, the characteristic power series $F^\dagger$ glues over all of weight space, so the slopes of $U_p$ on overconvergent modular forms are locally constant in families of weights.
  • The eigencurve is a genuine one-dimensional p-adic object: locally quasi-finite and flat over weight space, equidimensional of dimension one, and reduced.
  • Classical modular forms sit densely inside the eigencurve in the Zariski topology, so analytic interpolation results about Hecke eigenvalues can be converted into statements about classical forms and vice versa.
  • For definite quaternion algebras, the resulting eigencurve embeds as a union of irreducible components into the Coleman–Mazur eigencurve via a p-adic Jacquet–Langlands correspondence.
  • For $GL_n$ with $n>2$, classical points are not expected to be Zariski dense; instead, essentially self-dual classical points fill closed subsets of dimension $1+\lfloor n/2\rfloor$, while other components are genuinely p-adic objects.

Reading between the lines

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

  • I would draw a sharper moral than the notes state explicitly: any context with a compact operator and a clean family of Banach modules should yield an eigenvariety, so the main obstacle in new settings is not the machine but proving classicality of small-slope points.
  • The notes leave the Katz–Lubin theorem unproved; I would want a direct check of whether the bound $v < p/(p+1)$ is optimal, since a counterexample at the boundary would show exactly where the construction's radius of convergence breaks.
  • The $GL_n$ discussion suggests a testable prediction: the 'genuinely p-adic' components should carry pseudocharacters whose classical specializations are reducible or non-classical, and one could search for such components computationally via slope decompositions in small cohomological degree.
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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

0 major / 4 minor

Summary. The manuscript is a set of lecture notes from a Heidelberg spring school, explicitly disclaiming any original work. Section 2 recalls Banach spaces of overconvergent modular forms, the Katz–Lubin canonical subgroup theorem, the Up operator, Coleman's classicality theorem, and then uses the eigenvariety machine developed in [Lud24] to construct the Coleman–Mazur eigencurve as an adic space finite over a Fredholm spectral curve; it states and sketches proofs of local quasi-finite flatness over weight space, equidimensionality, reducedness, and Zariski-density/self-accumulation of classical points. Section 3 surveys other constructions of eigenvarieties; Section 4 treats definite quaternion algebras, including a p-adic Jacquet–Langlands closed immersion and étaleness of the weight map at regular small-slope classical points; Section 5 sketches overconvergent cohomology for GL_n and the resulting eigenvarieties.

Significance. If the exposition is faithful, these notes are a useful companion to [Lud24]: they connect the abstract eigenvariety machine to concrete geometric examples and state the standard properties of the eigencurve with pointers to the original literature. The notes are honest about their scope, explicitly attribute all results, and include helpful exercises. The main external input, Theorem 2.2.9 (Katz–Lubin), is a standard cited theorem; quoting it without proof is appropriate for lecture notes and does not create an internal gap. The only concrete mathematical error I located is a typo in the pseudocharacter identity in Proposition 2.6.6, which is local and does not affect the construction.

minor comments (4)
  1. [2.6] Proposition 2.6.6, displayed identity: the duplicated final term '+ T(g1g2g3)' should read '+ T(g3g2g1)' (cf. [BC09, Prop. 7.5.4]). As printed the identity is false for a general two-dimensional representation, since T(g1g2g3) is not generally equal to T(g3g2g1). This is a typo and does not change the construction, but the displayed statement should be corrected.
  2. [2.2.8] The notation 'M^{†,N}_k' appears where 'M^{†,v}_k(N)' is evidently intended; please harmonize the notation for the space of overconvergent modular forms.
  3. [2.2.10] Exercise 2.2.10 contains the typo 'q-expensions'; it should be 'q-expansions'.
  4. [References] The entries [H¨24] and [Lud24] list page ranges as 'pp. ?–?'; these should be updated if the volume pagination is known.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the notes are a citation-based exposition that constructs the eigencurve from external results, with no load-bearing self-citation and no premise defined by its own conclusion.

full rationale

I find no circular step. The notes explicitly disclaim originality in the abstract ('None of the contents are original work') and in Section 1 state that they apply results from Ludwig's lectures [Lud24], which are external to the author. The construction of the Coleman-Mazur eigencurve rests on cited prior results: the canonical subgroup theorem 2.2.9 is attributed to Katz and Lubin and is used to define the compact operator Up; the eigenvariety machine, spectral curves, Riesz theory, and characteristic power series are imported from [Lud24]; classicality is imported from Coleman [Col96]; and reducedness and density of classical points are proved using [Che05] and [Bel21]. None of these inputs is equivalent to the eigencurve's claimed properties, and no equation in the notes is defined in terms of its own conclusion. The author's own works [JN19a], [JN19b], [JN19c], and [NT21] appear only as incidental references (for terminology, comparisons, a pseudorepresentation lifting remark, and an application), and they carry no load in the derivation. The only concrete defect I located is a typo in Proposition 2.6.6, where the last pseudocharacter term is printed twice instead of giving the non-commutative partner term required by the identity; the correct identity is cited to [BC09, Prop. 7.5.4]. This is a transcription error and does not affect the derivation chain.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The notes introduce no new free parameters or entities. They rely on standard theorems and on the companion notes [Lud24] as background assumptions; these are cited, not proved.

assumptions (3)
  • domain assumption The Katz-Lubin canonical subgroup theorem (Theorem 2.2.9) is valid.
    Invoked in Section 2.2.8 to extend the canonical subgroup to Y^{≤v} and define Up.
  • domain assumption The spectral theory of compact operators on p-adic Banach spaces and the eigenvariety machine of [Lud24] are valid.
    Section 2.5 constructs the spectral curve and the eigenvariety by referring to [Lud24] as a black box.
  • domain assumption Coleman's classicality theorem (Theorem 2.2.12) holds for h < k-1.
    Used in Section 2.6.3 to prove density of classical points and in Proposition 2.6.5 to prove reducedness.

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Cite this review

Pith. "Pith review of Construction of eigenvarieties." pith.science (2026). https://pith.science/paper/T5B6Q7CW

@misc{pith2026241116880,
  author       = {Pith},
  title        = {Pith review of: Construction of eigenvarieties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T5B6Q7CW}},
  note         = {Machine review of arXiv:2411.16880}
}
read the original abstract

These are notes based on four lectures given at the Heidelberg spring school on non-archimedean geometry and eigenvarieties. None of the contents are original work. Our goal is to explain the construction of eigenvarieties in various different contexts, including the prototypical example of the Coleman--Mazur eigencurve. We will also discuss some of the common geometric properties of eigenvarieties.

Discussion (0). Continue with ORCID to comment.

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Reviewed August 12, 2026 · model on record in the stance chip above.