Interpolation of generalized Heegner classes along quaternionic Coleman families

Eris Rocha Walchek

arxiv: 2503.01749 · v2 · submitted 2025-03-03 · 🧮 math.NT

Interpolation of generalized Heegner classes along quaternionic Coleman families

Eris Rocha Walchek This is my paper

Pith reviewed 2026-05-23 01:25 UTC · model grok-4.3

classification 🧮 math.NT

keywords generalized Heegner classesquaternionic modular formsColeman familiesp-adic interpolationfinite slope familiesIwasawa theory

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The pith

Generalized Heegner classes from quaternionic modular forms are p-adically interpolated along Coleman families to yield big classes.

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

The paper extends an existing interpolation technique to quaternionic modular forms. It takes the generalized Heegner classes attached to individual forms and interpolates them p-adically as the forms move through a finite-slope Coleman family. This produces a single big class that specializes to each individual class at points of the family. A reader would care because these big classes are often the key objects needed to control Selmer groups or prove main conjectures in Iwasawa theory. The work applies the same steps previously used for classical modular forms, now in the quaternionic setting.

Core claim

We construct big generalized Heegner classes by interpolating p-adically the generalized Heegner classes associated to quaternionic modular forms along a Coleman (finite slope) family, following the approach introduced by Jetchev--Loeffler--Zerbes.

What carries the argument

P-adic interpolation of generalized Heegner classes along a quaternionic Coleman family, producing a big class that specializes correctly at each classical point.

If this is right

Big generalized Heegner classes exist in the quaternionic Coleman family setting.
The same interpolation steps used for classical forms carry over directly to quaternionic forms.
These big classes become available as input for further arithmetic constructions such as p-adic L-functions or Selmer group control.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The construction opens the possibility of proving main conjectures for abelian varieties with quaternionic multiplication using these big classes.
Similar interpolation may apply to other families of automorphic forms once the local conditions are checked.
Low-level numerical checks on small discriminant quaternionic forms could confirm that the interpolated classes match the expected specializations.

Load-bearing premise

The interpolation techniques of Jetchev--Loeffler--Zerbes extend without obstruction to the quaternionic modular forms and Coleman families setting.

What would settle it

An explicit Coleman family of quaternionic modular forms where the p-adic limit of the associated generalized Heegner classes either fails to exist or specializes to the wrong class at a classical point.

read the original abstract

We construct big generalized Heegner classes by interpolating $p$-adically the generalized Heegner classes associated to quaternionic modular forms along a Coleman (finite slope) family, following the approach introduced by Jetchev--Loeffler--Zerbes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Straightforward extension of Jetchev-Loeffler-Zerbes interpolation to quaternionic modular forms along Coleman families; useful if the details hold but hard to judge from abstract alone.

read the letter

The paper's main move is to build big generalized Heegner classes by p-adically interpolating the classes attached to quaternionic modular forms along a finite-slope Coleman family. It does this by following the Jetchev-Loeffler-Zerbes method directly, so the novelty is the adaptation to the quaternionic setting rather than a new technique. If the local conditions at p and the height pairing behave as expected, this gives a tool for working with main conjectures in the quaternionic case, which is a natural next step after the original work. The abstract is clear that it sticks to the prior approach without claiming major new ideas, which keeps the claim modest and focused. That is the part that lands cleanly. The soft spot is that we only have the abstract, so there is no way to check whether the quaternionic case introduces any real differences in the definition of the classes, the control theorems, or the interpolation step itself. The claim that the method extends without obstruction is plausible but needs the full write-up to confirm. No circularity shows up in the stated claim, and the citation to the earlier paper is appropriate. This is a specialized construction paper aimed at people already working on p-adic Heegner classes, Coleman families, and Iwasawa main conjectures for forms with quaternionic multiplication. A reader who has the Jetchev-Loeffler-Zerbes paper in hand will see exactly what is being reused and where the new input is. It is the kind of incremental but technically grounded work that deserves referee time to verify the adaptation is clean. I would send it to peer review rather than desk reject.

Referee Report

0 major / 2 minor

Summary. The paper constructs big generalized Heegner classes by p-adically interpolating the generalized Heegner classes attached to quaternionic modular forms along Coleman families of finite slope, following the approach of Jetchev--Loeffler--Zerbes.

Significance. If the interpolation extends without obstruction, the result supplies a tool for producing big classes in the quaternionic setting. This is a direct, credit-given extension of an existing technique and could support subsequent Iwasawa-theoretic or p-adic L-function applications once the details are verified.

minor comments (2)

Abstract: the statement that the construction 'follows the approach' is clear, but the abstract would be strengthened by naming the precise Coleman-family hypotheses (e.g., slope bounds, tame level, or local conditions at p) under which the interpolation is claimed to hold.
The manuscript should include a short comparison paragraph (perhaps in the introduction) highlighting any technical adjustments required by the quaternionic versus classical modular-form setting, even if they are minor.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript and for recommending minor revision. The referee's summary correctly describes the construction of big generalized Heegner classes via p-adic interpolation along quaternionic Coleman families, following Jetchev--Loeffler--Zerbes.

Circularity Check

0 steps flagged

No significant circularity; direct extension of independent prior construction

full rationale

The paper constructs big generalized Heegner classes via p-adic interpolation along quaternionic Coleman families by explicitly following the Jetchev--Loeffler--Zerbes approach. The cited work is by different authors and is treated as an external method being extended to a new setting (quaternionic modular forms). No self-citation load-bearing, self-definitional steps, fitted inputs renamed as predictions, or ansatz smuggling appear in the given claims. The derivation chain is self-contained as an application of prior independent techniques without internal reduction to the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no information on free parameters, axioms, or invented entities is provided.

pith-pipeline@v0.9.0 · 5552 in / 987 out tokens · 36718 ms · 2026-05-23T01:25:46.594637+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages · 1 internal anchor

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[Anc15] , Décomposition de motifs abéliens , Manuscripta Math. 146 (2015), no. 3-4, 307–328. MR 3312448 [BBG+79] J.-F. Boutot, L. Breen, P. Gérardin, J. Giraud, J.-P. Lab esse, J. S. Milne, and C. Soulé, Variétés de Shimura et fonctions L, Publications Mathématiques de l’Université Paris VII, vo l. 6, Université de Paris VII, U.E.R. de Mathématiques, Paris,

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[GZ86] B. H. Gross and D. B. Zagier, Heegner points and derivatives of L-series, Invent. Math. 84 (1986), no. 2, 225–320. [Han15] D. Hansen, Iwasawa theory of overconvergent modular forms, i: Critical p-adic l-functions, preprint, arXiv:1508.03982 (2015). [HB15] E. Hunter Brooks, Shimura curves and special values of p-adic L-functions, Int. Math. Res. Not...

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[How07] B. Howard, Variation of Heegner points in Hida families , Invent. Math. 167 (2007), no. 1, 91–128. [JL70] H. Jacquet and R. P. Langlands, Automorphic forms on GL(2), Lecture Notes in Mathematics, vol. Vol. 114, Springer-Verlag, Berlin-New York,

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Jetchev, D

MR 401654 [JLZ21] D. Jetchev, D. Loeﬄer, and S. L. Zerbes, Heegner points in Coleman families , Proc. Lond. Math. Soc. (3) 122 (2021), no. 1, 124–152. MR 4210260 [Kas99] P. L. Kassaei, p-adic modular forms over Shimura curves over Q, Ph.D. thesis, Massachusetts Institute of Technology,

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INTERPOLATION OF GENERALIZED HEEGNER CLASSES ALONG QUATER NIONIC COLEMAN F AMILIES 31 [Kas04] , P-adic modular forms over Shimura curves over totally real ﬁe lds, Compos. Math. 140 (2004), no. 2, 359–395. [Kat81] N. Katz, Serre-Tate local moduli , Algebraic surfaces (Orsay, 1976–78), Lecture Notes in Mat h., vol. 868, Springer, Berlin, 1981, pp. 138–202. ...

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MR 772569 [Kol90] V. A. Kolyvagin, Euler systems , The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, Birkhäuser Boston, Boston, MA, 1990, pp. 435–483. [LSZ22] D. Loeﬄer, C. Skinner, and S. L. Zerbes, Euler systems for GSp(4), J. Eur. Math. Soc. (JEMS) 24 (2022), no. 2, 669–733. MR 4382481 [Lun08] C. Lundkvist, Counterexamples regarding symmetr...

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Magrone, Generalized Heegner cycles and p-adic L-functions in a quaternionic setting , Ann

MR 3552987 [Mag22] P. Magrone, Generalized Heegner cycles and p-adic L-functions in a quaternionic setting , Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 23 (2022), no. 4, 1807–1870. MR 4553539 [Mil80] J. S. Milne, Étale cohomology, Princeton Mathematical Series, No. 33, Princeton Univers ity Press, Princeton, N.J.,

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Clay Math

MR 559531 [Mil05] , Introduction to Shimura Varieties , Harmonic analysis, the trace formula, and Shimura varieti es (Proc. Clay Math. Inst. 2003 Summer School, The Fields Inst. , Toronto, Canada, June 2–27, 2003), Clay Math. Proc., vol. 4, Amer. Math. Soc, Providence, 2005, pp. 2 65–378. [Mor81] Y. Morita, Reduction modulo P of Shimura curves , Hokkaido ...

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MR 2242284 [Nek95] J. Nekovář, On the p-adic height of Heegner cycles , Math. Ann. 302 (1995), no. 4, 609–686. [Nek00] , p-adic Abel-Jacobi maps and p-adic heights , The arithmetic and geometry of algebraic cycles (Banﬀ, AB, 1998), CRM Proc. Lecture Notes, vol. 24, Amer. Math. Soc. , Providence, RI, 2000, pp. 367–379. MR 1738867 [NO19] F. A. E. Nuccio Mor...

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MR 1128753 [Shi67] G. Shimura, Construction of class ﬁelds and zeta functions of algebraic c urves, Ann. of Math. (2) 85 (1967), 58–159. MR 204426 [Shi71] , Introduction to the arithmetic theory of automorphic funct ions, Publications of the Mathematical Society of Japan, No

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[1] [1]

146 (2015), no

[Anc15] , Décomposition de motifs abéliens , Manuscripta Math. 146 (2015), no. 3-4, 307–328. MR 3312448 [BBG+79] J.-F. Boutot, L. Breen, P. Gérardin, J. Giraud, J.-P. Lab esse, J. S. Milne, and C. Soulé, Variétés de Shimura et fonctions L, Publications Mathématiques de l’Université Paris VII, vo l. 6, Université de Paris VII, U.E.R. de Mathématiques, Paris,

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[2] [2]

Boutot and H

[BC91] J.-F. Boutot and H. Carayol, Uniformisation p-adique des courbes de Shimura: les théorèmes de Čerednik et de Drinfeld , Astérisque, no. 196-197, 1991, pp. 45–158. [BDP13] M. Bertolini, H. Darmon, and K. Prasanna, Generalized Heegner cycles and p-adic Rankin L-series, Duke Math. J. 162 (2013), no. 6, 1033–1148, With an appendix by Brian Conrad. M R ...

work page 1991

[3] [3]

[GZ86] B. H. Gross and D. B. Zagier, Heegner points and derivatives of L-series, Invent. Math. 84 (1986), no. 2, 225–320. [Han15] D. Hansen, Iwasawa theory of overconvergent modular forms, i: Critical p-adic l-functions, preprint, arXiv:1508.03982 (2015). [HB15] E. Hunter Brooks, Shimura curves and special values of p-adic L-functions, Int. Math. Res. Not...

work page internal anchor Pith review Pith/arXiv arXiv 1986

[4] [4]

Howard, Variation of Heegner points in Hida families , Invent

[How07] B. Howard, Variation of Heegner points in Hida families , Invent. Math. 167 (2007), no. 1, 91–128. [JL70] H. Jacquet and R. P. Langlands, Automorphic forms on GL(2), Lecture Notes in Mathematics, vol. Vol. 114, Springer-Verlag, Berlin-New York,

work page 2007

[5] [5]

Jetchev, D

MR 401654 [JLZ21] D. Jetchev, D. Loeﬄer, and S. L. Zerbes, Heegner points in Coleman families , Proc. Lond. Math. Soc. (3) 122 (2021), no. 1, 124–152. MR 4210260 [Kas99] P. L. Kassaei, p-adic modular forms over Shimura curves over Q, Ph.D. thesis, Massachusetts Institute of Technology,

work page 2021

[6] [6]

INTERPOLATION OF GENERALIZED HEEGNER CLASSES ALONG QUATER NIONIC COLEMAN F AMILIES 31 [Kas04] , P-adic modular forms over Shimura curves over totally real ﬁe lds, Compos. Math. 140 (2004), no. 2, 359–395. [Kat81] N. Katz, Serre-Tate local moduli , Algebraic surfaces (Orsay, 1976–78), Lecture Notes in Mat h., vol. 868, Springer, Berlin, 1981, pp. 138–202. ...

work page 2004

[7] [7]

MR 772569 [Kol90] V. A. Kolyvagin, Euler systems , The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, Birkhäuser Boston, Boston, MA, 1990, pp. 435–483. [LSZ22] D. Loeﬄer, C. Skinner, and S. L. Zerbes, Euler systems for GSp(4), J. Eur. Math. Soc. (JEMS) 24 (2022), no. 2, 669–733. MR 4382481 [Lun08] C. Lundkvist, Counterexamples regarding symmetr...

work page 1990

[8] [8]

Magrone, Generalized Heegner cycles and p-adic L-functions in a quaternionic setting , Ann

MR 3552987 [Mag22] P. Magrone, Generalized Heegner cycles and p-adic L-functions in a quaternionic setting , Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 23 (2022), no. 4, 1807–1870. MR 4553539 [Mil80] J. S. Milne, Étale cohomology, Princeton Mathematical Series, No. 33, Princeton Univers ity Press, Princeton, N.J.,

work page 2022

[9] [9]

Clay Math

MR 559531 [Mil05] , Introduction to Shimura Varieties , Harmonic analysis, the trace formula, and Shimura varieti es (Proc. Clay Math. Inst. 2003 Summer School, The Fields Inst. , Toronto, Canada, June 2–27, 2003), Clay Math. Proc., vol. 4, Amer. Math. Soc, Providence, 2005, pp. 2 65–378. [Mor81] Y. Morita, Reduction modulo P of Shimura curves , Hokkaido ...

work page 2003

[10] [10]

Nekovář, On the p-adic height of Heegner cycles , Math

MR 2242284 [Nek95] J. Nekovář, On the p-adic height of Heegner cycles , Math. Ann. 302 (1995), no. 4, 609–686. [Nek00] , p-adic Abel-Jacobi maps and p-adic heights , The arithmetic and geometry of algebraic cycles (Banﬀ, AB, 1998), CRM Proc. Lecture Notes, vol. 24, Amer. Math. Soc. , Providence, RI, 2000, pp. 367–379. MR 1738867 [NO19] F. A. E. Nuccio Mor...

work page 1995

[11] [11]

Shimura, Construction of class ﬁelds and zeta functions of algebraic c urves, Ann

MR 1128753 [Shi67] G. Shimura, Construction of class ﬁelds and zeta functions of algebraic c urves, Ann. of Math. (2) 85 (1967), 58–159. MR 204426 [Shi71] , Introduction to the arithmetic theory of automorphic funct ions, Publications of the Mathematical Society of Japan, No

work page 1967

[12] [12]

Iwanami Shoten, Publishers, Tokyo ; Princeton University Press, Princeton, N.J., 1971, Kanô Memorial Lectures, No

work page 1971

[13] [13]

Torzewski, Functoriality of motivic lifts of the canonical constructi on, Manuscripta Math

[Tor20] A. Torzewski, Functoriality of motivic lifts of the canonical constructi on, Manuscripta Math. 163 (2020), no. 1-2, 27–56. MR 4131990 F acultad de Ciencias, Universidad de la República, Iguá 422 5, 11400 Montevideo, Uruguay Email address : erocha@cmat.edu.uy

work page 2020