Explicit reciprocity laws for diagonal classes: higher level cases

Luca Marannino

arxiv: 2402.13648 · v3 · submitted 2024-02-21 · 🧮 math.NT

Explicit reciprocity laws for diagonal classes: higher level cases

Luca Marannino This is my paper

Pith reviewed 2026-05-24 04:12 UTC · model grok-4.3

classification 🧮 math.NT

keywords explicit reciprocity lawsdiagonal classesmodular eigenformsp-adic L-functionsordinary newformssupercuspidal formsprincipal seriesbalanced triples

0 comments

The pith

p-adic explicit reciprocity laws for balanced diagonal classes extend to triples where g and h are supercuspidal or ramified principal series at p.

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

The paper generalizes the p-adic explicit reciprocity laws for balanced diagonal classes, previously obtained by Darmon--Rotger and Bertolini--Seveso--Venerucci, to a broader setting. It treats geometric balanced triples (f, g, h) of modular eigenforms in which f is p-ordinary while g and h may both be supercuspidal at p or both ramified principal series at p. This matters to a sympathetic reader because the laws relate diagonal classes to p-adic L-values or regulators in more cases than before. The extension proceeds by verifying that the local conditions at p permit the same constructions used in the base cases without new obstructions. The result keeps the reciprocity statements intact under these relaxed local hypotheses at p.

Core claim

The p-adic explicit reciprocity laws for balanced diagonal classes hold for geometric balanced triples (f,g,h) of modular eigenforms where f is a p-ordinary newform and g and h are allowed to be both supercuspidal at p or both ramified principal series at p.

What carries the argument

The balanced diagonal classes attached to the triple (f,g,h), which carry the arithmetic data to which the reciprocity laws are applied.

If this is right

The reciprocity law applies when both g and h are supercuspidal at p.
The reciprocity law applies when both g and h are ramified principal series at p.
The diagonal class constructions remain valid and produce the same arithmetic objects under these local conditions at p.

Where Pith is reading between the lines

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

The same approach could be tested on triples with one supercuspidal and one principal-series form at p.
The enlarged range may supply new examples relating diagonal classes to p-adic regulators of motives attached to the triple.
The result suggests that further local conditions at p might be handled by similar compatibility arguments.

Load-bearing premise

The earlier reciprocity laws hold in the base cases, and the changed local conditions at p for g and h still allow the diagonal class constructions to extend compatibly.

What would settle it

An explicit triple (f,g,h) with g and h both supercuspidal at p for which the diagonal class fails to satisfy the predicted p-adic reciprocity relation with the corresponding L-value or p-adic L-function.

read the original abstract

We generalize the $p$-adic explicit reciprocity laws for balanced diagonal classes by Darmon--Rotger and Bertolini--Seveso--Venerucci to the case of geometric balanced triples $(f,g,h)$ of modular eigenforms where $f$ is a $p$-ordinary newform, while $g$ and $h$ are allowed to be (both) supercuspidal at $p$ or (both) ramified principal series at $p$.

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

This paper extends the p-adic reciprocity laws for diagonal classes to the cases where g and h are both supercuspidal or both ramified principal series at p.

read the letter

The main advance is filling in the local conditions at p that Darmon-Rotger and Bertolini-Seveso-Venerucci left out. The authors state that the balanced diagonal class constructions carry over when f is p-ordinary and g, h satisfy the new behaviors, and they claim no extra obstructions appear. That is the concrete contribution. The work adapts the prior frameworks to these settings rather than inventing new machinery, which is reasonable for a targeted extension. If the local computations at p check out, the result is useful for people tracking Euler systems and p-adic BSD-type statements. The dependence on the two cited papers is explicit and expected; nothing in the description suggests circularity. A soft spot is that the abstract gives no indication of how much new local analysis is required for the supercuspidal case versus the ramified principal series case, so the technical weight of the extension is hard to judge from the summary alone. The citation pattern is appropriate and points back to the right base results. This is written for specialists already working inside the arithmetic geometry of modular forms. A reader outside that circle will not get much from it. It deserves a serious referee to verify the local extensions and any new calculations at p.

Referee Report

0 major / 0 minor

Summary. The paper generalizes the p-adic explicit reciprocity laws for balanced diagonal classes, as established by Darmon--Rotger and Bertolini--Seveso--Venerucci, to geometric balanced triples (f,g,h) of modular eigenforms in which f is p-ordinary while both g and h are permitted to be supercuspidal at p or both ramified principal series at p.

Significance. If the stated generalization is established, the result extends the range of local conditions at p under which explicit reciprocity laws for diagonal classes are available. This broadens the applicability of the framework to a wider class of eigenform triples without altering the core construction or introducing new obstructions beyond those already handled in the base cases.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report and recommendation to accept the manuscript. The referee's summary accurately captures the scope of our generalization of the p-adic explicit reciprocity laws.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper presents a generalization of p-adic explicit reciprocity laws from the cited works of Darmon--Rotger and Bertolini--Seveso--Venerucci to new local behaviors (supercuspidal or ramified principal series) for g and h at p, with f p-ordinary. No self-citations appear, and the central claim introduces new constructions for the higher-level cases without any quoted reduction of a derived result to a fitted input, self-definition, or ansatz smuggled via the author's own prior work. The derivation is therefore self-contained against the external benchmarks it invokes, with no load-bearing step that collapses by construction to the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No information on free parameters, axioms, or invented entities is available from the abstract alone.

pith-pipeline@v0.9.0 · 5589 in / 1153 out tokens · 43408 ms · 2026-05-24T04:12:55.535394+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.2: logBK(κ(f,g,h))(ηφ=apf⊗ωg⊗ωh⊗tr+2) = (−1)k−2(r−k+2)!⋅a1(e˘f(TrMpt/M1pt(g×d(k−l−m)/2h))) under (f-ord)+(SC) and (F,1−T)-convenient
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Sections 3–4: filtered(φ,N)-modules, syntomic Abel-Jacobi, Nekovár-Nizioł syntomic cohomology with coefficients

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

[1]

In: Proc

arXiv:1907.10964 [math.NT].url:https://arxiv.org/abs/1907.10964. [FK88] Eberhard Freitag and Reinhardt Kiehl. ´Etale cohomology and the Weil conjecture. Vol. 13. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Translated from the German by Betty S. Waterhouse and William C. Waterhouse, With an historica...

work page doi:10.1007/978- 1907
[2]

de Rham comparison and Poincar´ e duality for rigid varieties

arXiv:2209.10289 [math.NT]. [KM85] Nicholas M. Katz and Barry Mazur.Arithmetic moduli of elliptic curves. Vol. 108. An- nals of Mathematics Studies. Princeton University Press, Princeton, NJ, 1985, pp. xiv+514. isbn: 0-691-08349-5; 0-691-08352-5.doi:10 . 1515 / 9781400881710.url:https : / / doi.org/10.1515/9781400881710. [LeS07] Bernard LeStum.Rigid cohom...

work page doi:10.1515/9781400881710 1985

[1] [1]

In: Proc

arXiv:1907.10964 [math.NT].url:https://arxiv.org/abs/1907.10964. [FK88] Eberhard Freitag and Reinhardt Kiehl. ´Etale cohomology and the Weil conjecture. Vol. 13. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Translated from the German by Betty S. Waterhouse and William C. Waterhouse, With an historica...

work page doi:10.1007/978- 1907

[2] [2]

de Rham comparison and Poincar´ e duality for rigid varieties

arXiv:2209.10289 [math.NT]. [KM85] Nicholas M. Katz and Barry Mazur.Arithmetic moduli of elliptic curves. Vol. 108. An- nals of Mathematics Studies. Princeton University Press, Princeton, NJ, 1985, pp. xiv+514. isbn: 0-691-08349-5; 0-691-08352-5.doi:10 . 1515 / 9781400881710.url:https : / / doi.org/10.1515/9781400881710. [LeS07] Bernard LeStum.Rigid cohom...

work page doi:10.1515/9781400881710 1985