Euler systems and the symmetric square of a Hida family

Debanjana Kundu; Jishnu Ray; Stefano Vigni

A non-trivial Euler system is built for the symmetric square of a Hida family of modular forms, yielding a divisibility toward the Iwasawa main conjecture.

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2026-07-15 04:12 UTC pith:XZKO4I7L

load-bearing objection Abstract-only: solid-looking Euler-system interpolation for Sym^{2} of a Hida family plus a main-conjecture divisibility; non-triviality and local conditions cannot be checked yet. the 2 major comments →

arxiv 2607.12679 v1 pith:XZKO4I7L submitted 2026-07-14 math.NT math.AG

Euler systems and the symmetric square of a Hida family

Debanjana Kundu , Jishnu Ray , Stefano Vigni This is my paper

classification math.NT math.AG MSC 11R2311F8011F6711G40

keywords Euler systemssymmetric squareHida familiesIwasawa main conjectureSelmer groupsmodular formsp-adic L-functions

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper constructs a non-trivial Euler system attached to the symmetric square of a p-adic Hida family of modular forms, for primes p at least 7. The system interpolates the known Euler system of Loeffler and Zerbes that was previously available only for the symmetric square of a single p-ordinary newform. Separately, the authors establish an algebraic functional equation relating dual Selmer groups in this Hida-family setting. Combining the new Euler system with that functional equation and with existing results of Büyükboduk and Ganguly on algebraic Rankin–Selberg p-adic L-functions produces a divisibility of characteristic ideals toward the Iwasawa main conjecture for the symmetric square of the Hida family. A sympathetic reader cares because Euler systems are among the few tools that systematically control Selmer groups, and the Hida-family version supplies p-adic continuity and modularity that single-form constructions lack; the resulting divisibility is therefore a step toward relating special values of p-adic L-functions to arithmetic invariants in a family.

Core claim

There exists a non-trivial Euler system for the symmetric square of a p-adic Hida family of modular forms that interpolates the Loeffler–Zerbes Euler system for ordinary newforms; together with an algebraic functional equation for the associated dual Selmer groups and work of Büyükboduk–Ganguly, this yields a divisibility toward the Iwasawa main conjecture for that symmetric square.

What carries the argument

The interpolated Euler system for the symmetric square of the Hida family: a collection of cohomology classes that specializes to the Loeffler–Zerbes classes at classical points and satisfies the Euler-factor relations needed to bound dual Selmer groups over the family.

Load-bearing premise

The interpolated classes remain non-trivial over the Hida family and meet the local conditions required to control dual Selmer groups without extra vanishing.

What would settle it

Exhibit a classical specialization of the Hida family at which the interpolated Euler-system classes vanish or fail the local Euler-factor conditions while the Loeffler–Zerbes classes are known to be non-zero, or compute that the resulting characteristic-ideal divisibility fails for an explicit ordinary newform of weight at least 2.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

Abstract-only: solid-looking Euler-system interpolation for Sym^{2} of a Hida family plus a main-conjecture divisibility; non-triviality and local conditions cannot be checked yet.

read the letter

This is an abstract-only read, so the letter is provisional. What the authors claim is a non-trivial Euler system for the symmetric square of a p-adic Hida family (p ≥ 7) that interpolates the Loeffler–Zerbes system for a single ordinary newform, plus an algebraic functional equation for the dual Selmer groups, and then a divisibility toward the Iwasawa main conjecture by feeding that into Büyükboduk–Ganguly. If the full text delivers, that is real progress inside the Euler-system / Hida-family toolkit, not a re-packaging of known single-form results.

Credit where it is due: the abstract is clean about the three pieces and about the external inputs (Loeffler–Zerbes for the classes, Büyükboduk–Ganguly for the algebraic Rankin–Selberg functional equations). There is no sign of circularity; the controlling classes are presented as interpolations of an existing construction rather than as objects defined from the Selmer groups they are meant to bound. That is the right shape for this literature.

The soft spot is exactly the one the stress-test flags, and it is an information deficit rather than a visible flaw. Everything load-bearing sits on the interpolation remaining non-zero in the appropriate Iwasawa cohomology and satisfying the local conditions at p and at the bad primes so that the Euler system actually controls the dual Selmer group. The abstract asserts non-triviality and the resulting divisibility but does not exhibit the local Euler-factor formulae or a non-vanishing criterion. Until those appear, one cannot tell whether the system stays useful or whether Büyükboduk–Ganguly applies without extra rank or vanishing obstructions in the symmetric-square setting. That is absence of evidence, not evidence of a hole.

Who it is for: people already working on Euler systems, Hida families, and main conjectures for motives attached to modular forms (especially weight-3 / Sym^{2} settings). A serious referee should see the full manuscript; the claims are important enough and the scaffolding is standard enough that desk rejection would be the wrong call. I would not cite it yet and would not bring an abstract to reading group, but I would accept it for peer review and read the paper when it is available.

Referee Report

2 major / 1 minor

Summary. The manuscript claims three results for a prime p ≥ 7: (i) construction of a non-trivial Euler system for the symmetric square of a p-adic Hida family of modular forms that interpolates the Loeffler–Zerbes Euler system attached to the symmetric square of a p-ordinary newform; (ii) an algebraic functional equation for the dual Selmer groups in this Hida-family setting; (iii) a divisibility toward the Iwasawa main conjecture for the symmetric square of the Hida family, obtained by combining (i)–(ii) with the functional equations of algebraic Rankin–Selberg p-adic L-functions due to Büyükboduk–Ganguly. Only the abstract was available for this review; the constructions, local conditions, interpolation maps, and intermediate lemmas could not be inspected.

Significance. If the claims hold, the paper would supply a standard high-value package in the Iwasawa theory of Hida families: an Euler system that interpolates a known ordinary construction, an algebraic functional equation controlling dual Selmer groups, and a resulting main-conjecture divisibility. The work sits squarely in the Loeffler–Zerbes / Büyükboduk–Lei–Loeffler–Zerbes circle of ideas and would extend those methods from individual ordinary forms to Hida families. Explicit credit is due for the stated interpolation of an external Euler system and for the use of an external algebraic functional equation; both are the right tools for the stated goal. Significance cannot be confirmed without the full text.

major comments (2)

[Abstract] Abstract (central existence claim): The load-bearing assertion is that the Hida-family interpolation of the Loeffler–Zerbes classes remains non-zero in the appropriate Iwasawa cohomology of the symmetric-square Galois representation and satisfies the local conditions (at p and at primes of bad reduction) needed to bound the dual Selmer group. The abstract states non-triviality and the resulting divisibility but supplies neither local Euler-factor formulae nor a non-vanishing criterion. Without those statements the central claim cannot be verified from the available text.
[Abstract] Abstract (appeal to Büyükboduk–Ganguly): The divisibility toward the Iwasawa main conjecture rests on applying the algebraic functional equations of Rankin–Selberg p-adic L-functions of Büyükboduk–Ganguly in the pure symmetric-square setting. The abstract does not address possible extra rank or vanishing obstructions that could arise when specialising from Rankin–Selberg to the symmetric square of a Hida family. This applicability step is load-bearing for the main-conjecture divisibility and must be checked in the full manuscript.

minor comments (1)

[Abstract] The abstract is clearly written and correctly situates the work relative to Loeffler–Zerbes and Büyükboduk–Ganguly. No further presentation issues can be assessed from the abstract alone.

Circularity Check

0 steps flagged

Abstract-only review: no circularity detectable; claims rest on external Euler systems and functional equations.

full rationale

Only the abstract is available. It states three contributions: (1) construction of a non-trivial Euler system for the symmetric square of a p-adic Hida family that interpolates the Loeffler–Zerbes Euler system for a p-ordinary newform; (2) an algebraic functional equation for dual Selmer groups; (3) a divisibility toward the Iwasawa main conjecture obtained by combining the Euler system with the functional equation and work of Büyükboduk–Ganguly. None of these steps is self-definitional: the controlling classes are claimed to interpolate an external construction (Loeffler–Zerbes), and the main-conjecture divisibility invokes an external algebraic functional equation (Büyükboduk–Ganguly). There is no fitted parameter renamed as a prediction, no uniqueness theorem imported solely from the authors, and no renaming of a known empirical pattern. The abstract supplies no equations that would allow one to exhibit a reduction of the form “Eq. X = Eq. Y by construction.” Consequently the derivation chain, as presented, is self-contained against external benchmarks and exhibits no circularity of the kinds enumerated in the instructions. Score 0 is the honest finding for an abstract-only review that contains no load-bearing self-reference.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

Abstract-only review: no free parameters are fitted to data. The work rests on standard background of Hida theory, Euler systems, and Iwasawa theory, plus the cited external results of Loeffler–Zerbes and Büyükboduk–Ganguly. No new particles or ad-hoc geometric objects are introduced; the Euler system is a constructed cohomology class, not an invented entity with independent physical evidence.

axioms (5)

domain assumption Existence and basic properties of p-adic Hida families of ordinary modular forms and their Galois representations (including ordinary projectors).
The whole construction is for the symmetric square of a Hida family; this is standard background invoked by the title and abstract.
domain assumption The Loeffler–Zerbes Euler system for the symmetric square of a p-ordinary newform exists and is non-trivial under the paper’s hypotheses.
The abstract states that the new system interpolates that construction; the result inherits its validity and non-triviality hypotheses.
domain assumption Functional equations of algebraic Rankin–Selberg p-adic L-functions as in Büyükboduk–Ganguly apply in the symmetric-square Hida setting used here.
The divisibility toward the main conjecture is explicitly built on that recent work.
domain assumption p ≥ 7 is prime; residual representations and local conditions are such that Euler-system machinery applies (no exceptional zeros or rank jumps that kill non-triviality).
Stated hypothesis p≥7; non-triviality of the constructed system is load-bearing and typically requires such arithmetic hypotheses.
standard math Standard formalism of dual Selmer groups, characteristic ideals, and Iwasawa main conjectures over Hida deformation rings.
Background language of the abstract’s second and third contributions.

pith-pipeline@v1.1.0-grok45 · 6014 in / 2777 out tokens · 29788 ms · 2026-07-15T04:12:04.817995+00:00 · methodology

0 comments

read the original abstract

Let $p\geq7$ be a prime number. We build a non-trivial Euler system for the symmetric square of a $p$-adic Hida family of modular forms interpolating the Euler system constructed by Loeffler-Zerbes for the symmetric square of a $p$-ordinary newform. As a second contribution, we prove an algebraic functional equation for dual Selmer groups in this setting. Finally, building on recent work by B\"uy\"ukboduk-Ganguly on functional equations of algebraic (Rankin-Selberg) $p$-adic $L$-functions, we prove a divisibility result towards the Iwasawa main conjecture for the symmetric square of a Hida family.

Euler systems and the symmetric square of a Hida family

Core claim

What carries the argument

Load-bearing premise

What would settle it

discussion (0)