Selmer stability in families of congruent Galois representations
Pith reviewed 2026-05-22 16:05 UTC · model grok-4.3
The pith
The number of level-raising modular forms g congruent to a fixed f modulo p with matching p-Selmer rank grows at least like X (log X)^{α-1}.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
Fix a residual Galois representation bar rho and a corresponding weight-2 newform f of optimal level. Let N_g denote the level of a weight-2 form g. Under mild assumptions on bar rho, the number of level-raising modular forms g congruent to f modulo p with N_g ≤ X such that the p-rank of the Selmer group of g equals the p-rank of the Selmer group of f is at least C X (log X)^{α-1} for an explicit constant C and some α > 0, as X tends to infinity.
What carries the argument
Greenberg's local conditions used to define the Selmer groups of the Galois representations attached to the congruent modular forms.
If this is right
- The p-rank of the Selmer group attached to the Galois representation remains invariant for a positive-density subset of the congruent forms in the level-raising family.
- Stability of Selmer p-ranks under congruence can be established without assuming the full strength of conjectures on ranks in twist families.
- The lower bound supplies an explicit positive exponent α that quantifies how many congruent forms inherit the same Selmer rank from the base form f.
- The method yields a partial extension of Ono-Skinner type results from elliptic curves to the setting of higher-weight modular forms and their Galois representations.
Where Pith is reading between the lines
- If the same stability holds when the base form f is allowed to vary in a larger family, one might obtain average-rank statements for all congruent modular forms of bounded level.
- The counting argument could be adapted to study the distribution of Selmer ranks across the full Hida family rather than just level-raising congruences.
- A matching upper bound on the number of forms where the rank jumps would give an asymptotic for the stable count itself.
Load-bearing premise
The residual Galois representation satisfies certain mild conditions that permit control of the level-raising congruences and the comparison of the associated Selmer groups.
What would settle it
A concrete residual representation bar rho for which the number of qualifying g with N_g ≤ X and matching p-Selmer rank grows slower than X (log X)^c for every positive c, or an explicit sequence of levels where the p-rank jumps for all but o(X / log X) many congruent forms.
read the original abstract
In this article I study the variation of Selmer groups in families of modular Galois representations that are congruent modulo a fixed prime $p \geq 5$. Motivated by analogies with Goldfeld's conjecture on ranks in quadratic twist families of elliptic curves, I investigate the stability of Selmer groups defined over $\mathbb{Q}$ via Greenberg's local conditions under congruences of residual Galois representations. Let $X$ be a positive real number. Fix a residual representation $\bar{\rho}$ and a corresponding modular form $f$ of weight $2$ and optimal level. I count the number of level-raising modular forms $g$ of weight $2$ that are congruent to $f$ modulo $p$, with level $N_g\leq X$, such that the $p$-rank of the Selmer groups of $g$ equals that of $f$. Under some mild assumptions on $\bar{\rho}$, I prove that this count grows at least as fast as $X (\log X)^{\alpha - 1}$ as $X \to \infty$, for an explicit constant $\alpha > 0$. The main result is a partial generalization of theorems of Ono and Skinner on rank-zero quadratic twists to the setting of modular forms and Selmer groups.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies the variation of p-Selmer groups (via Greenberg local conditions) attached to weight-2 modular forms in families of Galois representations congruent modulo a fixed prime p ≥ 5. Fixing a residual representation bar rho and a corresponding optimal-level form f, it counts the level-raising forms g congruent to f mod p with conductor N_g ≤ X for which the p-rank of the Selmer group of g equals that of f. Under mild assumptions on bar rho, the main theorem asserts that this count is ≫ X (log X)^{α−1} as X → ∞ for an explicit α > 0. The result is presented as a partial generalization of Ono–Skinner theorems on rank-zero quadratic twists.
Significance. If the central claim holds, the work supplies an asymptotic lower bound on the number of congruent modular forms whose Selmer ranks remain stable under level-raising, furnishing evidence toward stability phenomena analogous to Goldfeld’s conjecture but in the setting of congruent families rather than twists. It connects techniques from the arithmetic of modular forms (level-raising, congruences, Selmer groups) with prior results of Ono–Skinner and thereby contributes a new quantitative statement about the distribution of Selmer ranks in a thin but infinite family of representations.
major comments (2)
- [Main theorem statement and local conditions section] The proof of the main lower bound (presumably Theorem 1.1 or the statement in §1) requires that the Greenberg local condition at each new level-raising prime q does not increase the global Selmer rank relative to f. The manuscript must explicitly verify, under the stated mild assumptions on bar rho, that for a positive-density set of such q the local cohomology group H^1_f(Q_q, V_g) has the expected dimension (finite or unramified subspace) so that the local contribution remains compatible with rank stability. Without this verification the count X (log X)^{α−1} may include forms for which the Selmer rank jumps.
- [Assumptions paragraph and §2 (setup)] The mild assumptions on bar rho (irreducibility, fixed weight 2, optimality of the level of f) are invoked to control residual behavior at primes dividing N_f and at p, but the argument must also address the residual representation at the new prime q. If these assumptions do not guarantee that bar rho|G_{Q_q} remains irreducible or satisfies the necessary local conditions for a positive proportion of q, the density of admissible level-raising primes used to obtain the logarithmic factor may be smaller than claimed.
minor comments (2)
- [Introduction] The explicit value of the constant α should be stated in terms of the parameters of the setup (e.g., the dimension of the Selmer group of f or the number of primes dividing the level).
- [Notation and §3] Notation for the Selmer group (e.g., Sel_p(V_g) versus Sel^{Gr}(V_g)) should be made uniform throughout the text and compared with the conventions in the cited works of Greenberg and of Ono–Skinner.
Simulated Author's Rebuttal
We thank the referee for the careful reading of the manuscript and the constructive comments, which help clarify the local conditions and setup. We respond to each major comment below and indicate the planned revisions.
read point-by-point responses
-
Referee: [Main theorem statement and local conditions section] The proof of the main lower bound (presumably Theorem 1.1 or the statement in §1) requires that the Greenberg local condition at each new level-raising prime q does not increase the global Selmer rank relative to f. The manuscript must explicitly verify, under the stated mild assumptions on bar rho, that for a positive-density set of such q the local cohomology group H^1_f(Q_q, V_g) has the expected dimension (finite or unramified subspace) so that the local contribution remains compatible with rank stability. Without this verification the count X (log X)^{α−1} may include forms for which the Selmer rank jumps.
Authors: We agree that an explicit verification of the local conditions at the new primes q is needed to ensure the Selmer rank remains stable. Under the mild assumptions on the irreducible residual representation bar rho of weight 2, the admissible level-raising primes q are selected so that the congruence holds and the local representation at q is compatible with the Greenberg conditions (unramified or finite subspace). To make this rigorous, we will insert a short lemma in the local conditions section showing, via the Chebotarev density theorem, that a positive-density subset of such q has the expected local cohomology dimension, preventing rank jumps in the count. This will be added in the revised manuscript. revision: yes
-
Referee: [Assumptions paragraph and §2 (setup)] The mild assumptions on bar rho (irreducibility, fixed weight 2, optimality of the level of f) are invoked to control residual behavior at primes dividing N_f and at p, but the argument must also address the residual representation at the new prime q. If these assumptions do not guarantee that bar rho|G_{Q_q} remains irreducible or satisfies the necessary local conditions for a positive proportion of q, the density of admissible level-raising primes used to obtain the logarithmic factor may be smaller than claimed.
Authors: The global irreducibility of bar rho, together with the weight-2 assumption, ensures via Chebotarev that bar rho restricted to G_{Q_q} remains irreducible (or satisfies the required local conditions for level raising) for a positive-density set of primes q; the level-raising condition itself further restricts to a positive-density subset where the local behavior is controlled. The density of admissible q is therefore not reduced below the claimed positive density. To address the point explicitly, we will expand the assumptions paragraph in §2 with a brief discussion of the local residual representation at q and confirm that the logarithmic factor is preserved. This clarification will appear in the revised version. revision: yes
Circularity Check
No circularity: result derived from external theorems on modular forms and Selmer groups
full rationale
The paper states a lower bound on the count of level-raising forms with stable Selmer rank, explicitly under mild assumptions on the residual representation bar rho, and presents the result as a partial generalization of theorems by Ono and Skinner. No step reduces a prediction to a fitted parameter, self-definition, or load-bearing self-citation chain; the derivation chain invokes prior independent results on Galois representations and Greenberg local conditions without redefining the target count in terms of itself. The abstract and setup remain self-contained against external number-theoretic benchmarks.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption Mild assumptions on the residual representation bar rho
Lean theorems connected to this paper
-
IndisputableMonolith/Foundation/AbsoluteFloorClosure.leanreality_from_one_distinction unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
Theorem A: N_f(X) ≫ X (log X)^{α-1} with α = (p-3)/(p-1)^2 under mild assumptions on ρ-bar; uses Ω_ρ and Proposition 3.4 on dim Sel_Gr equality
-
IndisputableMonolith/Foundation/AlexanderDuality.leanalexander_duality_circle_linking unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
Lemma 3.2, 3.3, Proposition 3.4: β_ℓ(A_g)=0 for ℓ in Ω_ρ or dividing N_ρ when ℓ ≢ ±1 mod p
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.
discussion (0)
Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.