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Selmer stability in families of congruent Galois representations

This paper investigates the stability of Selmer groups in families of modular Galois representations congruent modulo a prime $p \geq 5$ , proving that the number of level-raising modular forms sharing the same Selmer $p$ -rank as a fixed form grows at least as fast as $X(\log X)^{\alpha-1}$ as the level bound $X$ tends to infinity.

Original authors: Anwesh Ray

Published 2026-03-11

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Anwesh Ray

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to solve a mystery about the hidden "fingerprint" of numbers. In the world of advanced mathematics, specifically number theory, there are objects called modular forms. Think of these as incredibly complex, rhythmic patterns or musical scores that encode deep secrets about prime numbers.

Associated with each of these musical scores is a Galois representation. If the modular form is the song, the Galois representation is the sheet music that tells you how the song behaves when you twist it, turn it, or look at it through different lenses.

The Big Question: Do the "Fingerprints" Stay the Same?

The paper by Anwesh Ray asks a fascinating question: If two different musical scores (modular forms) sound almost identical when played on a specific, low-quality radio (modulo a prime number $p$ ), do they share the same hidden structural secrets?

In math-speak, this is about congruence. If two forms are "congruent," it means their coefficients match up perfectly when you divide them by a specific prime number (like 5, 7, 11, etc.).

The author is investigating a specific structural secret called the Selmer group.

The Analogy: Imagine the Selmer group is a "stability score" or a "complexity meter" for the mathematical object. It tells you how many independent solutions exist for a certain equation related to that object.
The Goal: Ray wants to know: If I have a base song (Form $f$ ) and I find a new, slightly louder version of it (Form $g$ ) that sounds the same on the low-quality radio, does the new version have the exact same stability score?

The "Goldfeld" Inspiration

The paper is inspired by a famous guess (conjecture) made by a mathematician named Goldfeld. Goldfeld looked at elliptic curves (which are like twisted donuts defined by equations). He guessed that if you take one curve and create thousands of "twisted" versions of it, about half of them would have a complexity score of 0, and half would have a score of 1.

Ray is asking: Does this same "50/50 stability" happen for these modular forms when we twist them by raising their "level" (making the song more complex)?

The Method: Level Raising

To find these matching forms, Ray uses a technique called Level Raising (developed by Diamond and Taylor).

The Metaphor: Imagine you have a simple melody. "Level raising" is like taking that melody and adding a new, complex instrument to the orchestra. The new song is much longer and more detailed, but if you listen to it through a cheap radio (modulo $p$ ), it sounds exactly like the original simple melody.

Ray creates a massive family of these "new, complex songs" that are congruent to the original one. He then counts how many of them keep the same stability score (the same $p$ -rank of the Selmer group).

The Main Discovery

The paper proves a very strong result:
There are lots of these new, complex songs that keep the exact same stability score as the original.

In fact, the number of such songs grows very fast as you look at larger and larger numbers. The author provides a formula showing that the count of these "stable" forms grows at a rate of roughly:
$X \times (\text{something related to } \log X)$

This means that if you look at a huge range of numbers, you won't just find a few matches; you will find a dense forest of them.

Why Does This Matter?

Predictability in Chaos: Number theory can feel chaotic. This paper shows that even when you make these mathematical objects more complex (by raising the level), their core structural stability often remains unchanged if they share a basic "fingerprint."
Bridging Worlds: It connects the world of elliptic curves (donuts) with the world of modular forms (musical scores), showing that the rules governing their stability are surprisingly similar.
A New Tool: This gives mathematicians a way to predict the behavior of these groups without having to calculate them from scratch every time. If you know the "fingerprint" (the residual representation), you can be confident that many of its "twisted" cousins will share the same stability properties.

Summary in One Sentence

Anwesh Ray proves that if you take a mathematical "song" and create thousands of more complex versions that sound identical on a low-quality radio, a huge number of those complex versions will retain the exact same hidden structural stability as the original, confirming a pattern similar to what was guessed for elliptic curves decades ago.

1. Problem Statement and Motivation

The paper investigates the behavior of Selmer groups within families of modular Galois representations that are congruent modulo a prime $p \ge 5$ .

Context: The work is motivated by Goldfeld's Conjecture, which posits that in families of quadratic twists of an elliptic curve, 50% have analytic rank 0 and 50% have rank 1. A key feature of quadratic twist families is that the mod-2 Galois representation remains fixed ( $E[2] \cong E_d[2]$ ), implying a strong relationship between their 2-Selmer groups.
Generalization: Ray seeks to generalize this phenomenon to $p$ $p$ -congruences between modular forms. Specifically, given a residual Galois representation $\bar{\rho}$ $\overset{ρ}{ˉ}$ associated with a modular form $f$ $f$ , the author studies the set of other modular forms $g$ $g$ (of weight 2) such that:
1. $\bar{\rho}_g \cong \bar{\rho}_f$ (they are congruent modulo $p$ ).
2. They have different levels ( $N_g$ ).
3. The $p$ -rank of their Greenberg Selmer groups remains stable (i.e., $\dim_{\kappa} \text{Sel}^{\text{Gr}}(A_g/\mathbb{Q})[p] = \dim_{\kappa} \text{Sel}^{\text{Gr}}(A_f/\mathbb{Q})[p]$ ).
Goal: To prove that the number of such "stable" forms $g$ with level $N_g \le X$ grows at a specific rate as $X \to \infty$ .

2. Methodology and Framework

2.1. Setup and Notation

Field/Prime: $p \ge 5$ is a fixed prime. $K$ is a $p$ -adic field with ring of integers $\mathcal{O}$ , uniformizer $\varpi$ , and residue field $\kappa$ .
Representations: $\bar{\rho}: G_{\mathbb{Q}} \to \text{GL}_2(\kappa)$ is a surjective, odd residual representation with determinant $\bar{\chi}$ (mod- $p$ cyclotomic character).
Modular Forms: $f$ is a weight 2 Hecke eigenform of optimal level $N_f = N_{\bar{\rho}}$ such that $\bar{\rho}_f \cong \bar{\rho}$ .
Selmer Groups: The paper focuses on Greenberg Selmer groups ( $\text{Sel}^{\text{Gr}}$ ), defined via Greenberg's local conditions (unramified outside $p$ , and specific conditions at $p$ related to the ordinary filtration). These are compared to Bloch-Kato Selmer groups.

2.2. Level Raising and Congruences

The paper utilizes the Diamond-Taylor Level Raising Theorem (Theorem 2.4). This theorem states that if $\bar{\rho}$ arises from a newform of level $N$ , and $M$ is a level satisfying specific local conditions (related to the trace and determinant of Frobenius at primes dividing $M/N$ ), then there exists a newform $g$ of level $M$ congruent to $f$ modulo $p$ .

The author constructs a set of allowable levels by multiplying the optimal level $N_{\bar{\rho}}$ by square-free integers composed of primes from a specific set $\Omega_{\bar{\rho}}$ .

2.3. Stability Analysis via Local Invariants

The core technical machinery involves analyzing the difference between the Selmer group of the residual representation and the Selmer group of the $p$ -divisible module.

Key Lemma (2.7 & 3.2/3.3): The author defines local invariants $\beta_\ell(A_h) = \dim_\kappa \ker(h_\ell)$ , where $h_\ell$ maps the local cohomology of the residual representation to the $p$ -torsion of the full representation.
Stability Condition: The dimension of the Selmer group is stable if and only if these local obstructions $\beta_\ell$ vanish for all primes $\ell$ dividing the level.
The Set $\Omega_{\bar{\rho}}$ : The author defines a set of primes $\Omega_{\bar{\rho}}$ $Ω_{\overset{ρ}{ˉ}}$ such that for any $\ell \in \Omega_{\bar{\rho}}$ $ℓ \in Ω_{\overset{ρ}{ˉ}}$ :
1. $\ell \nmid N_{\bar{\rho}}p$ .
2. $\ell \not\equiv \pm 1 \pmod p$ .
3. The trace of Frobenius satisfies specific congruence conditions (specifically $\text{tr}(\bar{\rho}(\sigma_\ell)) = 0$ and $\det(\bar{\rho}(\sigma_\ell)) = -\ell$ in a specific basis).
Vanishing Result: Lemmas 3.2 and 3.3 prove that if the level-raising primes are chosen from $\Omega_{\bar{\rho}}$ (and no primes $\equiv \pm 1 \pmod p$ divide the original level), then $\beta_\ell(A_f) = \beta_\ell(A_g) = 0$ . This ensures the $p$ -ranks of the Selmer groups remain identical.

3. Key Contributions

Generalization of Ono-Skinner Results: The paper extends theorems by Ono and Skinner (originally regarding rank 0 quadratic twists of elliptic curves) to the setting of modular forms and Greenberg Selmer groups.
Explicit Density Constant: The author identifies a specific set of primes $\Omega_{\bar{\rho}}$ and calculates its Dirichlet density explicitly as:
$\alpha = \frac{p-3}{(p-1)^2}$
This constant governs the growth rate of the family of stable forms.
Proof of Stability: The paper provides a rigorous proof that for forms $g$ constructed via level raising at primes in $\Omega_{\bar{\rho}}$ , the $p$ -rank of the Selmer group is invariant under the congruence.

4. Main Results

Theorem A (Main Theorem):
Let $p \ge 5$ and assume the residual representation $\bar{\rho}$ satisfies the condition that no prime $\ell \equiv \pm 1 \pmod p$ divides its Artin conductor $N_{\bar{\rho}}$ . Let $N_f(X)$ be the number of modular forms $g$ of weight 2, level $N_g \le X$ , congruent to $f$ modulo $p$ , such that:
$\dim_{\kappa} \text{Sel}^{\text{Gr}}(A_g/\mathbb{Q})[p] = \dim_{\kappa} \text{Sel}^{\text{Gr}}(A_f/\mathbb{Q})[p]$
Then, as $X \to \infty$ :
$N_f(X) \gg X (\log X)^{\alpha - 1}$
where $\alpha = \frac{p-3}{(p-1)^2}$ .

Interpretation:
This result implies that there is a positive density of modular forms in the congruence class of $f$ that share the same Selmer rank. The growth rate is slightly slower than linear (due to the $(\log X)^{\alpha-1}$ factor), but it confirms that "stable" forms are abundant.

5. Significance

Analogy to Goldfeld's Conjecture: The paper provides a robust analogue to Goldfeld's conjecture in the context of modular forms. Just as Goldfeld's conjecture predicts a specific distribution of ranks in quadratic twist families, this work demonstrates that Selmer ranks are stable (and thus "frozen") for a significant subset of level-raised forms.
Iwasawa Theory Connections: By establishing the stability of Selmer groups under congruence, the results have implications for Iwasawa theory, where the behavior of Selmer groups over $\mathbb{Z}_p$ -extensions is central. If the residual Selmer rank is stable, it often constrains the Iwasawa invariants ( $\mu$ and $\lambda$ ) of the associated $p$ -adic $L$ -functions.
Methodological Advancement: The paper successfully adapts techniques from the study of elliptic curves (specifically the work of Mazur, Rubin, Klagsbrun, and Smith on 2-Selmer groups) to the more complex setting of $p$ -adic modular forms with $p \ge 5$ , handling the intricacies of Greenberg's local conditions and level-raising obstructions.

In summary, Ray's work establishes that Selmer stability is a generic phenomenon within families of congruent modular Galois representations, provided the level-raising primes are chosen from a set of positive density determined by the prime $p$ .