How can 7-adic Galois representations of elliptic curves be classified via Kummer descent?

The classification stalled on two level-49 curves

For an elliptic curve E over Q without complex multiplication, Serre's open image theorem says the ℓ-adic Galois representation ρE,ℓ∞ has open image in GL2(Zℓ). Mazur's Program B asks which subgroups actually occur. Rouse, Sutherland and Zureick-Brown carried this out up to the rational points of a short list of modular curves, including the curves X+ns(ℓ) for primes ℓ>17 and two genus-9 curves of level 49 that Furio and Lombardo denote X♯ns(49)=49.147.9.a.1 and X♯sp(49)=49.196.9.a.1.

Earlier work had already settled X+ns(27) and X+ns(49); Furio and Lombardo proved X+ns(49) has exactly seven rational points, all CM. The two genus-9 curves X♯ns(49) and X♯sp(49) remained. This paper computes their rational points, and that computation is what closes the 7-adic case of the classification.

Rational points become a superelliptic equation

The curves X♯ns(49) and X♯sp(49) admit natural degree-7 maps to the genus-zero curves X+ns(7) and X+sp(7). Furio and Lombardo showed that a non-CM rational point on either genus-9 curve yields coprime integers x, y and an integer w satisfying F(x,y)=kw⁷, with k∈{7,56} in the non-split case and k∈{1,7} in the split case. Here F is the homogenization of one of two cubics, fns(t)=t³−7t²+7t+7 and fsp(t)=t³−4t²+3t+1, both of which define the field K=Q(ζ7)+.

The exponent 7 in this equation is not arbitrary: it comes from primes of potentially multiplicative reduction, where the Tate curve makes inertia act unipotently and the shape of the level-49 group forces the valuation of F(x,y) to be divisible by 7. The cases with 7|k are exactly the ones Furio and Lombardo left open, and they are the cases that lead to a twist of the Klein quartic whose Jacobian has Mordell–Weil rank 3.

Kummer descent shrinks 49 classes to seven

Rather than attack the rank-3 Klein quartic twist directly, the authors descend on the superelliptic equation itself, in the style of Kummer. Over K the form F(x,y) is the norm of x−θy for a root θ of f, and since K has class number one, a solution yields x−θy=δω⁷ with ω∈K× and δ an S-unit. Only the class of δ in the Selmer group K(7,S)=O×K,S/O×7K,S matters, and comparing norms restricts it to an explicit subset Δ(θ,k) of 49 classes.

Each δ∈Δ(θ,k) produces a plane septic Cθ,δ of genus 15 over Q, and over K each of these becomes a diagonal septic, a twist of the Fermat curve u₀⁷+u₁⁷+u₂⁷=0. Remarkably, all six pairs (F,k) that arise lead to the same 49 twists of the Fermat septic. The Fermat septic is a cyclic unramified degree-7 cover of the Klein quartic Z: v₀³v₁+v₁³v₂+v₂³v₀=0, and this cover survives the twisting, so each septic maps onto a twist Zθ,δ of the Klein quartic. The 49 septics therefore map to at most 7 quartics up to isomorphism, and in fact to only three.

A 7-adic computation kills 42 of the 49 classes

The decisive local step is at the prime 7. When 7 divides F(x,y), a short computation in the completion of K at 7 shows that the relevant class vanishes: a homomorphism κ to F7 must vanish on it. This rules out 42 of the 49 classes δ∈Δ(θ,k). The remaining 7 all give quartics that are Q-isomorphic to one fixed quartic Z₀, and a classical change of coordinates of Klein, defined over K, gives an isomorphism over Q between Z₀ and the Klein quartic itself.

Conversely, for the other 42 classes the quartic Zθ,δ is not isomorphic to the Klein quartic over Q. So the 7-adic step is not a heuristic filter; it is exactly the condition that separates the surviving twists from the rest, and the cases 7|k are precisely those missing from the earlier work.

The Klein quartic's rational points rest on Fermat

The Klein quartic has only its three obvious rational points. Hurwitz showed that any other rational point would give a nontrivial solution of Fermat's equation of exponent 7, and Elkies later gave a direct proof through an elliptic quotient of rank zero. Pulling these three points back through the degree-7 cover produces the solutions listed in the paper's Theorem 1.

This is the paper's most striking structural point: the last open case of the 7-adic part of Mazur's Program B rests on Fermat's Last Theorem for exponent 7, proved by Lamé. The classification is therefore not merely completed by a new computation; it is shown to be logically downstream of a classical Diophantine theorem.

Six solutions, four of them non-CM

Theorem 1 states that for the three pairs (Fns,7), (Fns,56) and (Fsp,7), the integer solutions (x,y,w) of F(x,y)=kw⁷ with gcd(x,y)=1 all satisfy w=±1, and they are exactly the ones in the paper's Table 1. In the non-split case the solutions are ±(0,1,1), ±(7,1,1) and ±(7,3,−1); in the split case they are ±(1,−1,1), ±(5,2,−1) and ±(4,3,1).

These correspond to rational points t=x/y on X+ns(7) and X+sp(7): t=0, 7, 7/3 in the non-split case and t=−1, 5/2, 4/3 in the split case. The j-invariants are 0 at t=0 and t=−1, and non-CM at the other four points. The four non-CM points do not lift to rational points of X♯ns(49) or X♯sp(49), which is exactly what Theorem 2 asserts.

A Frobenius criterion excludes the remaining candidates

Theorem 2 states that every rational point of X♯ns(49) and X♯sp(49) is a cusp or a CM point, and more precisely that every non-cuspidal rational point of either curve has j-invariant 0. The proof of the non-CM part uses a group-theoretic criterion rather than the explicit equations for the maps X♯•(49)→X+•(7).

The criterion is this: let E be an elliptic curve over Q whose mod-49 Galois image is contained, up to conjugacy, in one of the groups G♯ns(49) or G♯sp(49), and let p≠7 be a prime of good reduction with trace of Frobenius ap. If 7∤ap(ap²−4p), then Frobp⁴⁸ acts on E[49](Q) as a scalar. Modulo 7 such a Frobenius lies in the Cartan subgroup, so its 48th power is 1 modulo 7, and the shape of the level-49 group forces this power to be scalar modulo 49. For each of the eight non-CM candidates listed in the paper's Table 4, the criterion fails at a small prime p, giving a contradiction. The CM points are handled by the same criterion together with a lemma of Furio and Lombardo in the non-split case.

The unconditional route bypasses a rank-3 conjecture

Furio and Lombardo had reduced the rational points on X♯ns(49) and X♯sp(49) to the rational points of a single plane quartic XE3(7), a twist of the Klein quartic whose Jacobian has Mordell–Weil rank 3, and they conjectured that this quartic has exactly four rational points. The present paper does not address that conjecture. Its proof is unconditional and takes a different route, so the classification result does not depend on the conjecture being true.

The paper also records that on September 18, 2026, Eray Karabiyik communicated a direct proof of the conjecture using quadratic Chabauty over number fields via an elliptic quotient over a number field of degree 21, and that on September 28, 2026, Nguyen Xuan Tho emailed a preprint with an AI-generated proof. These are noted as independent developments, not as inputs to the argument here.

The scope of the result is exactly the 7-adic case for non-CM elliptic curves over Q. Corollary 3 states that Theorem 1.7 of Furio and Lombardo holds unconditionally: for every elliptic curve E over Q without complex multiplication, the 7-adic image G=ρE,7∞(Gal Q) satisfies one of the first two alternatives of the Rouse–Sutherland–Zureick-Brown theorem, either XG has infinitely many rational points and G is one of the groups of Sutherland and Zywina, or G appears in the Rouse–Sutherland–Zureick-Brown table. The paper does not extend this to other primes ℓ, and the classification for general ℓ remains a separate question.

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