A local-defect lower bound for the 3-Selmer group over cyclic cubic fields

Definitions and mathematical statement

Let $E/{\mathbb{Q}}$ have a nonidentity point $P\in E({\mathbb{Q}})[3]$ and $\operatorname{rank}E({\mathbb{Q}})=0$. Let $K/{\mathbb{Q}}$ be cyclic of degree three, of conductor $f$, and let $\nu(f)$ count its distinct prime divisors. The conventional finite $3$-Selmer group is \[ \operatorname{Sel}_3(E/K)=\ker\left(H^1(K,E[3])\longrightarrow \prod_vH^1(K_v,E)\right), \] where the maps impose the local Kummer conditions. Chantal David's question is whether, for every such $E,K$, \[ 3^{\nu(f)-1}\mid\#\operatorname{Sel}_3(E/K), \quad\text{equivalently}\quad \dim_{{\mathbb{F}}_3}\operatorname{Sel}_3(E/K)\ge\nu(f)-1. \] The cardinality sign is part of the supplied verification's repair; divisibility is not applied to a group without taking its order.

The supplied partial theorem retains rational $3$-torsion and the cyclic cubic extension but does not require rank zero. Put $C=\langle P\rangle$, $E'=E/C$, and $\phi:E\to E'$. The $\phi$-Selmer group is the subgroup of $H^1(K,C)$ satisfying the local Kummer conditions from $E'(K_v)/\phi E(K_v)$. Identify $C$ with ${\mathbb{Z}}/3$ using $P$. The everywhere-unramified character space is \[ U_K=\operatorname{Hom}(\operatorname{Cl}_K,{\mathbb{F}}_3)\subseteq H^1(K,C),\qquad r_3(\operatorname{Cl}_K)=\dim_{{\mathbb{F}}_3}(\operatorname{Cl}_K/3\operatorname{Cl}_K). \] Define the localization defect \[ \delta(E,K)=r_3(\operatorname{Cl}_K) -\dim_{{\mathbb{F}}_3}(U_K\cap\operatorname{Sel}_\phi(E/K)). \] Let $N_E$ be the conductor of $E$, let $\omega(m)$ count distinct prime divisors of a positive integer $m$, and let $\rho(E,K)$ be the dimension of the span in $\operatorname{Cl}_K/3\operatorname{Cl}_K$ of the prime-ideal classes above rational primes dividing $3N_E$. The report asserts \[ 0\le\delta(E,K)\le\rho(E,K)\le2\omega(3N_E), \] that $\operatorname{Sel}_\phi(E/K)$ injects into $\operatorname{Sel}_3(E/K)$, using the dual kernel $\mu_3$ and the fact that cyclic cubic fields are totally real, and that genus theory gives $r_3(\operatorname{Cl}_K)\ge\nu(f)-1$. Consequently its reported bound is \[ \dim_{{\mathbb{F}}_3}\operatorname{Sel}_3(E/K) \ge\nu(f)-1-\delta(E,K) \ge\nu(f)-1-2\omega(3N_E). \] A negative right side is merely a vacuous dimension lower bound, not divisibility by a negative power interpreted as an integer. The exact requested bound follows under $\delta(E,K)=0$, in particular if every prime class above $3N_E$ belongs to $3\operatorname{Cl}_K$. This conditional statement does not assert vanishing of the defect for every $K$.

The historical remarks motivate the question by cyclic-cubic genus-theory class-number divisibility $3^{d-1}\mid h_K$, with $d$ the number of ramified discriminant primes, and report progress of the form $3^{d-c_E}$ with $c_E$ depending on $E$. They explicitly question whether that progress concerns an algebraic $L$-value, a $3$-primary Selmer group or a $3$-descent group, and mention computations of special $L$-values. Those computations are not silently treated as proofs for the finite Selmer group defined above. Poonen proposes analogues for other primes or base number fields, without a precise replacement statement.

Short English statement

Does every rank-zero elliptic curve over ${\mathbb{Q}}$ with rational three-torsion acquire a $3$-Selmer group of order divisible by $3^{\nu(f)-1}$ over a cyclic cubic field? The supplied report gives a weaker conductor-dependent lower bound and recovers the exact bound when a specified localization defect vanishes.

Sources

• Local UnsolvedMath registry, permanent ID $20000461$, source code catalogue entry; AIM algebraic number theory corpus, aim-algebraic-number-theory-notes.json, notes[112]. Attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0.
• AIM Rational and integral points on higher dimensional varieties, Question 28 attributed to Chantal David, with remarks including Poonen: https://aimath.org/WWN/qptsurface2/qptsurface2.pdf. The supplied verification reports that official HTML/PDF restore the lost cardinality sign; the $P\ne O$ and exponent typography are likewise made explicit. The remarks end with "Problem/".
• Original literature field is empty; no full reference or proof is supplied for the historical progress or the later local-defect theorem.
• Registry dated August 14, 2026: statement exact, status partially_solved, classification PARTIAL-PROGRESS; attempt 1 records problem status open, partial result, rigor proved, novelty confidence low. The defect decomposition and bounds are machine-generated research aids, not peer reviewed, requiring independent verification before citation.