The structure of $\operatorname{Ext}^1_{\mathcal{O}_K}(\mathbf{Z}/p\mathbf{Z},\mu_p)$ and $K$-rational $p$-torsion on elliptic curves with good reduction everywhere when the class-number and low-ramification hypotheses fail

For an elliptic curve with good reduction everywhere over a number field $K$, Yasuda proved that $K$-rational points of prime order $p$ cannot exist under two hypotheses: (A) $p$ does not divide the class number of the cyclotomic extension $K(\zeta_p)$, and (B) every prime of $K$ above $p$ has ramification index $<p-1$. The result was established in Yasuda (2008) for the number fields treated there, and in the general form proved in Yasuda (2012) and restated as Theorem 1.1 of Yasuda (2013) it holds for any number field $K$ and every prime $p\ge5$, even for curves whose bad reduction is confined to a prescribed set $B_{K,p}$ of primes not above $p$ (so $E$ has good reduction at every prime above $p$, and curves with good reduction everywhere are included). The proof computes the group $\operatorname{Ext}^1_{\mathcal{O}_K}(\mathbf{Z}/p\mathbf{Z},\mu_p)$ of extensions of the constant group scheme $\mathbf{Z}/p\mathbf{Z}$ by $\mu_p$ in the category of finite flat commutative group schemes over the ring of integers $\mathcal{O}_K$, using the Oort–Tate classification of order-$p$ group schemes, whose explicit normal forms and Honda–Dieudonné computation of the local Ext groups are available exactly in the low-ramification range $e<p-1$. The open core consists of the regimes the hypotheses exclude: the structure of the global Ext group and the associated torsion criterion when $p$ divides $\#\mathrm{Cl}(K(\zeta_p))$ or when some prime above $p$ has ramification index $\ge p-1$, together with the local Ext computation in that higher-ramification range and the complete accounting of unit-generated non-split extension classes $T_\varepsilon$. When (A) or (B) fails, the results of Yasuda (2012, 2013) do not determine the structure theorem; in the class-number regime torsion is known to occur, so the surviving question is the exact structure theorem and existence criterion, not blanket nonexistence.

Background

An elliptic curve $E$ over a number field $K$ has good reduction at a prime $\mathfrak q$ of $K$ if it extends to a smooth group scheme over the local ring at $\mathfrak q$ (equivalently, the reduced curve over the residue field is nonsingular), and it has good reduction everywhere when this holds at every prime. Curves with good reduction everywhere are exceptionally rigid, and classifying their rational torsion is a classical theme: over $K=\mathbf{Q}$ no such curve exists at all, while over quadratic and higher-degree fields a delicate arithmetic of examples and nonexistence theorems has developed. General torsion theory - the complete classification of torsion over $\mathbf{Q}$ (Mazur 1978) and the uniform bound on torsion order in terms of the degree of $K$ (Merel 1996) - says nothing specific about the everywhere-good-reduction locus.

Yasuda (2008) attacked the nonexistence side with group-scheme methods. The $p$-torsion of an elliptic curve with good reduction everywhere is carried by its Néron model, the canonical smooth group scheme over the ring of integers $\mathcal{O}_K$ extending $E$; its $p$-torsion subgroup scheme $E[p]$ is a finite flat commutative group scheme over $\mathcal{O}_K$ of rank $p^2$ (a group scheme is a scheme carrying a group structure; finite flat means its coordinate ring is a finite flat $\mathcal{O}_K$-algebra). If $E(K)$ contains a point $P$ of order $p$, then the constant group scheme $\mathbf{Z}/p\mathbf{Z}$, the group scheme of $p$ constant points, generated by $P$ appears inside $E[p]$ with quotient $\mu_p$ (the group scheme of $p$-th roots of unity); passing to the isogenous curve $E^*=E/\langle P\rangle$, which has the same primes of bad reduction, identifies $\mu_p$ with a subgroup scheme of $E^*[p]$ whose quotient is $\mathbf{Z}/p\mathbf{Z}$, so the way the étale part $\mathbf{Z}/p\mathbf{Z}$ is glued to the multiplicative part $\mu_p$ is controlled by extension classes $0\to\mu_p\to G\to\mathbf{Z}/p\mathbf{Z}\to0$. These classes form the group $\operatorname{Ext}^1_{\mathcal{O}_K}(\mathbf{Z}/p\mathbf{Z},\mu_p)$ computed in the category of finite flat commutative group schemes over $\mathcal{O}_K$. The prototypical non-split extensions are the group schemes $T_\varepsilon$ attached to a unit $\varepsilon\in\mathcal{O}_K^\times$ that is not a $p$-th power: each sits in such a sequence, and $T_\varepsilon\cong T_{\varepsilon'}$ only when $\varepsilon/\varepsilon'$ is a $p$-th power of a unit; Schoof (2003) systematically exploited these schemes in his study of abelian varieties with good reduction everywhere over cyclotomic fields.

Locally at a prime $\mathfrak p$ above $p$, the whole calculus rests on the Oort–Tate classification (Tate–Oort 1970) of finite flat group schemes of order $p$ over a discrete valuation ring with residue characteristic $p$: explicit normal forms exist precisely when the ramification index $e_{\mathfrak p}$, the exponent of $\mathfrak p$ in the factorization of $p\mathcal{O}_K$, satisfies $e_{\mathfrak p}<p-1$, and in that range Honda's theory of Dieudonné modules (the linear-algebraic classification of $p$-divisible groups over the residue field) computes the local Ext groups. Globally, flat cohomology ties $\operatorname{Ext}^1_{\mathcal{O}_K}(\mathbf{Z}/p\mathbf{Z},\mu_p)$ to the arithmetic of the cyclotomic extension $K(\zeta_p)$ obtained by adjoining a primitive $p$-th root of unity: its classes are governed by the class group $\mathrm{Cl}(K(\zeta_p))$ - the finite abelian group measuring failure of unique factorization in that field's ring of integers - and by $p$-th-power classes of units.

The nonexistence theorem, in the general form proved in Yasuda (2012) and restated as Theorem 1.1 of Yasuda (2013), reads: let $K$ be a number field, let $p\ge5$ be a prime such that $e_{\mathfrak p}<p-1$ for every prime $\mathfrak p$ of $K$ above $p$, and let $E/K$ have $B_{K,p}$-reduction - bad reduction only at primes of $K$ whose residue characteristic $\ell$ and residue degree $f$ satisfy $\ell\ne p$ and $\ell^f\not\equiv\pm1\bmod p$; in particular $E$ has good reduction at every prime above $p$, and curves with good reduction everywhere are included. If $p\nmid\#\mathrm{Cl}(K(\zeta_p))$, then $E(K)$ has no point of order $p$. Hypothesis (B) is exactly the range of validity of the Oort–Tate normal forms and of the Honda–Dieudonné computation of the local Ext groups, and hypothesis (A) is exactly the condition that kills the class-group contribution to the global Ext group, so the theorem stops precisely where its machinery stops. The unresolved territory it leaves is the structure of the global Ext group and the torsion question when (A) or (B) fails, the local Ext computation in the higher-ramification range $e\ge p-1$, and a complete description of how the unit group of $\mathcal{O}_K$ contributes non-split classes in general number fields.

Problem Statement

Determine the group $\operatorname{Ext}^1_{\mathcal{O}_K}(\mathbf{Z}/p\mathbf{Z},\mu_p)$ - the extensions of the constant group scheme $\mathbf{Z}/p\mathbf{Z}$ by the multiplicative group scheme $\mu_p$ in the category of finite flat commutative group schemes over $\mathcal{O}_K$ - for a general number field $K$ and prime $p$, in the regimes excluded by Yasuda's hypotheses:

(i) the class-number regime, where $p$ divides $\#\mathrm{Cl}(K(\zeta_p))$;

(ii) the higher-ramification regime, where some prime $\mathfrak p$ of $K$ above $p$ has ramification index $e_{\mathfrak p}\ge p-1$, which requires the local Ext computation beyond the Oort–Tate/Dieudonné normal-form framework that is valid only for $e_{\mathfrak p}<p-1$;

(iii) required only as an ingredient of (i) and (ii), the complete accounting of the non-split extension classes $T_\varepsilon$ generated by units $\varepsilon\in\mathcal{O}_K^\times$ that are not $p$-th powers, i.e. the exact contribution of the global unit group to the Ext group over general number fields.

The question is the structure of $\operatorname{Ext}^1_{\mathcal{O}_K}(\mathbf{Z}/p\mathbf{Z},\mu_p)$ in each of the regimes (i) and (ii), together with the resulting torsion criterion; regime (iii) enters only insofar as it contributes to (i) or (ii). The target is a structure theorem for the global Ext group precise enough to yield a criterion for exactly when an elliptic curve with good reduction everywhere over $K$ can carry a $K$-rational point of order $p$. Known results delimit the answer: nonexistence holds whenever (A) and (B) do (Yasuda 2012/2013), while torsion does occur in the class-number regime, where Yasuda (2013) constructs curves with good reduction everywhere and a rational point of order $5$ over quadratic fields such as $K=\mathbf{Q}(\sqrt{26})$, for which $5\mid\#\mathrm{Cl}(K(\zeta_5))$.

References

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