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Dear MO,

Let $p$ be a prime and let $E/\mathbb{Q}_p$ be an elliptic curve. Suppose that $E/\mathbb{Q}_p$ has good ordinary reduction at $p$. In his 1972 paper Propriétés galoisiennes des points d'ordre fini des courbes elliptiques'' (more specifically, see Corollaire, in p. 274), Serre shows along the way that the inertia subgroup of $\operatorname{Gal}(\mathbb{Q}_p(E[p])/\mathbb{Q}_p)$ is, with respect to a suitable basis of $E[p]$, isomorphic to either a matrix group of the form {$[\ast\ 0; 0\ 1]$} or {$[\ast\ \ast; 0\ 1]$}. 1]$} as a subgroup of$\operatorname{GL}(2,\mathbb{F}_p)$. After this result, Serre remarks that he doesn't know of any simple criterion that would determine whether one is in the first case or the second case. Question: Nowadays, do we know of a criterion to tell whether one is in the first case or the second case? A more concrete question: Here is the particular example that I am working with: Let$E/\mathbb{Q}$be 1225h1'' in Cremona's tables, given by $$E : y^2 + xy + y = x^3 + x^2 - 8x + 6.$$ This curve has a rational$37$-isogeny and therefore$\operatorname{Gal}(\mathbb{Q}(E[37])/\mathbb{Q})$is a Borel subgroup of$\operatorname{GL}(2,\mathbb{F}_{37})$. The curve$E$has good ordinary reduction at$p=37$and I am trying to find out whether the ramification index of$37$in the extension$\mathbb{Q}(E[37])/\mathbb{Q}$is just$\varphi(36)$\varphi(37)$ or rather $\varphi(36)\cdot \varphi(37)\cdot 37$, where $\varphi$ is the Euler phi function.

The $37$th division polynomial of $E/\mathbb{Q}$ has degree $684$ and it factors (over $\mathbb{Q}[x]$) as a product of $4$ polynomials of degrees $6$, $6$, $6$ and $666$, respectively. The extension of degree $666$ is, well, diabolically large and I can't find the ramification at $37$ computationally... or at least I don't know how to!

Thanks for your help!

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# Ramification in p-division fields associated to elliptic curves with good ordinary reduction

Dear MO,

Let $p$ be a prime and let $E/\mathbb{Q}_p$ be an elliptic curve. Suppose that $E/\mathbb{Q}_p$ has good ordinary reduction at $p$. In his 1972 paper Propriétés galoisiennes des points d'ordre fini des courbes elliptiques'' (more specifically, see Corollaire, in p. 274), Serre shows along the way that the inertia subgroup of $\operatorname{Gal}(\mathbb{Q}_p(E[p])/\mathbb{Q}_p)$ is, with respect to a suitable basis of $E[p]$, isomorphic to either a matrix group of the form {$[\ast\ 0; 0\ 1]$} or {$[\ast\ \ast; 0\ 1]$}. After this result, Serre remarks that he doesn't know of any simple criterion that would determine whether one is in the first case or the second case.

Question: Nowadays, do we know of a criterion to tell whether one is in the first case or the second case?

A more concrete question: Here is the particular example that I am working with: Let $E/\mathbb{Q}$ be 1225h1'' in Cremona's tables, given by $$E : y^2 + xy + y = x^3 + x^2 - 8x + 6.$$ This curve has a rational $37$-isogeny and therefore $\operatorname{Gal}(\mathbb{Q}(E[37])/\mathbb{Q})$ is a Borel subgroup of $\operatorname{GL}(2,\mathbb{F}_{37})$. The curve $E$ has good ordinary reduction at $p=37$ and I am trying to find out whether the ramification index of $37$ in the extension $\mathbb{Q}(E[37])/\mathbb{Q}$ is just $\varphi(36)$ or rather $\varphi(36)\cdot 37$, where $\varphi$ is the Euler phi function.

The $37$th division polynomial of $E/\mathbb{Q}$ has degree $684$ and it factors (over $\mathbb{Q}[x]$) as a product of $4$ polynomials of degrees $6$, $6$, $6$ and $666$, respectively. The extension of degree $666$ is, well, diabolically large and I can't find the ramification at $37$ computationally... or at least I don't know how to!

Thanks for your help!