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I'm looking at the following proposition (II.$4.4$) from Silverman's Advanced Topics in the Arithmetic of Elliptic Curves:

Let $K$ be a number field, $\mathfrak P$ a maximal ideal of $K$, and let $E_1/K$ and $E_2/K$ be elliptic curves with good reduction at $\mathfrak P$. Then the natural reduction map $$\hom(E_1,E_2)\to\hom(\tilde E_1,\tilde E_2),\ \ \ \ \phi\mapsto\tilde\phi$$ is injective and preserves degrees.

Silverman doesn't explicitly define this reduction map, I guess he just assumes it's obvious, but I don't quite see how we could always define this. To define the reduction $\tilde E$, we consider $E$ as an elliptic curve over the completion of $K$ with respect to $\mathfrak P$, call it $K_{\mathfrak P}$, then rewrite the equation defining $E$ as an equation over the ring of integers $R$ of $K_{\mathfrak P}$, then take the reduction of this equation modulo our maximal ideal $\mathscr M$ to get a curve $\tilde E$ defined over $k=R/\mathscr M$.

In the case of a morphism $\phi:E_1\to E_2$, how would we define this $\tilde\phi$? If $\phi$ is defined over $\overline K$, we can't really guarantee that it'll be defined over $K_{\mathfrak P}$, can we?

I suppose we could define $\tilde\phi$ by the equation

$$\tilde\phi(\tilde P)=\widetilde{\phi(P)}.$$

How can we be sure this is well-defined? What we would need to check is that if $P\in E_1$ with $\tilde P=\tilde O$, then $\widetilde{\phi(P)}=\tilde O$. But also, don't we only define $\tilde P$ when $P\in E_1(K_{\mathfrak P})$? So how could we be sure that all elements of $\tilde E_1(\bar k)$ are of the form $\tilde P$ for some $P$?

I know there's some correspondence between unramified extensions of $L$ and arbitrary extensions of $k$, so maybe to determine $\tilde\phi$ on all elements of $\tilde E_1(\bar k)$ we just need to be able to define $\phi$ over $K^{nr}$ where $K^{nr}$ is the maximal unramified extension of $K$?

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  • $\begingroup$ (if $K/ \mathbb{Q}$) why not take a maximal ideal $\mathfrak{p} \subset \overline{\mathbb{Z}}$ above $\mathfrak{P}$, such that $E_1(\mathcal{O}_L/(\mathfrak{p} \cap \mathcal{O}_L))$ is well-defined ? $\endgroup$
    – reuns
    Jul 9, 2017 at 4:58

1 Answer 1

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As you've surmised, describing a "natural" reduction map with the technology Silverman has developed up to this point in the book is messy at best. To discuss these things more sanely and cleanly, it's best to use the existence and basic properties of Néron models (to which Chapter IV of the book is dedicated). After replacing $K$ by its completion at the given prime, we may as well assume $K$ is a complete discretely valued field with ring of integers $\mathscr{O}_K$, and that $E_1,E_2$ are elliptic curves over $K$.

The Néron model $\mathscr{E}$ of an elliptic curve $E$ over $K$ is a separated, smooth $\mathscr{O}_K$-group scheme of finite type with generic fiber $E$ which satisfies the following universal property (sometimes called the Néron mapping property): for any smooth $\mathscr{O}_K$-scheme $T$, the map $\mathrm{Hom}_{\mathscr{O}_K}(T,\mathscr{E})\to\mathrm{Hom}_K(T_K,E)$ (restriction to generic fibers) is bijective. In particular, $\mathscr{E}$ is functorial in its generic fiber $E$ on the category of smooth $\mathscr{O}_K$-schemes. Moreover, the special fiber $\mathscr{E}\times_{\mathscr{O}_K}k$ of $\mathscr{E}$ is a smooth $k$-group scheme of finite type (here $k$ is the residue field of $K$). Thus the Néron model furnishes a canonical, functorial "reduction" of $E$ (note that $\mathscr{E}\times_{\mathscr{O}_K}k$ is smooth, but may not be proper, i.e. it may not be an elliptic curve over $k$).

You can get a surprising amount of mileage just taking for granted the existence of Néron models and their basic properties without actually delving into the technical details of how they are constructed. In the context of your question, we can immediately make sense of Silverman's "reduction map" as follows. Let $\mathscr{E}_1$ and $\mathscr{E}_2$ be the Néron models of $E_1$ and $E_2$ over $\mathscr{O}_K$. The Néron mapping property ensures that there is a canonical bijection $\mathrm{Hom}_{\mathscr{O}_K}(\mathscr{E}_1,\mathscr{E}_2)\to\mathrm{Hom}_K(E_1,E_2)$. On the other hand, any map of $\mathscr{O}_K$-schemes $\mathscr{E}_1\to\mathscr{E}_2$ induces, by extension of the base from $\mathscr{O}_K$ to its residue field $k$, a map of $k$-schemes $\mathscr{E}_1\times_{\mathscr{O}_K}k\to\mathscr{E}_2\times_{\mathscr{O}_K}k$. So, if we take the inverse of the first bijection and compose it with base change to the special fiber we get Silverman's map

$$\mathrm{Hom}_K(E_1,E_2)\to\mathrm{Hom}_k(\mathscr{E}_1\times_{\mathscr{O}_K}k,\mathscr{E}_2\times_{\mathscr{O}_K}k)$$.

Notice that we have this map without any assumptions on the reduction types of $E_1$ and $E_2$. The Néron model allows intrinsic definitions (without resorting to any equations) of the reduction types. Good reduction means that the Néron model is proper, or equivalently, an elliptic curve over $\mathscr{O}_K$, and this is implies that the special fiber is an elliptic curve over the residue field. In this context one can then prove injectivity of the map above.

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  • $\begingroup$ Thanks! I'll have to revisit this more carefully once I get through Chapter IV. $\endgroup$ Jul 9, 2017 at 18:02

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