Why special fiber of Neron model of elliptic curve is group variety?

Question

Let $E$ be an elliptic curve over local field $K$, whose ring of integers is $R$. Let $ε$ be an Neron model of $E$, and $ε'$ be it's special fiber.

Then, why $ε'$ is a group variety ?

Does this depend on some special condition of neron model? Or is the 'Special fiber of group scheme is always group variety'holds in general?

Thank you in advance.

Answer 1

The special fiber of a group scheme is always a group scheme. One way to think about this is follows.

Fact (Yoneda's lemma): Let $S$ be a scheme and $G$ an $S$-scheme. Then, the map $$(m,i,e)\mapsto (R\mapsto (m(R),i(R),e(R))$$ is a bijection $$\left\{\begin{matrix}\text{group }S-\text{scheme}\\ \text{structures }(m,i,e)\end{matrix}\right\}\to \left\{\begin{matrix}\text{Factorizations }\begin{matrix}\mathbf{Sch}_S & --> & > \mathbf{Grp}\\ & G\searrow& \uparrow\\ & & > \mathbf{Set}\end{matrix}\end{matrix}\right\}.$$

In words this says that to give a group scheme structure on $G\to S$ is the same thing as giving a functorial group structure on $G(X)$ for every $S$-scheme $X$.

Well, if $T\to S$ is any map of schemes, then one naturally gets a group $T$-scheme on $G_T$. Why? Because, we need to give a fucntorial group structure on $\mathrm{Hom}_T(Y,G_T)$ for every $T$-scheme $Y$. But, by the definition of the fiber product, this is functorially in bijection with $\mathrm{Hom}_S(Y,G)$ where, here, $Y$ is considered an $S$-scheme through the composition $Y\to T\to S$. But, $\mathrm{Hom}_S(Y,G)$ has a group scheme structure since $G$ is a group $S$-scheme. We win.

Answer 2

Question:"Then, why $ε′$ is a group variety? Does this depend on some special condition of neron model? Or is the 'Special fiber of group scheme is always group variety'holds in general? Thank you in advance."

Answer: Assume for simplicity you consider an affine $R$-group scheme $\epsilon:=Spec(A)$ with $R \rightarrow A$ a commutative unital $R$-algebra with $K:=K(R)$ the quotient field of $R$. Let $\epsilon ':=Spec(K \otimes_R A)$ be the special fiber over $K$ and let $K'$ be any commutative unital $K$-algebra. There is by Frobenius reciprocity an equality

$$Hom_{R-alg}(A,K') \cong Hom_{K-alg}(K\otimes_R A, K')$$

of sets and since $\epsilon (K'):=Hom_{R-alg}(A,K')\cong \epsilon'(K')$ equals the $K'$-points of the group scheme $\epsilon$, it follows $\epsilon'(K')$ has canonically the structure of an abelian group for any map $K \rightarrow K'$. Hence the affine scheme $\epsilon'$ is an affine $K$-group scheme by definition.

Note: For the special fiber $\epsilon'$ to be a "group variety" (in the sense of Hartshorne, Ch I) the base field $K$ should be algebraically closed of characteristic zero. Since $K\otimes_R A$ is a Hopf algebra it follows from Thm.11.4 (Waterhouse, "Introduction to affine group schemes") that $A$ is reduced. By Thm.11.6 in the same book it follows $Spec(K\otimes_R A)$ is smooth.

I recommend the Waterhouse book as an introduction to the study of affine $k$-group schemes. It requires some knowledge on commutative algebra at the level of Atiyah-Macdonalds book (or Matsumura's book).