$K$-groups and dual graphs of special fibers

Question

Let $p$ be a prime number, let $E$ be an elliptic curve defined over $\mathbb{Q}_p$. Let $\mathcal{E}_p$ be the special fiber of the Néron model of $E$ over $\mathbb{Z}_p$ and let $\mathcal{C}_p$ be the special fiber of the minimal regular model of $E$ over $\mathbb{Z}_p$

In their 1986 paper "$K_2$ and $L$-functions of elliptic curves - computer calculations" Bloch and Grayson (p. 82, 83) state that the first $K$-group $K_1'(\mathcal{E}_p)$ is non-torsion iff $E$ has split multiplicative reduction at $p$. In this case we have $K_1'(\mathcal{E}_p)\cong \mathbb{Z}\oplus\mathrm{torsion}$ and their argument is that this follows because the first homology of the dual graph of $\mathcal{C}_p$ is isomorphic to $\mathbb{Z}$. But the only proof of this fact that I've seen (in the thesis of Rolshausen) does not use the dual graph at all. Note that the rank of the first homology of the dual graph is equal to the toric rank $t_p$ of (the connected component of) $\mathcal{E}_p$.

(1) How can you use the dual graph of $\mathcal{C}_p$ to show that the rank of $K_1'(\mathcal{E}_p)$ is equal to $t_p$?

Presumably such a proof would use étale cohomology which unfortunately I don't know enough about.

Now suppose that $C$ is a (smooth, projective) curve of positive genus defined over $\mathbb{Q}_p$. Let $\mathcal{C}_p$ be the special fiber of the minimal regular model of $C$ over $\mathbb{Z}_p$. Let $t_p$ be the toric rank of (the connected component of the special fiber of) the Néron model of $\mathrm{Jac}(C)$ over $\mathbb{Z}_p$. As above, this is equal to the rank of the first homology of the dual graph of $\mathcal{C}_p$.

(2) Is it known that the rank of $K_1'(\mathcal{C}_p)$ is equal to $t_p$?

Answer 1

I'll draw the connection in the case of where the special fiber $\mathcal{C}_p$ is a triangle of three crossing copies of $\mathbb{P}^1$. Let $Z$ be the closed subscheme of $\mathcal{C}_p$ consisting of the three singularities (with reduced induced structure), and $U\cong\bigsqcup_3\mathbb{G}_m$ its open complement. Applying the localization sequence for $K'$-theory (and noting that $U$ and $Z$ are regular) gives an exact sequence $$ K_1(Z) \to K_1'(\mathcal{C}_p) \to K_1(U)\to K_0(Z). $$ First, $K_1(Z)$ is torsion and thus does not contribute to the question. Now $K_1(U)\cong\mathbb{Z}^3$ and $K_0(Z)\cong\mathbb{Z}^3$, and if you work out the boundary map, you see that the complex $K_1(U)\to K_0(Z)$ is exactly the complex computing the homology of the dual graph of $\mathcal{C}_p$. I guess it is an exercise to do the remaining list of possible special fibers of the minimal regular model.

Alternatively, one can identify $K_1'(\mathcal{C}_p)_{\mathbb{Q}}$ with étale Borel-Moore homology (as in Section 2 of this paper of Kondo and Yasuda.)