Shioda-Tate formula

The main references for this part are the works of Shioda and Tate[2], [4], [5].

Let $k$ be a field and let $\overline{k}$ bea fixed algebraic closure of $k$. Let $ℰ$ be an elliptic surface over a curve $C/k$ and let $K=k(C)$ be the function field of $C$. Let$\overline{ℰ}=ℰ(\overline{k})$ (or more precisely$\overline{ℰ}=ℰ{\times }_{\mathrm{Spec}k}\mathrm{Spec}\overline{k}$). The Néron-Severi group of$\overline{ℰ}$, denoted by $\mathrm{NS}(\overline{ℰ})$, is by definition the group ofdivisors on $\overline{ℰ}$ modulo algebraic equivalence. Underthe previous assumptions, $\mathrm{NS}(\overline{ℰ})$ is a finitely generated abeliangroup (this is a consequence of the so-called ‘theorem of thebase’ which can be found in [1]). The Néron-Severi groupof $ℰ$, denoted by $\mathrm{NS}(ℰ)$, is simply the image of the group ofdivisors on $ℰ$ in $\mathrm{NS}(\overline{ℰ})$. Let $T\subset \mathrm{NS}(ℰ)$ be the subgroupgenerated by the image of the zero-section ${\sigma }_0$ and all theirreducible components of the fibers of $\pi$. $T$ is sometimescalled the “trivial part” of $\mathrm{NS}(ℰ)$.

Theorem (Shioda-Tate formula).

For each $t\mathrm{\in }C$ let $n_t$ be the number ofirreducible components on the fiber at $t$, i.e. ${\pi }^{\mathrm{-}\mathrm{1}}\mathit{}\mathrm{(}t\mathrm{)}$.Then:

${\mathrm{rank}}_{\mathbb{Z}}(ℰ/K)$$=$${\mathrm{rank}}_{\mathbb{Z}}(\mathrm{NS}(ℰ))-{\mathrm{rank}}_{\mathbb{Z}}(T)$

$=$${\mathrm{rank}}_{\mathbb{Z}}(\mathrm{NS}(ℰ))-2-{\displaystyle \sum _{t\in C}}(n_t-1).$