# 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 $\overline{k}$ be a fixed algebraic closure of $k$. Let $\mathcal{E}$ be an elliptic surface over a curve $C/k$ and let $K=k(C)$ be the function field of $C$. Let $\overline{\mathcal{E}}=\mathcal{E}(\overline{k})$ (or more precisely $\overline{\mathcal{E}}=\mathcal{E}\times_{\operatorname{Spec}k}\operatorname{% Spec}\overline{k}$). The Néron-Severi group of $\overline{\mathcal{E}}$, denoted by $\operatorname{NS}(\overline{\mathcal{E}})$, is by definition the group of divisors on $\overline{\mathcal{E}}$ modulo algebraic equivalence. Under the previous assumptions, $\operatorname{NS}(\overline{\mathcal{E}})$ is a finitely generated abelian group (this is a consequence of the so-called ‘theorem of the base’ which can be found in [1]). The Néron-Severi group of $\mathcal{E}$, denoted by $\operatorname{NS}(\mathcal{E})$, is simply the image of the group of divisors on $\mathcal{E}$ in $\operatorname{NS}(\overline{\mathcal{E}})$. Let $T\subset\operatorname{NS}(\mathcal{E})$ be the subgroup generated by the image of the zero-section $\sigma_{0}$ and all the irreducible components of the fibers of $\pi$. $T$ is sometimes called the “trivial part” of $\operatorname{NS}(\mathcal{E})$.

###### Theorem (Shioda-Tate formula).

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

 $\displaystyle\operatorname{rank}_{\mathbb{Z}}(\mathcal{E}/K)$ $\displaystyle=$ $\displaystyle\operatorname{rank}_{\mathbb{Z}}(\operatorname{NS}(\mathcal{E}))-% \operatorname{rank}_{\mathbb{Z}}(T)$ $\displaystyle=$ $\displaystyle\operatorname{rank}_{\mathbb{Z}}(\operatorname{NS}(\mathcal{E}))-% 2-\sum_{t\in C}(n_{t}-1).$

## References

• 1 S. Lang, Fundamentals of Diophantine Geometry, Springer-Verlag (1983).
• 2 T. Shioda, On elliptic modular surfaces, J. Math. Soc. Japan 24 (1972), 20-59.
• 3 T. Shioda, An Explicit Algorithm for Computing the Picard Number of Certain Algebraic Surfaces, Amer. J. Math. 108 (1986), 415-432.
• 4 T. Shioda, On the Mordell-Weil Lattices, Commentarii Mathematici Universitatis Sancti Pauli, Vol 39, No. 2, 1990, pp. 211-239.
• 5 J. Tate, On the conjectures of Birch and Swinnerton-Dyer and a geometric analog, Séminaire Bourbaki, 9, Soc. Math. France, Paris, 1966, Exp. No. 306, 415-440, 1995.
Title Shioda-Tate formula ShiodaTateFormula 2013-03-22 15:34:22 2013-03-22 15:34:22 alozano (2414) alozano (2414) 4 alozano (2414) Theorem msc 14J27