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 k¯ be a fixed algebraic closureMathworldPlanetmath of k. Let be an elliptic surface over a curve C/k and let K=k(C) be the function field of C. Let ¯=(k¯) (or more precisely ¯=×SpeckSpeck¯). The Néron-Severi group of ¯, denoted by NS(¯), is by definition the group of divisors on ¯ modulo algebraic equivalence. Under the previous assumptions, NS(¯) is a finitely generatedMathworldPlanetmath abelian groupMathworldPlanetmath (this is a consequence of the so-called ‘theorem of the base’ which can be found in [1]). The Néron-Severi group of , denoted by NS(), is simply the image of the group of divisors on in NS(¯). Let TNS() be the subgroupMathworldPlanetmathPlanetmath generated by the image of the zero-section σ0 and all the irreducible components of the fibers of π. T is sometimes called the “trivial part” of NS().

Theorem (Shioda-Tate formula).

For each tC let nt be the number of irreducible components on the fiber at t, i.e. π-1(t). Then:

rank(/K) = rank(NS())-rank(T)
= rank(NS())-2-tC(nt-1).


  • 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
Canonical name ShiodaTateFormula
Date of creation 2013-03-22 15:34:22
Last modified on 2013-03-22 15:34:22
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 4
Author alozano (2414)
Entry type Theorem
Classification msc 14J27