## Kyoto Journal of Mathematics

- Kyoto J. Math.
- Volume 55, Number 4 (2015), 687-772.

### Structure of Tate–Shafarevich groups of elliptic curves over global function fields

#### Abstract

The structure of the Tate–Shafarevich groups of a class of elliptic curves over global function fields is determined. These are known to be finite abelian groups and hence they are direct sums of finite cyclic groups where the orders of these cyclic components are invariants of the Tate–Shafarevich group. This decomposition of the Tate–Shafarevich groups into direct sums of finite cyclic groups depends on the behaviour of Drinfeld–Heegner points on these elliptic curves. These are points analogous to Heegner points on elliptic curves over the rational numbers.

#### Article information

**Source**

Kyoto J. Math., Volume 55, Number 4 (2015), 687-772.

**Dates**

Received: 6 December 2012

Revised: 25 June 2014

Accepted: 8 September 2014

First available in Project Euclid: 25 November 2015

**Permanent link to this document**

https://projecteuclid.org/euclid.kjm/1448460077

**Digital Object Identifier**

doi:10.1215/21562261-3157730

**Mathematical Reviews number (MathSciNet)**

MR3479308

**Zentralblatt MATH identifier**

1378.11064

**Subjects**

Primary: 11G05: Elliptic curves over global fields [See also 14H52] 11G09: Drinfelʹd modules; higher-dimensional motives, etc. [See also 14L05] 11G20: Curves over finite and local fields [See also 14H25] 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture [See also 14G10] 14G10: Zeta-functions and related questions [See also 11G40] (Birch- Swinnerton-Dyer conjecture) 14G17: Positive characteristic ground fields 14G25: Global ground fields 14H52: Elliptic curves [See also 11G05, 11G07, 14Kxx]

**Keywords**

Elliptic curves Tate–Shafarevich groups function fields

#### Citation

Brown, M. L. Structure of Tate–Shafarevich groups of elliptic curves over global function fields. Kyoto J. Math. 55 (2015), no. 4, 687--772. doi:10.1215/21562261-3157730. https://projecteuclid.org/euclid.kjm/1448460077