On the density of abelian surfaces with Tate- Shafarevich group of order five times a square
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2013
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866916443256258560 |
|---|---|
| author | Keil, Stefan Kloosterman, Remke |
| author_facet | Keil, Stefan Kloosterman, Remke |
| contents | Let A=E_1xE_2 be be the product of two elliptic curves over QQ, both having a rational five torsion point P_i. Set B=A/<(P_1,P_2)>. In this paper we give an algorithm to decide whether the Tate-Shafarevich group of the abelian surface B has square order or order five times a square, assuming that we can find a basis for the Mordell-Weil groups of both E_i, and that the Tate-Shafarevich groups of the E_i are finite.
We considered all pairs (E_1,E_2), with prescribed bounds on the conductor and the coefficients on a minimal Weierstrass equation. In total we considered around 20.0 million of abelian surfaces of which 49.16% have a Tate-Shafarevich group of non-square order. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1303_3500 |
| institution | arXiv |
| publishDate | 2013 |
| record_format | arxiv |
| spellingShingle | On the density of abelian surfaces with Tate- Shafarevich group of order five times a square Keil, Stefan Kloosterman, Remke Number Theory Let A=E_1xE_2 be be the product of two elliptic curves over QQ, both having a rational five torsion point P_i. Set B=A/<(P_1,P_2)>. In this paper we give an algorithm to decide whether the Tate-Shafarevich group of the abelian surface B has square order or order five times a square, assuming that we can find a basis for the Mordell-Weil groups of both E_i, and that the Tate-Shafarevich groups of the E_i are finite. We considered all pairs (E_1,E_2), with prescribed bounds on the conductor and the coefficients on a minimal Weierstrass equation. In total we considered around 20.0 million of abelian surfaces of which 49.16% have a Tate-Shafarevich group of non-square order. |
| title | On the density of abelian surfaces with Tate- Shafarevich group of order five times a square |
| topic | Number Theory |
| url | https://arxiv.org/abs/1303.3500 |