### Abstract

We study abelian varieties defined over function fields of curves in positive characteristic p, focusing on their arithmetic in the system of Artin-Schreier extensions. First, we prove that the L-function of such an abelian variety vanishes to high order at the center point of its functional equation under a parity condition on the conductor. Second, we develop an Artin-Schreier variant of a construction of Berger. This yields a new class of Jacobians over function fields for which the Birch and Swinnerton-Dyer conjecture holds. Third, we give a formula for the rank of the Mordell-Weil groups of these Jacobians in terms of the geometry of their fibers of bad reduction and homomorphisms between Jacobians of auxiliary Artin-Schreier curves. We illustrate these theorems by computing the rank for explicit examples of Jacobians of arbitrary dimension g, exhibiting Jacobians with bounded rank and others with unbounded rank in the tower of Artin-Schreier extensions. Finally, we compute the Mordell-Weil lattices of an isotrivial elliptic curve and a family of non-isotrivial elliptic curves. The latter exhibits an exotic phenomenon whereby the angles between lattice vectors are related to point counts on elliptic curves over finite fields. Our methods also yield new results about supersingular factors of Jacobians of Artin-Schreier curves.

Original language | English (US) |
---|---|

Pages (from-to) | 8553-8595 |

Number of pages | 43 |

Journal | Transactions of the American Mathematical Society |

Volume | 368 |

Issue number | 12 |

DOIs | |

State | Published - Jan 1 2016 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Transactions of the American Mathematical Society*,

*368*(12), 8553-8595. https://doi.org/10.1090/tran6641

**Arithmetic of abelian varieties in artin-schreier extensions.** / Pries, Rachel; Ulmer, Douglas.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 368, no. 12, pp. 8553-8595. https://doi.org/10.1090/tran6641

}

TY - JOUR

T1 - Arithmetic of abelian varieties in artin-schreier extensions

AU - Pries, Rachel

AU - Ulmer, Douglas

PY - 2016/1/1

Y1 - 2016/1/1

N2 - We study abelian varieties defined over function fields of curves in positive characteristic p, focusing on their arithmetic in the system of Artin-Schreier extensions. First, we prove that the L-function of such an abelian variety vanishes to high order at the center point of its functional equation under a parity condition on the conductor. Second, we develop an Artin-Schreier variant of a construction of Berger. This yields a new class of Jacobians over function fields for which the Birch and Swinnerton-Dyer conjecture holds. Third, we give a formula for the rank of the Mordell-Weil groups of these Jacobians in terms of the geometry of their fibers of bad reduction and homomorphisms between Jacobians of auxiliary Artin-Schreier curves. We illustrate these theorems by computing the rank for explicit examples of Jacobians of arbitrary dimension g, exhibiting Jacobians with bounded rank and others with unbounded rank in the tower of Artin-Schreier extensions. Finally, we compute the Mordell-Weil lattices of an isotrivial elliptic curve and a family of non-isotrivial elliptic curves. The latter exhibits an exotic phenomenon whereby the angles between lattice vectors are related to point counts on elliptic curves over finite fields. Our methods also yield new results about supersingular factors of Jacobians of Artin-Schreier curves.

AB - We study abelian varieties defined over function fields of curves in positive characteristic p, focusing on their arithmetic in the system of Artin-Schreier extensions. First, we prove that the L-function of such an abelian variety vanishes to high order at the center point of its functional equation under a parity condition on the conductor. Second, we develop an Artin-Schreier variant of a construction of Berger. This yields a new class of Jacobians over function fields for which the Birch and Swinnerton-Dyer conjecture holds. Third, we give a formula for the rank of the Mordell-Weil groups of these Jacobians in terms of the geometry of their fibers of bad reduction and homomorphisms between Jacobians of auxiliary Artin-Schreier curves. We illustrate these theorems by computing the rank for explicit examples of Jacobians of arbitrary dimension g, exhibiting Jacobians with bounded rank and others with unbounded rank in the tower of Artin-Schreier extensions. Finally, we compute the Mordell-Weil lattices of an isotrivial elliptic curve and a family of non-isotrivial elliptic curves. The latter exhibits an exotic phenomenon whereby the angles between lattice vectors are related to point counts on elliptic curves over finite fields. Our methods also yield new results about supersingular factors of Jacobians of Artin-Schreier curves.

UR - http://www.scopus.com/inward/record.url?scp=84991000213&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84991000213&partnerID=8YFLogxK

U2 - 10.1090/tran6641

DO - 10.1090/tran6641

M3 - Article

AN - SCOPUS:84991000213

VL - 368

SP - 8553

EP - 8595

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 12

ER -