Biases in the prime number race of function fields

We give a construction of the genus field for Kummer [Formula: see text]-cyclic extensions of rational congruence function fields, where [Formula: see text] is a prime number. First, we compute the genus field of a field contained in a cyclotomic function field, and then for the general case. This generalizes the result obtained by Peng for a Kummer [Formula: see text]-cyclic extension. Finally, we study the extension [Formula: see text], for [Formula: see text], [Formula: see text] abelian extensions of [Formula: see text].

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The -parity conjecture for abelian varieties over function fields of characteristic

Compositio Mathematica ◽

10.1112/s0010437x13007501 ◽

2014 ◽

Vol 150 (4) ◽

pp. 507-522 ◽

Cited By ~ 1

Author(s):

Fabien Trihan ◽

Seidai Yasuda

Keyword(s):

Finite Field ◽

Zeta Function ◽

Prime Number ◽

Abelian Variety ◽

Function Field ◽

Function Fields ◽

Selmer Group ◽

Smooth Variety ◽

Tate Conjecture ◽

Parity Conjecture

AbstractLet $A/K$ be an abelian variety over a function field of characteristic $p>0$ and let $\ell $ be a prime number ($\ell =p$ allowed). We prove the following: the parity of the corank $r_\ell $ of the $\ell $-discrete Selmer group of $A/K$ coincides with the parity of the order at $s=1$ of the Hasse–Weil $L$-function of $A/K$. We also prove the analogous parity result for pure $\ell $-adic sheaves endowed with a nice pairing and in particular for the congruence Zeta function of a projective smooth variety over a finite field. Finally, we prove that the full Birch and Swinnerton-Dyer conjecture is equivalent to the Artin–Tate conjecture.

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The Abstract Prime Number Theorem For Algebraic Function Fields

Analytic Number Theory ◽

10.1007/978-1-4612-3464-7_32 ◽

1990 ◽

pp. 529-558 ◽

Cited By ~ 3

Author(s):

Wen-Bin Zhang

Keyword(s):

Prime Number ◽

Function Fields ◽

Algebraic Function ◽

Prime Number Theorem ◽

Algebraic Function Fields

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GENUS FIELDS OF CYCLIC l-EXTENSIONS OF RATIONAL FUNCTION FIELDS

International Journal of Number Theory ◽

10.1142/s1793042113500243 ◽

2013 ◽

Vol 09 (05) ◽

pp. 1249-1262 ◽

Cited By ~ 2

Author(s):

VÍCTOR BAUTISTA-ANCONA ◽

MARTHA RZEDOWSKI-CALDERÓN ◽

GABRIEL VILLA-SALVADOR

Keyword(s):

Prime Number ◽

Rational Function ◽

Function Field ◽

Function Fields ◽

Cyclic Extension ◽

Class Field ◽

Dirichlet Characters ◽

Genus Field ◽

Hilbert Class Field

We give a construction of genus fields for Kummer cyclic l-extensions of rational congruence function fields, l a prime number. First we find this genus field for a field contained in a cyclotomic function field using Leopoldt's construction by means of Dirichlet characters and the Hilbert class field defined by Rosen. The general case follows from this. This generalizes the result obtained by Peng for a cyclic extension of degree l.

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