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# Weak Arithmetic Equivalence

Published:2014-11-03
Printed: Mar 2015
• Guillermo Mantilla-Soler,
Universidad de los Andes, Carrera 1 N. 18A-10, Bogotá, Colombia
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## Abstract

Inspired by the invariant of a number field given by its zeta function, we define the notion of weak arithmetic equivalence and show that under certain ramification hypotheses, this equivalence determines the local root numbers of the number field. This is analogous to a result of Rohrlich on the local root numbers of a rational elliptic curve. Additionally, we prove that for tame non-totally real number fields, the integral trace form is invariant under arithmetic equivalence.
 Keywords: arithmeticaly equivalent number fields, root numbers
 MSC Classifications: 11R04 - Algebraic numbers; rings of algebraic integers 11R42 - Zeta functions and $L$-functions of number fields [See also 11M41, 19F27]

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