• Corpus ID: 119634253

Reconstructing global fields from Dirichlet L-series

@article{Cornelissen2017ReconstructingGF,
  title={Reconstructing global fields from Dirichlet L-series},
  author={Gunther Cornelissen and Bart de Smit and Xin Li and Matilde Marcolli and Harry Smit},
  journal={arXiv: Number Theory},
  year={2017}
}
We prove that two global fields are isomorphic if and only if there is an isomorphism of groups of Dirichlet characters that preserves L-series. 
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Additive structure of totally positive quadratic integers
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References

SHOWING 1-10 OF 21 REFERENCES
Reconstructing global fields from dynamics in the abelianized Galois group
We study a dynamical system induced by the Artin reciprocity map for a global field. We translate the conjugacy of such dynamical systems into various arithmetical properties that are equivalent to
Algebraic Number Theory
I: Algebraic Integers.- II: The Theory of Valuations.- III: Riemann-Roch Theory.- IV: Abstract Class Field Theory.- V: Local Class Field Theory.- VI: Global Class Field Theory.- VII: Zeta Functions
On the Field-theoreticity of Homomorphisms between the Multiplicative Groups of Number Fields
We discuss the eld-theoreticity of homomorphisms between the multiplicative groups of number elds. We prove that, for instance, for a given isomorphism between the multiplicative groups of number
Quantum Statistical Mechanics, L-Series and Anabelian Geometry I: Partition Functions
The zeta function of a number field can be interpreted as the partition function of an associated quantum statistical mechanical (QSM) system, built from abelian class field theory.
Isomorphisms of Galois Groups of Algebraic Function Fields
THEOREM. If there exists a topological isomorphism v: G1 G2, there corresponds a unique isomorphism of fields z: Q, I Q2 such that U(gJ = rgl 1 for every g, e G1. An analogous theorem for algebraic
On Artin L-functions and Gassmann Equivalence for Global Function Fields
In this paper we present an approach to study arithmetical properties of global function fields by working with Artin L-functions. In particular we recall and then extend a criteria of two function
Curves, dynamical systems, and weighted point counting
  • G. Cornelissen
  • Mathematics
    Proceedings of the National Academy of Sciences
  • 2013
TLDR
The method of proof is to show that this is equivalent to the respective class field theories of the curves being isomorphic as dynamical systems, in a sense that the result solves the analogue of the isospectrality problem for curves over finite fields.
Galois group of the maximal abelian extension over an algebraic number field
The aim of the present work is to determine the Galois group of the maximal abelian extension Ω A over an algebraic number field Ω of finite degree, which we fix once for all. Let Z be a continuous
Endomorphisms of abelian varieties over finite fields
Almost all of the general facts about abelian varieties which we use without comment or refer to as "well known" are due to WEIL, and the references for them are [12] and [3]. Let k be a field, k its
...
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