On Artin L-functions and Gassmann Equivalence for Global Function Fields

Abstract

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 fields to be arithmetically equivalent in terms of Artin L-functions of representations associated to the common normal closure of those fields. We provide few examples of such non-isomorphic fields and also discuss an algorithm to construct many such examples by using torsion points on elliptic curves. Finally, we will show how to apply our results in order to distinguish two global fields by a finite list of Artin L-functions.Acknowledgements: I would like to thank both my advisors, namely professor Bart de Smit and professor Karim Belabas, for their useful advices during the project.

Twisted isospectrality, homological wideness and isometry

“…In this form, the statement of the proposition is similar to a number-theoretical result of Solomatin [73] that inspired our proof below.…”

“…Our method of proof for Theorem A, presented in Section 1-5, is based on a similar construction of Solomatin [73] for algebraic function fields, which in turn is based on number theoretical work of Bart de Smit in [23]. The analogy between number theory and spectral differential geometry was pioneered by Sunada [74] (see also the survey [77]), and the importance of representation theoretical techniques was pointed out early on by Sunada [75] and Pesce [65].…”

Characterization of global fields by Dirichlet L-series

“…On the other hand, it is not possible to copy the proof of Theorem 10.1 for function fields, as this would force fixing a rational subfield F q ptq inside both K and L (that plays the role of Q in the number field proof), for which there are infinitely many, non-canonical, choices. However, Theorem 10.1 does hold in the relative setting of separable geometric extensions of a fixed rational function field of characteristic not equal to 2, compare [19]. It is unclear to us whether the analogue of Theorem 10.1 holds for a global function field without fixing a rational subfield.…”

“…The current paper not only presents full and simplified proofs, but also uses only classical methods from number theory (class field theory, Chebotarev, Grünwald-Wang, and inverse Galois theory). In [19], reconstruction of field extensions of a fixed rational function field from relative L-series is treated from the point of view of the method in sections 9 and 10 of this paper.…”

Arithmetic equivalence for non-geometric extensions of global function fields

“…The second part of this work (Section 4) justifies the need for the previous generalization by presenting an actual example of two non-geometric extensions K and L of F q (T ) which are arithmetically equivalent and non isomorphic over F q (T ) but such that they are not equivalent over F q 2 (T ). In fact, known examples of non-geometric extensions equivalent over F q (T ) have been already obtained in [CKVdZ10] and by Solomatin [Sol16] but considering equivalent geometric extensions over F q r (T ), which therefore are equivalent over F q (T ). Instead, for every prime number p we present a couple (K(p), K ′ (p)) of equivalent and non-isomorphic extensions of F 2 (T ) which contain F 4 (T ) but are not equivalent over this bigger field; furthermore, we will show that K(p) is not equivalent to K(q) if p = q and p ≥ 5, so that the examples are distinct from each other.…”