Fields of definition of function fields and hurwitz families — groups as galois groups

AbstractLetE/kbe a function field over an infinite field of constants. Assume thatE/k(x) is a separable extension of degree greater than one such that there exists a place of degree one ofk(x) ramified inE. LetK/kbe a function field. We prove that there exist infinitely many nonisomorphic separable extensionsL/Ksuch that [L:K]=[E:k(x)] andAutkL=AutKL≅Autk(x)E.

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Introduction

Berkeley Lectures on p-adic Geometry ◽

10.23943/princeton/9780691202082.003.0001 ◽

2020 ◽

pp. 1-6

Author(s):

Peter Scholze ◽

Jared Weinstein

Keyword(s):

Moduli Spaces ◽

Function Fields ◽

Number Fields ◽

Important Special Case ◽

Langlands Correspondence ◽

Key Innovation ◽

Perfectoid Spaces ◽

Definition Of ◽

Special Case

This introductory chapter provides an overview of Drinfeld's work on the global Langlands correspondence over function fields. Whereas the global Langlands correspondence is largely open in the case of number fields K, it is a theorem for function fields, due to Drinfeld and L. Lafforgue. The key innovation in this case is Drinfeld's notion of an X-shtuka (or simply shtuka). The Langlands correspondence for X is obtained by studying moduli spaces of shtukas. A large part of this course is about the definition of perfectoid spaces and diamonds. There is an important special case where the moduli spaces of shtukas are classical rigid-analytic spaces. This is the case of local Shimura varieties. Some examples of these are the Rapoport-Zink spaces.

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An arithmetic proof of Pop`s Theorem concerning Galois groups of function fields over number fields.

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crll.1996.478.107 ◽

1996 ◽

Vol 1996 (478) ◽

pp. 107-126

Keyword(s):

Function Fields ◽

Number Fields ◽

Galois Groups

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On definable Galois groups and the strong canonical base property

Journal of Mathematical Logic ◽

10.1142/s0219061317500027 ◽

2017 ◽

Vol 17 (01) ◽

pp. 1750002

Author(s):

Daniel Palacín ◽

Anand Pillay

Keyword(s):

Finite Rank ◽

Galois Groups ◽

Strong Form ◽

Current Paper ◽

Base Property ◽

Canonical Base ◽

Definition Of

In [E. Hrushovski, D. Palacín and A. Pillay, On the canonical base property, Selecta Math. (N.S.) 19(4) (2013) 865–877], Hrushovski and the authors proved, in a certain finite rank environment, that rigidity of definable Galois groups implies that [Formula: see text] has the canonical base property in a strong form; “internality to” being replaced by “algebraicity in”. In the current paper, we give a reasonably robust definition of the “strong canonical base property” in a rather more general finite rank context than [E. Hrushovski, D. Palacín and A. Pillay, On the canonical base property, Selecta Math. (N.S.) 19(4) (2013) 865–877], and prove its equivalence with rigidity of the relevant definable Galois groups. The new direction is an elaboration on the old result that [Formula: see text]-based groups are rigid.

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