scholarly journals On the construction of Galois extensions of function fields and number fields

1974 ◽  
Vol 207 (2) ◽  
pp. 99-120 ◽  
Author(s):  
Kuang-yen Shih
Keyword(s):  
Function Fields ◽  
Number Fields ◽  
Galois Extensions

Introduction

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.

On a certain analogy between algebraic number fields and function fields

2001 ◽  
pp. 512-515
Author(s):  
Ichiro Satake ◽  
Genjiro Fujisaki ◽  
Kazuya Kato ◽  
Masato Kurihara ◽  
Shoichi Nakajima
Keyword(s):  
Algebraic Number ◽  
Function Fields ◽  
Number Fields ◽  
Algebraic Number Fields ◽  
And Function

scholarly journals Number fields and function fields: coalescences, contrasts and emerging applications

2015 ◽  
Vol 373 (2040) ◽  
pp. 20140315 ◽  
Author(s):  
J. P. Keating ◽  
Z. Rudnick ◽  
T. D. Wooley
Keyword(s):  
Finite Field ◽  
Function Fields ◽  
Analytic Number Theory ◽  
Number Fields ◽  
Past Half Century ◽  
Recent Developments ◽  
Substantial Progress ◽  
And Function ◽  
Past Half ◽  
Field Setting

The similarity between the density of the primes and the density of irreducible polynomials defined over a finite field of q elements was first observed by Gauss. Since then, many other analogies have been uncovered between arithmetic in number fields and in function fields defined over a finite field. Although an active area of interaction for the past half century at least, the language and techniques used in analytic number theory and in the function field setting are quite different, and this has frustrated interchanges between the two areas. This situation is currently changing, and there has been substantial progress on a number of problems stimulated by bringing together ideas from each field. We here introduce the papers published in this Theo Murphy meeting issue, where some of the recent developments are explained.

scholarly journals ALGEBRAIC DIVISIBILITY SEQUENCES OVER FUNCTION FIELDS

2012 ◽  
Vol 92 (1) ◽  
pp. 99-126 ◽  
Author(s):  
PATRICK INGRAM ◽  
VALÉRY MAHÉ ◽  
JOSEPH H. SILVERMAN ◽  
KATHERINE E. STANGE ◽  
MARCO STRENG
Keyword(s):  
Function Field ◽  
Function Fields ◽  
Number Fields ◽  
Lucas Sequences ◽  
Primitive Divisors ◽  
Irreducible Elements ◽  
Primitive Divisor

AbstractIn this note we study the existence of primes and of primitive divisors in function field analogues of classical divisibility sequences. Under various hypotheses, we prove that Lucas sequences and elliptic divisibility sequences over function fields defined over number fields contain infinitely many irreducible elements. We also prove that an elliptic divisibility sequence over a function field has only finitely many terms lacking a primitive divisor.

scholarly journals Finite presentability of arithmetic groups over global function fields

1987 ◽  
Vol 30 (1) ◽  
pp. 23-39 ◽  
Author(s):  
Helmut Behr
Keyword(s):  
Small Groups ◽  
Function Fields ◽  
Algebraic Groups ◽  
Number Fields ◽  
Arithmetic Groups ◽  
Finitely Generated ◽  
Global Function ◽  
Global Function Fields ◽  
Finite Presentability ◽  
Finitely Presented

Arithmetic subgroups of reductive algebraic groups over number fields are finitely presentable, but over global function fields this is not always true. All known exceptions are “small” groups, which means that either the rank of the algebraic group or the set S of the underlying S-arithmetic ring has to be small. There exists now a complete list of all such groups which are not finitely generated, whereas we onlyhave a conjecture which groups are finitely generated but not finitely presented.

Partial Euler Products on the Critical Line

2005 ◽  
Vol 57 (2) ◽  
pp. 267-297 ◽  
Author(s):  
Keith Conrad
Keyword(s):  
Asymptotic Behavior ◽  
Elliptic Curve ◽  
Riemann Hypothesis ◽  
Critical Line ◽  
Function Fields ◽  
Number Fields ◽  
Euler Product ◽  
Second Moments ◽  
Precise Sense ◽  
Euler Products

AbstractThe initial version of the Birch and Swinnerton-Dyer conjecture concerned asymptotics for partial Euler products for an elliptic curve L-function at s = 1. Goldfeld later proved that these asymptotics imply the Riemann hypothesis for the L-function and that the constant in the asymptotics has an unexpected factor of. We extend Goldfeld's theorem to an analysis of partial Euler products for a typical L-function along its critical line. The general phenomenon is related to second moments, while the asymptotic behavior (over number fields) is proved to be equivalent to a condition that in a precise sense seemsmuch deeper than the Riemann hypothesis. Over function fields, the Euler product asymptotics can sometimes be proved unconditionally.

Sign in / Sign up

Export Citation Format

Share Document