Iwasawa main conjecture for the Carlitz cyclotomic extension and applications

Bruno Anglès; Andrea Bandini; Francesc Bars; Ignazio Longhi

doi:10.1007/s00208-019-01875-8

Iwasawa main conjecture for the Carlitz cyclotomic extension and applications

Bruno Anglès, Andrea Bandini, Francesc Bars^*, Ignazio Longhi

^*Corresponding author for this work

School of Mathematics and Physics

Research output: Contribution to journal › Article › peer-review

6 Citations (Scopus)

Abstract

We prove an Iwasawa Main Conjecture for the class group of the p-cyclotomic extension F of the function field F_q(θ) (p is a prime of F_q[θ]), showing that its Fitting ideal is generated by a Stickelberger element. We use this and a link between the Stickelberger element and a p-adic L-function to prove a close analog of the Ferrero–Washington Theorem for F and to provide information on the p-adic valuations of the Bernoulli-Goss numbers β(j) (i.e., on the values of the Carlitz-Goss ζ-function at negative integers).

Original language	English
Pages (from-to)	475-523
Number of pages	49
Journal	Mathematische Annalen
Volume	376
Issue number	1-2
DOIs	https://doi.org/10.1007/s00208-019-01875-8
Publication status	Published - 1 Feb 2020

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Anglès, B., Bandini, A., Bars, F., & Longhi, I. (2020). Iwasawa main conjecture for the Carlitz cyclotomic extension and applications. Mathematische Annalen, 376(1-2), 475-523. https://doi.org/10.1007/s00208-019-01875-8

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title = "Iwasawa main conjecture for the Carlitz cyclotomic extension and applications",

abstract = "We prove an Iwasawa Main Conjecture for the class group of the p-cyclotomic extension F of the function field Fq(θ) (p is a prime of Fq[θ]), showing that its Fitting ideal is generated by a Stickelberger element. We use this and a link between the Stickelberger element and a p-adic L-function to prove a close analog of the Ferrero–Washington Theorem for F and to provide information on the p-adic valuations of the Bernoulli-Goss numbers β(j) (i.e., on the values of the Carlitz-Goss ζ-function at negative integers).",

author = "Bruno Angl{\`e}s and Andrea Bandini and Francesc Bars and Ignazio Longhi",

note = "Funding Information: All the authors thank the MTM 2009-10359, which funded a workshop on Iwasawa theory for function fields in 2010 and supported the authors during their stay in Barcelona in the summer of 2013, and the CRM (Centre de Recerca Matem{\`a}tica, Bellaterra, Barcelona) for providing a nice environment to work on this project. The fourth author thanks NCTS/TPE for support to travel to Barcelona in summer 2013. Publisher Copyright: {\textcopyright} 2019, Springer-Verlag GmbH Germany, part of Springer Nature.",

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doi = "10.1007/s00208-019-01875-8",

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T1 - Iwasawa main conjecture for the Carlitz cyclotomic extension and applications

AU - Anglès, Bruno

AU - Bandini, Andrea

AU - Bars, Francesc

AU - Longhi, Ignazio

N1 - Funding Information: All the authors thank the MTM 2009-10359, which funded a workshop on Iwasawa theory for function fields in 2010 and supported the authors during their stay in Barcelona in the summer of 2013, and the CRM (Centre de Recerca Matemàtica, Bellaterra, Barcelona) for providing a nice environment to work on this project. The fourth author thanks NCTS/TPE for support to travel to Barcelona in summer 2013. Publisher Copyright: © 2019, Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2020/2/1

Y1 - 2020/2/1

N2 - We prove an Iwasawa Main Conjecture for the class group of the p-cyclotomic extension F of the function field Fq(θ) (p is a prime of Fq[θ]), showing that its Fitting ideal is generated by a Stickelberger element. We use this and a link between the Stickelberger element and a p-adic L-function to prove a close analog of the Ferrero–Washington Theorem for F and to provide information on the p-adic valuations of the Bernoulli-Goss numbers β(j) (i.e., on the values of the Carlitz-Goss ζ-function at negative integers).

AB - We prove an Iwasawa Main Conjecture for the class group of the p-cyclotomic extension F of the function field Fq(θ) (p is a prime of Fq[θ]), showing that its Fitting ideal is generated by a Stickelberger element. We use this and a link between the Stickelberger element and a p-adic L-function to prove a close analog of the Ferrero–Washington Theorem for F and to provide information on the p-adic valuations of the Bernoulli-Goss numbers β(j) (i.e., on the values of the Carlitz-Goss ζ-function at negative integers).

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U2 - 10.1007/s00208-019-01875-8

DO - 10.1007/s00208-019-01875-8

M3 - Article

AN - SCOPUS:85070217638

SN - 0025-5831

VL - 376

SP - 475

EP - 523

JO - Mathematische Annalen

JF - Mathematische Annalen

IS - 1-2

ER -

Iwasawa main conjecture for the Carlitz cyclotomic extension and applications

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