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Iwasawa main conjecture for the Carlitz cyclotomic extension and applications

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Abstract

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

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Notes

  1. We shall not distinguish between a prime ideal of A, like \(\mathfrak {p}\), and the place of F corresponding to it.

  2. The map \(\omega _\mathfrak {p}\) can also be defined as the morphism of \(\mathbb {F}\)-algebras such that \(v_\mathfrak {p}(\theta -\omega _{\mathfrak {p}}(\theta ))\geqslant 1\): it satisfies \(\omega _\mathfrak {p}(a)\equiv a \pmod {\mathfrak {p}}\) and corresponds to the choice of a root of \(\pi _{\mathfrak {p}}\) in \(\overline{\mathbb {F}}\) (because \(\pi _\mathfrak {p}=\pi _\mathfrak {p}(\theta )\in A=\mathbb {F}[\theta ]\) and we have \(\pi _\mathfrak {p}(\theta )\equiv \pi _\mathfrak {p}(\omega _{\mathfrak {p}}(\theta ))\pmod \mathfrak {p}\), therefore \(\pi _\mathfrak {p}(\omega _{\mathfrak {p}}(\theta ))\equiv 0\)).

  3. By R-valued distributions on a locally profinite group G we mean the linear functionals on the space of compactly supported locally constant functions \(G\rightarrow R\).

  4. Readers are warned that the notation in [41] is different from ours: our \(S_n(j)\) becomes \(S_n(-j)\) in [41].

  5. This definition of \(\mathbb {S}_\mathfrak {p}\) - the same as in [39, §5.5(b)] - differs from the one in [21], where the factor \(\mathbb {C}_\mathfrak {p}^*\) is missing. We decided to insert this factor in order to emphasize the symmetry with \(\mathbb {S}_\infty \).

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Acknowledgements

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.

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Authors and Affiliations

  1. LMNO, Université de Caen BP 5186, 14032, Caen Cedex, France

    Bruno Anglès

  2. Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127, Pisa, Italy

    Andrea Bandini

  3. Departament de Matemàtiques, Facultat de Ciencies, Universitat Autònoma de Barcelona, 08193, Bellaterra, Barcelona, Catalonia, Andorra

    Francesc Bars

  4. Department of Mathematical Sciences, Xi’an Jiaotong-Liverpool University, 111 Ren Ai Road, Dushu Lake Higher Education Town, Suzhou Industrial Park, Suzhou, 215123, Jiangsu, China

    Ignazio Longhi

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  1. Bruno Anglès
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  2. Andrea Bandini
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  3. Francesc Bars
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  4. Ignazio Longhi
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Correspondence to Francesc Bars.

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Communicated by Toby Gee.

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F. Bars supported by MTM2016-75980-P.

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Anglès, B., Bandini, A., Bars, F. et al. Iwasawa main conjecture for the Carlitz cyclotomic extension and applications. Math. Ann. 376, 475–523 (2020). https://doi.org/10.1007/s00208-019-01875-8

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