On generalized main conjectures and $p$-adic Stark conjectures for Artin motives
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- by Alexandre Maksoud
- Trans. Amer. Math. Soc.
- DOI: https://doi.org/10.1090/tran/9131
- Published electronically: April 3, 2024
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Abstract:
Given an odd prime number $p$ and a $p$-stabilized Artin representation $\rho$ over $\mathbb {Q}$, we introduce a family of $p$-adic Stark regulators and we formulate an Iwasawa-Greenberg main conjecture and a $p$-adic Stark conjecture which can be seen as an explicit strengthening of conjectures by Perrin-Riou and Benois in the context of Artin motives. We show that these conjectures imply the $p$-part of the Tamagawa number conjecture for Artin motives at $s=0$ and we obtain unconditional results on the torsionness of Selmer groups. We also relate our new conjectures with various main conjectures and variants of $p$-adic Stark conjectures that appear in the literature. In the case of monomial representations, we prove that our conjectures are essentially equivalent to some newly introduced Iwasawa-theoretic conjectures for Rubin-Stark elements. We derive from this a $p$-adic Beilinson-Stark formula for finite-order characters of an imaginary quadratic field in which $p$ is inert.
Along the way, we prove that the Gross-Kuz’min conjecture unconditionally holds for abelian extensions of imaginary quadratic fields.
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Bibliographic Information
- Alexandre Maksoud
- Affiliation: Fakultät für mathematik, Paderborn University, Warburger Str. 100, 33098 Paderborn, Germany
- MR Author ID: 1553362
- ORCID: 0000-0003-4976-4768
- Email: maksoud.alexandre@gmail.com
- Received by editor(s): January 31, 2023
- Received by editor(s) in revised form: June 29, 2023
- Published electronically: April 3, 2024
- Additional Notes: This research was supported by the Luxembourg National Research Fund, Luxembourg, INTER/ANR/18/12589973 GALF
- © Copyright 2024 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
- MSC (2020): Primary 14J33, 18G80, 57K20
- DOI: https://doi.org/10.1090/tran/9131