arXiv subjects

Alexandre Maksoud

Publications and source records attributed to Alexandre Maksoud.

6 recordsLinked to original sources

arXiv2025

The eigencurve at crystalline points with scalar Frobenius and Gross-Stark regulators

A complete description of the local geometry of the $p$-adic eigencurve at $p$-irregular classical weight one cusp forms is given in the cases where the usual $R=T$ methods fall short. As an application, we show that the ordinary $p$-adic \'etale cohomology group attached to the tower of elliptic modular curves $X_1(Np^r)$ is not free over the Hecke algebra, when localized at a $p$-irregular weight one point.

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arXiv2023

A canonical generator for congruence ideals of Hida families

We construct canonical adjoint $p$-adic $L$-functions generating the congruence ideal attached to Hida families using Ohta's pairing. We show that these $p$-adic $L$-functions, suitably modified by certain Euler factors, are interpolated by a regular element of Hida's universal ordinary Hecke algebra. We also relate them to characteristic series of primitive adjoint Selmer groups.

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arXiv2022

$\mathscr{L}$-invariants of Artin motives

We compute Benois $\mathscr{L}$-invariants of weight $1$ cuspforms and of their adjoint representations and show how this extends Gross' $p$-adic regulator to Artin motives which are not critical in the sense of Deligne. Benois' construction depends on the choice of a regular submodule which is well understood when the representation is $p$-regular, as it then amounts to the choice of a ``motivic'' $p$-refinement. The situation is dramatically different in the $p$-irregular case, where the regular submodules are parametrized by a flag variety and thus depend on continuous parameters. We are nevertheless able to show in some examples, how Hida theory and the geometry of the eigencurve can be used to detect a finite number of choices of arithmetic and ``mixed-motivic'' significance.

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arXiv2022

On the rank of Leopoldt's and Gross's regulator maps

We generalize Waldschmidt's bound for Leopoldt's defect and prove a similar bound for Gross's defect for an arbitrary extension of number fields. As an application, we prove new cases of Gross's finiteness conjecture (also known as the Gross-Kuz'min conjecture) beyond the classical abelian case, and we show that Gross's $p$-adic regulator has at least half of the conjectured rank. We also describe and compute non-cyclotomic analogues of Gross's defect.

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arXiv2021

On generalized Iwasawa main conjectures and $p$-adic Stark conjectures for Artin motives

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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arXiv2018

Th\'eorie d'Iwasawa des motifs d'Artin et des formes modulaires de poids 1

Let $p$ be an odd prime. We study the structure of the cyclotomic Greenberg-Selmer group attached to a general irreducible Artin motive over $\mathbb{Q}$ endowed with an ordinary $p$-stabilization. Under the Leopoldt and the weak $p$-adic Schanuel Conjectures, we show that it is of torsion over the Iwasawa algebra. Under mild hypotheses on $p$ we compute the constant term of its characteristic series in terms of a $p$-adic regulator and we highlight an extra zeros phenomenon. We then focus on Artin motives attached to classical weight one modular forms, to which our preceding results apply unconditionally. We formulate an Iwasawa Main Conjecture in this context and prove one divisibility using a Theorem of Kato.

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