L-invariants of Artin motives

in Annales mathématiques du Québec (2022)

We compute Benois 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 ... [more ▼]

We compute Benois 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. [less ▲]

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

E-print/Working paper (2021)

We continue the study of Selmer groups associated with an Artin representation endowed with a p-stabilization which was initiated in arXiv:1811.05368. We formulate a main conjecture and an extra zeros ... [more ▼]

We continue the study of Selmer groups associated with an Artin representation endowed with a p-stabilization which was initiated in arXiv:1811.05368. We formulate a main conjecture and an extra zeros conjecture at all unramified odd primes p, which are shown to imply the p-part of the Tamagawa number conjecture for Artin motives at s=0. We also relate our new conjectures with various cyclotomic Iwasawa main conjectures and p-adic Stark conjectures that appear in the literature. In particular, they provide a natural interpretation for recent conjectures on p-adic L-functions attached to (the adjoint of) a weight one modular form. 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. [less ▲]

Théorie d'Iwasawa des motifs d'Artin et des formes modulaires de poids 1

E-print/Working paper (2019)

Let p be an odd prime. We study the structure of the cyclotomic Greenberg-Selmer group attached to a general irreducible Artin motive over Q endowed with an ordinary p-stabilization. Under the Leopoldt ... [more ▼]

Let p be an odd prime. We study the structure of the cyclotomic Greenberg-Selmer group attached to a general irreducible Artin motive over 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. [less ▲]