Annales Mathematiques du Quebec最新文献

Let K be an imaginary quadratic field where the prime p splits. Our goal in this article is to prove results towards the Iwasawa main conjectures for p-nearly-ordinary families associated to (mathrm {GL}_2times mathrm {Res}_{K/mathbb {Q}}mathrm {GL}_1) with a minimal set of assumptions. The main technical input is an improvement on the locally restricted Euler system machinery that allows the treatment of residually reducible cases, which we apply with the Beilinson–Flach Euler system.

The rank one Gross conjecture for Deligne–Ribet p-adic L-functions was solved in works of Darmon-Dasgupta-Pollack and Ventullo by the Eisenstein congruence among Hilbert modular forms. The purpose of this paper is to prove an analogue of the Gross conjecture for the Katz p-adic L-functions attached to imaginary quadratic fields via the congruences between CM forms and non-CM forms. The new ingredient is to apply the p-adic Rankin–Selberg method to construct a non-CM Hida family which is congruent to a Hida family of CM forms at the (1+varepsilon ) specialization.

This paper begins a new approach to the r-trace formula, without removing the nontempered contribution to the spectral side. We first establish an invariant trace formula whose discrete spectral terms are weighted by automorphic L-functions. This involves extending the results of Finis, Lapid, and Müller on the continuity of the coarse expansion of Arthur’s noninvariant trace formula to the refined expansion, and then to the invariant trace formula, while incorporating the use of basic functions at unramified places.

We introduce multi-torsion, a spectral invariant generalizing Ray–Singer analytic torsion. We define multi-torsion for compact manifolds with a certain local geometric product structure that gives a bigrading on differential forms. We prove that multi-torsion is metric-independent in a suitable sense. Our definition of multi-torsion is inspired by an interpretation of each of analytic torsion and the eta invariant as a regularized integral of a closed differential form on a space of metrics on a vector bundle or on a space of elliptic operators. We generalize the Stokes’ theorem argument explaining the dependence of torsion and eta on the geometric data used to define them to the local product setting to prove our metric-independence theorem for multi-torsion.

We extend the definition of the bounded reduction property to endomorphisms of automatic group and find conditions for it to hold. We study endomorphisms with L-quasiconvex image and prove that those with finite kernel satisfy a synchronous version of the bounded reduction property. Finally, we use these techniques to prove L-quasiconvexity of the equalizer of two endomorphisms under certain (strict) conditions.

We ask several questions about substitution maps in the Robba ring. These questions are motivated by p-adic Hodge theory and the theory of p-adic dynamical systems. We provide answers to those questions in special cases, thereby generalizing results of Kedlaya, Colmez, and others.

We study a p-adic Maass–Shimura operator in the context of Mumford curves defined by [15]. We prove that this operator arises from a splitting of the Hodge filtration, thus answering a question in [15]. We also study the relation of this operator with generalized Heegner cycles, in the spirit of [1, 4, 19, 28].

Let (ell ) be a rational prime. Previously, abelian (ell )-towers of multigraphs were introduced which are analogous to (mathbb {Z}_{ell })-extensions of number fields. It was shown that for towers of bouquets, the growth of the (ell )-part of the number of spanning trees behaves in a predictable manner (analogous to a well-known theorem of Iwasawa for (mathbb {Z}_{ell })-extensions of number fields). In this paper, we extend this result to abelian (ell )-towers over an arbitrary connected multigraph (not necessarily simple and not necessarily regular). In order to carry this out, we employ integer-valued polynomials to construct power series with coefficients in (mathbb {Z}_ell ) arising from cyclotomic number fields, different than the power series appearing in the prequel. This allows us to study the special value at (u=1) of the Artin–Ihara L-function, when the base multigraph is not necessarily a bouquet.

We show that extended graph 4-manifolds (as defined by Frigerio–Lafont–Sisto in [12]) do not support Einstein metrics.