A new geometric construction of a family of Galois representations associated to modular forms: research announcement (Algebraic Number Theory and Related Topics 2014)

"Algebraic Number Theory and Related Topics 2014". December 1～5, 2014. edited by Takeshi Tsuji, Hiroki Takahashi and Yuichiro Hoshi. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.This is a research announcement of the results in [Mih15]. For an odd prime p dividing an integer N ≥ 5, we define an inverse system of sheaves of torsion Zp-modules on a modular curve of level Γ1(N). The representation of Gal(Q/Q) associated to any cuspidal eigenform is obtained as a twist of a quotient of its cohomology. We construct a family of representations of Gal(Q/Q) associated to cuspidal eigenforms of finite bounded slope as a quotient of a twist of a scalar extension of its cohomology

Banach halos and short isometries

The aim of this article is twofold. First, we develop the notion of a Banach halo, similar to that of a Banach ring, except that the usual triangular inequality is replaced by the inequality $|a + b| \leq (|a| , |b|)_p$ involving the p-norm for some $p \in]0, +\infty]$, or by the inequality $|a+b|\leq C\max(|a|,|b|)$. This allows us to have a flow of powers on Banach halos and to work, e.g., with the square of the usual absolute value on $\mathbb{Z}$. Then we define and study the group of short isometries of normed involutive coalgebras over a base commutative Banach halo. An aim of this theory is to define a representable group $K_n\subset {\rm GL}_n$ whose points with values in $\mathbb{R}$ give $O_n(\mathbb{R})$ and whose points with values in $\mathbb{Q}_p$ give GL$_n(\mathbb{Z}_p)$, giving to the analogy between these two groups a kind of geometric explanation