Overconvergent Eichler–Shimura isomorphisms

Abstract

Given a prime $p\gt 2$, an integer $h\geq 0$, and a wide open disk $U$ in the weight space $ \mathcal{W} $ of ${\mathbf{GL} }_{2} $, we construct a Hecke–Galois-equivariant morphism ${ \Psi }_{U}^{(h)} $ from the space of analytic families of overconvergent modular symbols over $U$ with bounded slope $\leq h$, to the corresponding space of analytic families of overconvergent modular forms, all with ${ \mathbb{C} }_{p} $-coefficients. We show that there is a finite subset $Z$ of $U$ for which this morphism induces a $p$-adic analytic family of isomorphisms relating overconvergent modular symbols of weight $k$ and slope $\leq h$ to overconvergent modular forms of weight $k+ 2$ and slope $\leq h$.

Rankin–Eisenstein classes in Coleman families

“…However, B 1 (G, M) is manifestly finitely generated as an A-module, and any finitely generated submodule of an orthonormalisable A-Banach module is closed [8,Lemma 2.8]. This proves (1). Parts (2) and (3) now follow from the open image theorem [10, Proposition I.…”

“…Parts (2) and (3) now follow from the open image theorem [10, Proposition I. 1.3], which shows that any continuous surjective map between Q p -Banach spaces has a continuous section (and, in particular, a continuous bijection between Q p -Banach spaces must be a homeomorphism).…”

“…Definition 4.5. 1 We say M has a slope λ decomposition if we can write it as a direct sum of B U -modules…”

“…[30]). 1 Mathematics Institute, University of Warwick, Zeeman Building, Coventry CV4 7AL, UK , 2 Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK.…”

A 0.5 (half) overconvergent Eichler-Shimura isomorphism

“…In this article we continue our study of the overconvergent Eichler-Shimura isomorphisms started in [3]. Let X (N , p) → X denote the pair consisting of the modular curves of level := 1 (N ) ∩ 0 ( p), respectively 1 (N ), over Q p seen as rigid analytic curves, where the map is forget the level p-structure, let 0 ≤ w ≤ p/( p + 1) be a rational number and let K be a finite extension of Q p such that there is an element p w of K with v( p w ) = w. We denote by X (w) ⊂ X , respectively X (w) ⊂ X (N , p) the strict neighborhood of the ordinary locus, respectively the strict neighborhood of the connected component of the ordinary locus containing the cusp ∞ in X and X (N , p) of width w (see Sect.…”