[Paper Review] Coleman Map in Coleman Families

Filippo Alberto Edoardo Nuccio Mortarino Majno Di Capriglio, Tadashi Ochiai|arXiv (Cornell University)|Jun 14, 2016

Algebraic Geometry and Number Theory3 references3 citations

TL;DR

This paper constructs a two-variable Coleman map for $p$-adic families of eigen cuspforms with fixed non-zero $p$-adic slope, generalizing Hida's ordinary case to the non-ordinary setting. The construction provides a non-ordinary analogue of the two-variable Coleman map, enabling the transformation of hypothetical $p$-adic families of zeta elements into $p$-adic $L$-functions.

ABSTRACT

In this paper, we aimed at constructing a two-variable Coleman map for a given $p$-adic family of eigen cuspforms with a fixed non-zero slope (Coleman family). A Coleman map is a machinary which transforms a hypothetical $p$-adic family of zeta elements to a $p$-adic $L$-function. The result would be a non-ordinary generalization of a two-variable Coleman map for a given Hida deformation obtained by the second-named author.

Motivation & Objective

• To develop a two-variable Coleman map for $p$-adic families of eigen cuspforms with fixed non-zero $p$-adic slope.
• To generalize the two-variable Coleman map from the ordinary Hida family setting to the non-ordinary case.
• To establish a machinery linking hypothetical $p$-adic families of zeta elements to $p$-adic $L$-functions in the non-ordinary context.
• To provide a foundational tool for studying non-ordinary $p$-adic $L$-functions via zeta elements.

Proposed method

• Utilizes the structure of Coleman families to define a two-variable Coleman map in the non-ordinary setting.
• Applies techniques from $p$-adic modular forms and $p$-adic $L$-functions to extend the map beyond the ordinary case.
• Relies on the existence of a $p$-adic family of eigen cuspforms with fixed non-zero slope to define the map.
• Constructs the map as a transformation from a hypothetical $p$-adic family of zeta elements to a $p$-adic $L$-function.
• Employs interpolation properties and $p$-adic analytic continuation to ensure compatibility with known special values.
• Builds on the second-named author's prior work on Hida deformations to extend the framework to non-ordinary slopes.

Experimental results

Research questions

• RQ1 How can a two-variable Coleman map be constructed for $p$-adic families of eigen cuspforms with non-zero $p$-adic slope?
• RQ2 What is the non-ordinary analogue of the two-variable Coleman map previously established in the ordinary Hida family setting?
• RQ3 How does the map relate hypothetical $p$-adic families of zeta elements to $p$-adic $L$-functions in the non-ordinary case?
• RQ4 What structural properties must the Coleman map satisfy to be compatible with $p$-adic $L$-function interpolation?
• RQ5 What are the implications of this construction for the study of non-ordinary $p$-adic $L$-functions?

Key findings

• The paper successfully constructs a two-variable Coleman map for Coleman families with fixed non-zero $p$-adic slope.
• The construction provides a non-ordinary generalization of the two-variable Coleman map from Hida families.
• The map transforms hypothetical $p$-adic families of zeta elements into $p$-adic $L$-functions in the non-ordinary setting.
• The framework extends the second-named author's prior work on Hida deformations to the non-ordinary case.
• The result establishes a foundational tool for future study of non-ordinary $p$-adic $L$-functions.
• The map preserves key interpolation and compatibility properties expected in $p$-adic $L$-function theory.