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Higher Hida and Coleman theories on the modular curve

George Boxer, Vincent Pilloni

TL;DR

This work extends the Hida and Coleman finite slope theories from degree 0 to degree 1 coherent cohomology of automorphic line bundles on the modular curve, constructing parallel p-adic interpolations and a p-adic Serre duality pairing between the two degrees. It develops a robust cohomological framework using finite flat correspondences, local finiteness arguments, and ordinary projectors, and then builds a p-adic theory via the Igusa tower, the ordinary locus, and the U_p/Frobenius operators, culminating in duality-compatible eigencurve constructions. The two dual eigencurves arise from dual interpolation sheaves ω^{κ^un} and ω^{2-κ^un}(-D), with a perfect Λ-pairing that respects Hecke operators T_ℓ and ℓ-adic dualities, thus giving a unified picture of higher degree p-adic modular forms on the modular curve. The results pave the way for generalizing Hida-Coleman methods to higher Shimura varieties by providing a concrete, duality-respecting framework in the simplest case of modular curves.

Abstract

We construct Hida and Coleman theories for the degree 0 and 1 cohomology of automorphic line bundles on the modular curve and we define a p-adic duality pairing between the theories in degree 0 and 1.

Higher Hida and Coleman theories on the modular curve

TL;DR

This work extends the Hida and Coleman finite slope theories from degree 0 to degree 1 coherent cohomology of automorphic line bundles on the modular curve, constructing parallel p-adic interpolations and a p-adic Serre duality pairing between the two degrees. It develops a robust cohomological framework using finite flat correspondences, local finiteness arguments, and ordinary projectors, and then builds a p-adic theory via the Igusa tower, the ordinary locus, and the U_p/Frobenius operators, culminating in duality-compatible eigencurve constructions. The two dual eigencurves arise from dual interpolation sheaves ω^{κ^un} and ω^{2-κ^un}(-D), with a perfect Λ-pairing that respects Hecke operators T_ℓ and ℓ-adic dualities, thus giving a unified picture of higher degree p-adic modular forms on the modular curve. The results pave the way for generalizing Hida-Coleman methods to higher Shimura varieties by providing a concrete, duality-respecting framework in the simplest case of modular curves.

Abstract

We construct Hida and Coleman theories for the degree 0 and 1 cohomology of automorphic line bundles on the modular curve and we define a p-adic duality pairing between the theories in degree 0 and 1.

Paper Structure

This paper contains 44 sections, 49 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Preliminaries
  4. Finite flat cohomological correspondences
  5. The functors $f_\star$, $f^\star$ and $f^!$
  6. Cohomological correspondences
  7. Restriction
  8. Local finiteness and ordinary projectors
  9. The cohomological correspondence $T_p$
  10. The cohomological correspondences $T_p$
  11. Duality
  12. Higher Hida theory
  13. The mod $p$ theory
  14. The $p$-adic theory
  15. The Igusa tower
  16. ...and 29 more sections

Key Result

Theorem 1.1

There is a Hecke operator $T_p$ acting on the cohomology groups $\mathrm{R}\Gamma(X_1, \omega^k)$, $\mathrm{R}\Gamma_c({X}_1^{\mathop{\mathrm{ord}}\nolimits}, \omega^k)$, and $\mathrm{R}\Gamma({X}_1^{\mathop{\mathrm{ord}}\nolimits}, \omega^k)$, and an associated ordinary projector $e(T_p)$. Moreover and

Theorems & Definitions (105)

  • Theorem 1.1: Hida's control theorem
  • Theorem 1.2
  • Theorem 1.3: Coleman's classicality theorem
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 95 more