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Nearly higher Coleman theory and p-adic L-functions for $\mathrm{GSp}(4) \times \mathrm{GL}(2)$ and $\mathrm{GSp}(4) \times \mathrm{GL}(2) \times \mathrm{GL}(2)$

Andrew Graham, Rob Rockwood

Abstract

We construct four-variable $p$-adic $L$-functions for the spin Galois representation of a Siegel modular form of genus 2 twisted by the Galois representation of a cuspidal modular form as the modular forms vary in Coleman families. The main ingredient is the construction of a space of nearly overconvergent modular forms in the coherent cohomology of the Siegel threefold, extending the spaces of overconvergent modular forms appearing in higher Coleman theory. In addition to this, we construct $p$-adic distributions interpolating the Gan-Gross-Prasad automorphic periods for $(\mathrm{GSpin}(5), \mathrm{GSpin}(4))$ which, conditional on the local and global Gan-Gross-Prasad conjectures for this pair of groups, provides a construction of "square-root" $p$-adic $L$-functions for $\mathrm{GSp}(4) \times \mathrm{GL}(2) \times \mathrm{GL}(2)$ as the automorphic forms vary in Coleman families.

Nearly higher Coleman theory and p-adic L-functions for $\mathrm{GSp}(4) \times \mathrm{GL}(2)$ and $\mathrm{GSp}(4) \times \mathrm{GL}(2) \times \mathrm{GL}(2)$

Abstract

We construct four-variable -adic -functions for the spin Galois representation of a Siegel modular form of genus 2 twisted by the Galois representation of a cuspidal modular form as the modular forms vary in Coleman families. The main ingredient is the construction of a space of nearly overconvergent modular forms in the coherent cohomology of the Siegel threefold, extending the spaces of overconvergent modular forms appearing in higher Coleman theory. In addition to this, we construct -adic distributions interpolating the Gan-Gross-Prasad automorphic periods for which, conditional on the local and global Gan-Gross-Prasad conjectures for this pair of groups, provides a construction of "square-root" -adic -functions for as the automorphic forms vary in Coleman families.

Paper Structure

This paper contains 56 sections, 34 theorems, 223 equations.

Table of Contents

  1. Introduction
  2. Statement of main results
  3. $p$-adic $L$-functions for $\operatorname{GSp}_4 \times \operatorname{GL}_2$
  4. $p$-adic $L$-functions for $\operatorname{GSp}_4 \times \operatorname{GL}_2 \times \operatorname{GL}_2$
  5. Method of proof
  6. Acknowledgements
  7. Notation
  8. Groups and representations
  9. Algebraic groups
  10. Parabolic subgroups and Weyl groups
  11. Open orbit representatives
  12. Algebraic representation theory
  13. Algebraic nearly representations
  14. p-adic representation theory
  15. p-adic nearly representations for G
  16. ...and 41 more sections

Key Result

Theorem 1

There exists a locally analytic distribution $\mathscr{L}_p \in \mathscr{D}^{\operatorname{la}}(\mathbb{Z}_p^{\times}, \mathscr{O}_{U \times V})$ such that for all $z \colonequals (k_x, c_y) \in \Upsilon(\underline{\pi}, \underline{\sigma})$, all integers $j \in \operatorname{Crit}(\underline{\pi}_{ where: Moreover, the three-variable $p$-adic $L$-function on $U \times V$ given by $(k_x, c_y) \ma

Theorems & Definitions (120)

  • Remark 1.0.1
  • Theorem 1
  • Remark 1.1.1
  • Remark 1.1.2
  • Theorem 2
  • Theorem 3
  • Remark 1.1.3
  • Remark 2.1.1
  • Remark 2.1.2
  • Definition 2.2.3
  • ...and 110 more