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Hirzebruch-Zagier cycles in $p$-adic families and adjoint $L$-values

Antonio Cauchi, Marc-Hubert Nicole, Giovanni Rosso

Abstract

Let $E/F$ be a quadratic extension of totally real number fields. We show that the generalized Hirzebruch-Zagier cycles arising from the associated Hilbert modular varieties can be put in $p$-adic families. As an application, using the theory of base change, we give a geometric construction of the multivariable $p$-adic adjoint $L$-function twisted by the Hecke character of $E/F$, attached to Hida families of Hilbert modular forms over $F$.

Hirzebruch-Zagier cycles in $p$-adic families and adjoint $L$-values

Abstract

Let be a quadratic extension of totally real number fields. We show that the generalized Hirzebruch-Zagier cycles arising from the associated Hilbert modular varieties can be put in -adic families. As an application, using the theory of base change, we give a geometric construction of the multivariable -adic adjoint -function twisted by the Hecke character of , attached to Hida families of Hilbert modular forms over .

Paper Structure

This paper contains 40 sections, 43 theorems, 307 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Relation to previous literature
  4. Structure of the manuscript
  5. Acknowledgements
  6. Preliminaries on algebraic representations and their local systems
  7. Groups and varieties
  8. Representations
  9. Branching laws
  10. Cohomology
  11. Preliminaries on Hilbert modular forms and their L-functions
  12. Hilbert modular forms and their differential forms
  13. Hilbert cusp forms and their differential forms
  14. The Fourier expansion
  15. The Hecke algebra
  16. ...and 25 more sections

Key Result

Theorem 1.1

There is a Big Hirzebruch--Zagier class Let $\underline{k} \in {\mathbb{Z}}^{\Sigma_E}$ be such that $k_\sigma \equiv 0$ (mod $2$) and $k_\sigma = k_{\sigma'}$ for all $\sigma \in \Phi$. Then,

Theorems & Definitions (127)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Definition 2.6
  • Lemma 2.7
  • proof
  • ...and 117 more