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Evaluation of Bianchi Rigid Meromorphic Cocycles at Big ATR Points

Lennart Gehrmann, Xavier Guitart, Marc Masdeu

TL;DR

This work delivers a practical framework to evaluate Bianchi rigid meromorphic cocycles at big ATR points, providing the first numerical confirmation of their algebraicity and revealing patterns analogous to Borcherds products at big CM points. By developing the spinor-norm and field-of-definition theory, extending rigid Meromorphic cocycle constructions to multiple cusps, and implementing an overconvergent, p-adic algorithm, the authors compute and certify algebraicity of values at big ATR points for K = ℚ(i). The results extend known evidence from small RM/CM points, demonstrate instances where values lie in nontrivial algebraic extensions of ℚ(i), and reveal BY-like divisibility phenomena in the primes appearing in norms. Overall, the paper strengthens the conjectural reciprocity framework for rigid meromorphic cocycles and provides a robust computational toolkit for exploring their arithmetic at ATR-type special points.

Abstract

We develop the tools required to effectively evaluate the Bianchi rigid meromorphic cocycles introduced by Darmon-Gehrmann-Lipnowski at big ATR points, and use them to obtain the first numerical verification of the conjectured algebraicity of these special values. Moreover, our computations suggest that these special values exhibit behaviour analogous to that of the special values of Borcherds products on Hilbert modular surfaces at big CM points.

Evaluation of Bianchi Rigid Meromorphic Cocycles at Big ATR Points

TL;DR

This work delivers a practical framework to evaluate Bianchi rigid meromorphic cocycles at big ATR points, providing the first numerical confirmation of their algebraicity and revealing patterns analogous to Borcherds products at big CM points. By developing the spinor-norm and field-of-definition theory, extending rigid Meromorphic cocycle constructions to multiple cusps, and implementing an overconvergent, p-adic algorithm, the authors compute and certify algebraicity of values at big ATR points for K = ℚ(i). The results extend known evidence from small RM/CM points, demonstrate instances where values lie in nontrivial algebraic extensions of ℚ(i), and reveal BY-like divisibility phenomena in the primes appearing in norms. Overall, the paper strengthens the conjectural reciprocity framework for rigid meromorphic cocycles and provides a robust computational toolkit for exploring their arithmetic at ATR-type special points.

Abstract

We develop the tools required to effectively evaluate the Bianchi rigid meromorphic cocycles introduced by Darmon-Gehrmann-Lipnowski at big ATR points, and use them to obtain the first numerical verification of the conjectured algebraicity of these special values. Moreover, our computations suggest that these special values exhibit behaviour analogous to that of the special values of Borcherds products on Hilbert modular surfaces at big CM points.

Paper Structure

This paper contains 17 sections, 6 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Spinor norms and the field of definition
  3. The spinor norm
  4. The field of definition
  5. Local spinor norms
  6. Proof of Proposition \ref{['fieldofdefprop']}
  7. Rigid meromorphic cocycles for the Gaussian rationals
  8. Symmetric spaces
  9. Modular symbols
  10. Convergence criterion
  11. Overconvergent algorithm
  12. Evaluation at special points
  13. Evaluation at small RM and CM points
  14. Evaluation at ATR points
  15. Algebraicity conjecture
  16. ...and 2 more sections

Key Result

Proposition 2.3

The field of definition $E_{\Phi_\chi}$ of $\Phi_\chi$ is equal to $\mathbb{Q}$ unless $\chi$ is non-trivial, $K=K_\chi$, and $p$ is split in $K$, in which case $E_{\Phi_\chi}=K_\chi$.

Theorems & Definitions (15)

  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • Lemma 3.1
  • ...and 5 more