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A symmetric $p$-adic symbol for triples of modular forms

Wissam Ghantous

TL;DR

This work defines a symmetric $p$-adic triple symbol $(f,g,h)_{p}$ for triples of modular forms, derived from the Garrett-Rankin triple product $p$-adic $L$-function and realized via the $p$-adic Abel–Jacobi map. It establishes cyclic symmetry with explicit sign patterns depending on the parities of the weights and provides a robust computational framework by extending Lauder’s ordinary projection algorithm to nearly overconvergent and nonzero-slope contexts. The authors deliver practical methods to compute the symbol and related Poincaré pairings, including a Rankin–Cohen bracket-based approach to handle projections of nearly overconvergent forms and an eigenspace projection technique for general $U_p$-spectra. The work connects p-adic L-values, diagonal cycles, and Abel–Jacobi theory, with computational demonstrations that validate symmetry properties and offer tools for higher-weight forms. Overall, it advances both the theoretical symmetry structure and the computational toolkit for $p$-adic invariants of triples of modular forms, with potential applications to p-adic periods and the arithmetic of diagonal cycles.

Abstract

In 2014, Darmon and Rotger defined the Garrett-Rankin triple product $p$-adic $L$- function and related it to the image of certain diagonal cycles under the $p$-adic Abel- Jacobi map. We introduce a new $p$-adic triple symbol based on this $p$-adic $L$- function and show that it satisfies symmetry relations, when permuting the three input modular forms. We also provide computational examples illustrating this symmetry property. To do so, we extend Lauder's algorithm to allow for ordinary projections of nearly overconvergent modular forms -- not just overconvergent modular forms -- as well as certain projections over spaces of non-zero slope. Our work also gives an efficient method to calculate certain Poincaré pairings in higher weight, which may be of independent interest.

A symmetric $p$-adic symbol for triples of modular forms

TL;DR

This work defines a symmetric -adic triple symbol for triples of modular forms, derived from the Garrett-Rankin triple product -adic -function and realized via the -adic Abel–Jacobi map. It establishes cyclic symmetry with explicit sign patterns depending on the parities of the weights and provides a robust computational framework by extending Lauder’s ordinary projection algorithm to nearly overconvergent and nonzero-slope contexts. The authors deliver practical methods to compute the symbol and related Poincaré pairings, including a Rankin–Cohen bracket-based approach to handle projections of nearly overconvergent forms and an eigenspace projection technique for general -spectra. The work connects p-adic L-values, diagonal cycles, and Abel–Jacobi theory, with computational demonstrations that validate symmetry properties and offer tools for higher-weight forms. Overall, it advances both the theoretical symmetry structure and the computational toolkit for -adic invariants of triples of modular forms, with potential applications to p-adic periods and the arithmetic of diagonal cycles.

Abstract

In 2014, Darmon and Rotger defined the Garrett-Rankin triple product -adic - function and related it to the image of certain diagonal cycles under the -adic Abel- Jacobi map. We introduce a new -adic triple symbol based on this -adic - function and show that it satisfies symmetry relations, when permuting the three input modular forms. We also provide computational examples illustrating this symmetry property. To do so, we extend Lauder's algorithm to allow for ordinary projections of nearly overconvergent modular forms -- not just overconvergent modular forms -- as well as certain projections over spaces of non-zero slope. Our work also gives an efficient method to calculate certain Poincaré pairings in higher weight, which may be of independent interest.
Paper Structure (16 sections, 14 theorems, 101 equations)

This paper contains 16 sections, 14 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Modular forms
  4. The $U_p$ operator
  5. Ordinary projections of overconvergent modular forms
  6. Algorithmic methods
  7. Ordinary projections of nearly overconvergent modular forms
  8. Eigenspace projections
  9. Triples of modular forms
  10. The Garrett-Rankin triple product $p$-adic $L$-function
  11. A new $p$-adic triple symbol $(f,g,h)_{p}$
  12. Symmetry properties of $(f,g,h)_{p}$
  13. Examples
  14. Computing Poincaré pairings
  15. Symmetry relations for even weights
  16. ...and 1 more sections

Key Result

Proposition 2.1

The space of overconvergent modular forms of weight $k \ge 2$, growth condition $r \in B$, $\text{ord}_p(r)\not =0$, and level $\Gamma_1(N)$ is given by where by $\lim_{i \to \infty} b_i=0$, we mean that the $q$-expansion of $b_i$ is more and more divisible by $p$ as $i$ goes to infinity.

Theorems & Definitions (37)

  • Proposition 2.1
  • Remark 2.2
  • Theorem 2.3: Proposition 4.3, Col95
  • Proposition 2.4: Remark 3.2.2 in Urb14
  • Proposition 2.5: Lemma 3.3.4 in Urb14
  • Theorem 2.6: Lemma 2.7 in DR14
  • Remark 2.7
  • Remark 2.8
  • Theorem 3.1
  • Corollary 3.2
  • ...and 27 more