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Explicit reciprocity laws for diagonal classes: higher level cases

Luca Marannino

TL;DR

The paper proposes a generalized framework for p-adic explicit reciprocity laws of diagonal classes in triple products of modular forms, allowing a $p$-ordinary form $f$ and either supercuspidal or ramified principal-series partners $g,h$ at $p$. It develops étale and syntomic formalisms, tying diagonal Galois cohomology classes to $p$-adic periods via a Bloch-Kato logarithm evaluated on a distinguished de Rham class, and provides a concrete formula relating this period to a traced product of $g$ and $h$ with a Serre derivative. The main results are fully established in weight $(2,2,2)$ and are conditional in general balanced weights due to current limitations in cohomology with semistable coefficients, with a clear path outlined through forthcoming syntomic-coefficient theories. The work connects to $p$-adic $L$-functions and diagonal-class constructions, suggesting applications to anticyclotomic Iwasawa theory and the study of diagonal interactions at inert primes, thereby enriching the arithmetic of triple product p-adic phenomena.

Abstract

We generalize the $p$-adic explicit reciprocity laws for balanced diagonal classes by Darmon-Rotger and Bertolini-Seveso-Venerucci to the case of geometric balanced triples $(f,g,h)$ of modular eigenforms where $f$ is a $p$-ordinary newform, while $g$ and $h$ are allowed to be (both) supercuspidal at $p$ or (both) ramified principal series at $p$.

Explicit reciprocity laws for diagonal classes: higher level cases

TL;DR

The paper proposes a generalized framework for p-adic explicit reciprocity laws of diagonal classes in triple products of modular forms, allowing a -ordinary form and either supercuspidal or ramified principal-series partners at . It develops étale and syntomic formalisms, tying diagonal Galois cohomology classes to -adic periods via a Bloch-Kato logarithm evaluated on a distinguished de Rham class, and provides a concrete formula relating this period to a traced product of and with a Serre derivative. The main results are fully established in weight and are conditional in general balanced weights due to current limitations in cohomology with semistable coefficients, with a clear path outlined through forthcoming syntomic-coefficient theories. The work connects to -adic -functions and diagonal-class constructions, suggesting applications to anticyclotomic Iwasawa theory and the study of diagonal interactions at inert primes, thereby enriching the arithmetic of triple product p-adic phenomena.

Abstract

We generalize the -adic explicit reciprocity laws for balanced diagonal classes by Darmon-Rotger and Bertolini-Seveso-Venerucci to the case of geometric balanced triples of modular eigenforms where is a -ordinary newform, while and are allowed to be (both) supercuspidal at or (both) ramified principal series at .
Paper Structure (24 sections, 17 theorems, 156 equations)

This paper contains 24 sections, 17 theorems, 156 equations.

Table of Contents

  1. Introduction and statement of the main results
  2. Main results
  3. Comparison with the existing literature and proof strategy
  4. An overview on possible applications
  5. Balanced diagonal classes and the étale Abel-Jacobi map
  6. De Rham cohomology of modular curves
  7. Étale cohomology of modular curves
  8. The étale Abel-Jacobi map
  9. Construction of the invariant [DET]
  10. Some p-adic Hodge theory
  11. Filtered (phi,N)-modules and Galois representations
  12. Study of the relevant filtered [(phi,N)]-modules
  13. The syntomic Abel-Jacobi map
  14. Nekovář-Nizioł syntomic cohomology
  15. Nekovář-Nizioł syntomic cohomology with coefficients
  16. ...and 9 more sections

Key Result

Theorem 1.2

Let $(f,g,h)$ be a triple of modular forms satisfying Assumption introass and the conditions ($f$-ord) and (SC) as above. If the triple $(f,g,h)$ is moreover $(F,1-T)$-convenient in the sense of Definition expconvtriple for some finite extension $F/\mathbb{Q}_p$, then is equal to This formula is fully proven if the triple of weights of $(f,g,h)$ is $(2,2,2)$ and is conditional on the expectation

Theorems & Definitions (91)

  • Theorem 1.2: cf. Theorem \ref{['explicirreclawconj']}
  • Remark 1.3
  • Remark 1.4
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.7
  • Remark 2.8
  • Definition 2.9
  • ...and 81 more