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Non-reductive cycles and twisted arithmetic transfers for Shimura curves

Zhiyu Zhang

TL;DR

The note provides a largely expository account of recent progress on constructing new cycles on Shimura varieties and Rapoport–Zink spaces, including the twisted arithmetic fundamental lemma ($ ext{TAFL}$) and arithmetic transfers ($ ext{TAT}$), within the twisted GGP framework. It couples global methods (Hecke translations, geometric theta series, and Fourier–Jacobi cycles) with local analyses on Rapoport–Zink spaces (mirabolic cycles) and derives a seven-step program linking derived orbital integrals to arithmetic intersections via holomorphic generating functions and modularity. The work emphasizes non-reductive structures (parabolic, mirabolic) and uses global–local trace formula techniques to establish the arithmetic analogs of the relative Langlands program, with applications to twisted Asai motives and $p$-adic Beilinson–Bloch–Kato-type conjectures. The approach demonstrates how archimedean and non-archimedean analyses, together with modularity phenomena, yield a robust framework for proving refined transfer identities and height pairings in the arithmetic Langlands landscape.

Abstract

In this largely expository note, we explain some recent progress on new cycles on Shimura varieties and Rapoport-Zink spaces, (twisted) arithmetic fundamental lemma, and arithmetic analogs of relative Langlands program. We explain related formulations of arithmetic twisted Gan-Gross-Prasad conjecture, the proof of twisted AFL and certain arithmetic transfers.

Non-reductive cycles and twisted arithmetic transfers for Shimura curves

TL;DR

The note provides a largely expository account of recent progress on constructing new cycles on Shimura varieties and Rapoport–Zink spaces, including the twisted arithmetic fundamental lemma () and arithmetic transfers (), within the twisted GGP framework. It couples global methods (Hecke translations, geometric theta series, and Fourier–Jacobi cycles) with local analyses on Rapoport–Zink spaces (mirabolic cycles) and derives a seven-step program linking derived orbital integrals to arithmetic intersections via holomorphic generating functions and modularity. The work emphasizes non-reductive structures (parabolic, mirabolic) and uses global–local trace formula techniques to establish the arithmetic analogs of the relative Langlands program, with applications to twisted Asai motives and -adic Beilinson–Bloch–Kato-type conjectures. The approach demonstrates how archimedean and non-archimedean analyses, together with modularity phenomena, yield a robust framework for proving refined transfer identities and height pairings in the arithmetic Langlands landscape.

Abstract

In this largely expository note, we explain some recent progress on new cycles on Shimura varieties and Rapoport-Zink spaces, (twisted) arithmetic fundamental lemma, and arithmetic analogs of relative Langlands program. We explain related formulations of arithmetic twisted Gan-Gross-Prasad conjecture, the proof of twisted AFL and certain arithmetic transfers.

Paper Structure

This paper contains 15 sections, 1 theorem, 57 equations.

Table of Contents

  1. L-functions and cycles on Shimura varieties
  2. Producing cycles: Hecke translations, Weil representations and mirabolic cycles
  3. Shimura varieties and cycles from Hecke translations
  4. Global cycles: geometric theta series and twisted Fourier-Jacobi cycles
  5. Local cycles: mirabolic cycles on Rapoport-Zink spaces
  6. L-functions and ATGGP conjecture via twisted Fourier-Jacobi cycles
  7. Proof of TAFL: holomorphic Fourier expansion and local-global relations
  8. Twisted arithmetic fundamental lemma (TAFL) and arithmetic transfers (TAT): formulations
  9. Step I: globalization of intersections of cycles and supports
  10. Step II: non-archimedean intersections and uniformizations
  11. Step III: globalization of orbital integrals and analytic generating functions from relative trace formulas
  12. Step IV: archimedean intersections, archimedean orbital integrals and holomorphic differences
  13. Step V: modularity of geometric theta series and (holomorphic) modification of $1$-cycles
  14. Step VI: matching totally positive Fourier coefficients by induction, simple known cases and modularity
  15. Step VII: shrinking supports and from global identities to local TAFLs

Key Result

Theorem 2.2

Assume that $n>2$. When $n=2$, we have a similar decomposition of the endoscopic part of $H^1$ and Albanese.

Theorems & Definitions (8)

  • Definition 2.1: Relative translation cycles
  • Theorem 2.2: Liu's formula Liu-Fourier-Jacobi
  • Definition 2.3
  • Conjecture 3.1: Twisted GGP conjecture TwistedGGP
  • Conjecture 3.2: Arithmetic twisted GGP conjecture Proof-Invent-cycles-TAFL
  • Definition 4.1
  • Conjecture 4.2
  • Definition 4.3