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Finite Analytic Characters and Tannakian Source Reconstruction: A Finite Tensor–Quiver Framework for Beyond Endoscopy

Primarily AI-generated textHuman understanding: some partsmath.NT — Number Theorymath.AG — Algebraic Geometrymath.RT — Representation Theory

Contributed by Shawn Schafer ↗

Version 1 / Oct 14, 2026 / CC BY 4.0

Abstract

This paper organizes finite tensor–quiver reconstruction for reductive source data. For a fixed reductive Zariski-closed subgroup H of a reductive complex group G and a faithful finite-dimensional representation, the standard mixed-tensor reconstruction theorem and Noetherianity imply that finitely many H-invariant tensors have common stabilizer H. Classical quiver invariant theory then implies that finitely many traces of oriented cycles separate closed orbits, including semisimple realizations. For GL₂/ℚ, we give a restricted application of Cheng’s symmetric-square limiting distribution. After a finite-level cuspidal block has been specified and a target constituent selected by an interpolating Hecke test function built from that block’s Hecke characters, ratios of trace-formula distributions recover the first two Satake power sums at auxiliary unramified places, provided Cheng’s hypotheses hold and the full selected coefficient is nonzero. This is post-selection recovery, not a method for identifying the target constituent, and applies only to the nonzero-coefficient visible subset.

Provenance statement

This paper was developed through a human-directed, substantially AI-assisted research and writing process. GPT 5.6 Luna and GPT 6 were used iteratively to support mathematical research and exploration, test algebraic identities and proof steps, search for and compare related literature, formulate and refine arguments, and draft and revise the manuscript. AI assistance was also used to edit the abstract, organize the exposition, and prepare and check the LaTeX source and compiled PDF. The author directed the research tasks, set the intended scope, and guided the development and revision of the manuscript. The mathematical checking included AI-assisted examination of theorem statements, proof dependencies, algebraic calculations, and the hypotheses and limitations of the automorphic application. Build and consistency checks were used to identify compilation, citation, reference, and layout problems. These procedures do not establish mathematical correctness or constitute independent peer review. The work has not undergone an independent human specialist audit of every proof. The author accepts responsibility for the submitted manuscript, including its claims, attribution, and any errors or gaps that may remain. This statement is intended to make the substantial role of AI tools in the research, mathematical checking, and writing process explicit.

Tools used

OpenAI
ChatGPTVersion 5.6 Luna
OpenAI
ChatGPTVersion 6

References

  1. C. Breuil, B. Conrad, F. Diamond, and R. Taylor, On the modularity of elliptic curves over Q\mathbf Q: Wild 3-adic exercises , J. Amer. Math. Soc. 14 (2001), no. 4, 843--939.
  2. L. Clozel and P. Delorme, Pseudo-coefficients et cohomologie des groupes de Lie réductifs réels , C. R. Acad. Sci. Paris Sér. I Math. 300 (1985), no. 12, 385--387.
  3. J.-P. Labesse, Pseudo-coefficients très cuspidaux et K-théorie , Math. Ann. 291 (1991), no. 4, 607--616.
  4. D. A. Vogan, Jr., Representations of Real Reductive Lie Groups , Progress in Mathematics 15, Birkhäuser, 1981.
  5. S. A. Altuğ, Beyond endoscopy via the trace formula: I. Poisson summation and isolation of special representations , Compos. Math. 151 (2015), no. 10, 1791--1820.
  6. S. A. Altuğ, Beyond endoscopy via the trace formula---III: The standard representation , J. Inst. Math. Jussieu 19 (2020), 1349--1387.
  7. Y. Cheng, Beyond endoscopy for the symmetric square representation: The simple trace formula case , arXiv:2607.25383v1 (28 July 2026).arXiv
  8. Y. Cheng, Beyond endoscopy for $\GL_2$ over Q\mathbf Q with ramification 5: cancellation theory , arXiv:2608.15553 (16 August 2026).arXiv
  9. P. P. Warade, Jacquet--Zagier treatment of the beyond endoscopy trace formula for $\GL_2$ , arXiv:2609.01990 (2 September 2026).arXiv
  10. H. Jacquet and J. A. Shalika, On Euler products and the classification of automorphic representations. I , Amer. J. Math. 103 (1981), no. 3, 499--558.
Show all 26 referencesShow fewer references
  1. H. Jacquet and J. A. Shalika, On Euler products and the classification of automorphic forms. II , Amer. J. Math. 103 (1981), no. 4, 777--815.
  2. T. Deng and D. Wu, A trace formula derivation of the functional equation for symmetric square LL-functions , arXiv:2606.15729v2 (16 July 2026).arXiv
  3. F. Diamond and J. Shurman, A First Course in Modular Forms , Graduate Texts in Mathematics 228, Springer, 2005.
  4. P. Deligne and J. S. Milne, Tannakian categories , in Hodge Cycles, Motives, and Shimura Varieties , Lecture Notes in Math. 900, Springer, 1982, 101--228.
  5. L. Le Bruyn and C. Procesi, Semisimple representations of quivers , Trans. Amer. Math. Soc. 317 (1990), 585--598.
  6. Y. Sakellaridis, Transfer operators and Hankel transforms between relative trace formulas, I: Character theory , Adv. Math. 394 (2022), 108010.
  7. Y. Sakellaridis, Transfer operators and Hankel transforms between relative trace formulas, II: Rankin--Selberg theory , Adv. Math. 394 (2022), 108039.
  8. A. Schrijver, Tensor subalgebras and First Fundamental Theorems in invariant theory , J. Algebra 319 (2008), 1305--1319.
  9. M. Harris, K.-W. Lan, R. Taylor, and J. A. Thorne, On the rigid cohomology of certain Shimura varieties , Res. Math. Sci. 3 (2016), 37.
  10. J. Newton and J. A. Thorne, Symmetric power functoriality for Hilbert modular forms , Ann. of Math. (2) 203 (2026), no. 1, 283--347.
  11. I. Varma, Local-global compatibility for regular algebraic cuspidal automorphic representations when ℓ≠p\ell\neq p , Forum Math. Sigma 12 (2024), e21.
  12. P. Scholze, On torsion in the cohomology of locally symmetric varieties , Ann. of Math. (2) 182 (2015), no. 3, 945--1066.
  13. R. Beuzart-Plessis, Y. Liu, W. Zhang, and X. Zhu, Isolation of cuspidal spectrum, with application to the Gan--Gross--Prasad conjecture , Ann. of Math. 194 (2021), no. 2, 519--584; arXiv:1912.07169.arXiv
  14. T. Finis, E. Lapid, and W. Müller, On the spectral side of Arthur's trace formula -- absolute convergence , Ann. of Math. 174 (2011), no. 1, 173--195.
  15. Y. Takanashi, A note on the Sauvageot density principle , manuscripta math. 176 (2025), 57; arXiv:2503.17958.arXiv
  16. S. W. Shin and N. Templier, On fields of rationality for automorphic representations , Compos. Math. 150 (2014), no. 12, 2003--2053.

Version history

  1. v1Submitted by Shawn SchaferInitial depositCurrentOct 14, 2026