HEXAGONbeta

Finite Projective Source Geometry for GL2: Abel Resolvents and Central-Jet Degree Laws

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

Contributed by Shawn Schafer ↗

Version 1 / Oct 15, 2026 / CC BY 4.0

Abstract

Starting from the two trace functions p1 = Tr(A) and p2 = Tr(A^2) for semisimple A in GL2(C), we develop the associated projective source geometry. The coordinate y = t^2/d - 1 identifies the adjoint character, and normalized Haar measure on a specified closed subgroup H of SO(3) pushes forward to a probability measure nu_H. We describe this measure through harmonic multiplicities, moments, and its Stieltjes transform, and prove that it determines H up to conjugacy within the closed-subgroup class of SO(3). A positive Abel kernel resums the projective character tower and gives a resolvent formula whose endpoint limit recovers the mass of the identity atom. We give explicit Abel transforms for the finite rotation groups and positive-dimensional benchmark sources, and derive a finite-image Frobenius-prime Abel limit from Chebotarev. For central-jet interpolation, we prove two exact degree laws: the cyclic-versus-torus minimum degree is max(n, M+1), while the normalized tetrahedral detector against S4 and A5 has minimum degree 3 for M = 1, 2 and M+2 for M >= 3. We also compute the integral image of the tetrahedral averaging functional, obtaining 2Z for every M >= 2. All results are stated with their hypotheses. The paper does not claim a general infinite-spectrum Beyond Endoscopy trace formula or transfer of central-jet vanishing to orbital kernels.

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. S. Schafer, Finite Analytic Characters and Tannakian Source Reconstruction: A Finite Tensor--Quiver Framework for Beyond Endoscopy , Hexagon preprint, identifier hexagon:2610.00245v1 , version 1, October 14, 2026. https://hexagonmath.org/2610.00245v1.hexagon
  2. C. de Boor and A. Ron, The least solution for the polynomial interpolation problem , Math. Z. 210 (1992), no. 3, 347--378. DOI: 10.1007/BF02571803 (https://doi.org/10.1007/BF02571803).DOI
  3. T. Sauer, Polynomial interpolation of minimal degree , Numer. Math. 78 (1997), no. 1, 59--85. DOI: 10.1007/s002110050304 (https://doi.org/10.1007/s002110050304).DOI
  4. M. Gasca and T. Sauer, On bivariate Hermite interpolation with minimal degree polynomials , SIAM J. Numer. Anal. 37 (2000), no. 3, 772--798. DOI: 10.1137/S0036142998338873 (https://doi.org/10.1137/S0036142998338873).DOI
  5. C. A. Micchelli and Q. Sun, Interpolating filters with prescribed zeros and their refinable functions , Commun. Pure Appl. Anal. 6 (2007), no. 3, 789--808. DOI: 10.3934/cpaa.2007.6.789 (https://doi.org/10.3934/cpaa.2007.6.789).DOI
  6. S. Nozawa and M. Uno, On sharp characters of rank 2 with rational values , J. Algebra 286 (2005), no. 2, 325--340. DOI: 10.1016/j.jalgebra.2004.12.018 (https://doi.org/10.1016/j.jalgebra.2004.12.018).DOI
  7. J. A. Wolf, Spaces of Constant Curvature , 6th ed., AMS Chelsea Publishing, Providence, RI, 2011. DOI: 10.1090/chel/372 (https://doi.org/10.1090/chel/372).DOI
  8. Jinpeng An, Jiu-Kang Yu, and Jun Yu, On the dimension datum of a subgroup and its application to isospectral manifolds , J. Differential Geom. 94 (2013), no. 1, 59--85. DOI: 10.4310/JDG/1361889061 (https://doi.org/10.4310/JDG/1361889061).DOI
  9. J. Yu, A compactness result for dimension datum , Int. Math. Res. Not. IMRN 2015, no. 19, 9438--9449. DOI: 10.1093/imrn/rnu229 (https://doi.org/10.1093/imrn/rnu229).DOI
  10. M. Larsen, Rigidity in the invariant theory of compact Lie groups , Int. Math. Res. Not. IMRN 2015, no. 18, 8357--8371. DOI: 10.1093/imrn/rnu196 (https://doi.org/10.1093/imrn/rnu196).DOI
Show all 15 referencesShow fewer references
  1. G. Plonka and M. Tasche, Prony methods for recovery of structured functions , GAMM-Mitteilungen 37 (2014), no. 2, 239--258. DOI: 10.1002/gamm.201410011 (https://doi.org/10.1002/gamm.201410011).DOI
  2. R. P. Langlands, Beyond endoscopy , in Contributions to Automorphic Forms, Geometry, and Number Theory , Johns Hopkins Univ. Press, 2004, 611--697. The author-preferred version is available at https://publications.ias.edu/node/91.link
  3. A. Beauville, Finite Subgroups of PGL2(K)\mathrm{PGL}_2(K) , Contemporary Mathematics 522, American Mathematical Society, Providence, RI, 2010, 23--29. https://math.univ-cotedazur.fr/~beauvill/pubs/PGL%282%29.pdf.link
  4. J.-P. Serre, Quelques applications du théorème de densité de Chebotarev , Publ. Math. IHÉS 54 (1981), 123--201. DOI: 10.1007/BF02698692 (https://doi.org/10.1007/BF02698692).DOI
  5. P. P. Warade, Jacquet--Zagier treatment of the beyond endoscopy trace formula for GL2\mathrm{GL}_2 , arXiv:2609.01990, version 1 (2026). https://arxiv.org/abs/2609.01990v1.arXiv

Version history

  1. v1Submitted by Shawn SchaferInitial depositCurrentOct 15, 2026