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March 27, 20260 citationsOpen Access

Genus-Spectral Correspondence From Algebraic Curves to Operator Spectra

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TMThierry Marechal

Key Points

  • To explore the relationship between the geometric genus of algebraic curves and the spectral properties of differential operators.
  • Construct canonical operators for algebraic curves with rational coefficients.
  • Analyze the eigenvalue distribution of these operators based on the genus of the curves.
  • Investigate correlations between the genus and spectral capacity limits.
  • Genus greater than 1 implies finite spectral capacity for corresponding operators.
  • Provides proofs of Faltings-type theorems related to finiteness.
  • Establishes connections between Diophantine geometry and spectral theory.

Abstract

Universal correspondence between geometric genus of algebraic curves and spectral density bounds of associated differential operators. For curves C/ℚ with genus g, constructs canonical operators HC whose eigenvalue distribution is constrained by g. Key result: genus g > 1 (hyperbolic) forces finite spectral capacity, providing operator-theoretic proofs of Faltings-type finiteness theorems. Unifies Diophantine geometry and spectral theory. Applications to Fermat-Catalan, abc conjecture, elliptic curves, modular curves.

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Cite This Study

Thierry Marechal (2026) studied this question.

synapsesocial.com/papers/69c620ab15a0a509bde192f4https://doi.org/10.5281/zenodo.19221460
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