Intensional semantics for comparison problems in arithmetic geometry
Abstract
This work proposes a semantic environment for arithmetic geometry in which passage between presentations carries an explicit weight. Morphism costs induce a Lawvere generalized distance by taking infima over transports, and heights satisfy the corresponding transport bounds. A labeled quantitative (2,1)-category retains invertible $2$-morphisms as coherent comparisons between transports. Its vertical groupoids admit a transport interpretation inspired by the identity types of Voevodsky's C-systems. If a height factors through the underlying object, then a vertical comparison has equal endpoints, whatever its transport cost. We formulate an intensional Szpiro inequality for elliptic packages, where the discriminant height transports along a morphism with a controlled model defect, and we relate a bound of this form to the abc conjecture. The defect remains visible between non-minimal and normalized presentations, while conductor complexity stays attached to the underlying curve. We then treat the Birch and Swinnerton-Dyer conjecture in its pre-modularity form, using strong point-count asymptotics with an explicit convergence requirement. The refined constant retains separately typed arithmetic factors. A finite differential correction makes the period independent of the chosen differential and links its normalization to model transport. Within the declared finite-descent theory, Tate--Shafarevich finiteness reduces to finitely many closed prime columns of Selmer shadows and a uniform cutoff. For y2=x3−q2x with q≡3pmod8 prime, two-isogeny descent closes the two-primary column uniformly. The statements admit expression in ACA0 relative to certified descent and Mordell--Weil data, with certified Cauchy names for the period and regulator.
Related
- One Arithmetic Gadget, Two Problems
- The Siegel-Weil formula in geometry and arithmetic
- Definite integral Albert algebras: arithmetic and finite geometry
- The Operator–Spectral Framework of ZEBTS: A Unified Structure for Geometry, Arithmetic, and the Riemann Hypothesis
- Geometry-induced formation and merging of multiple dead zones in diffusion-reaction problems
BibTeX
@misc{laniewski2026intensional,
title = {{Intensional semantics for comparison problems in arithmetic geometry}},
author = {R. Laniewski},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.13609},
eprint = {2608.13609},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.13609},
}
RIS
TY - UNPB
TI - Intensional semantics for comparison problems in arithmetic geometry
AU - R. Laniewski
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2608.13609
UR - https://arxiv.org/abs/2608.13609
AB - This work proposes a semantic environment for arithmetic geometry in which passage between presentations carries an explicit weight. Morphism costs induce a Lawvere generalized distance by taking infima over transports, and heights satisfy the corresponding transport bounds. A labeled quantitative (2,1)-category retains invertible $2$-morphisms as coherent comparisons between transports. Its vertical groupoids admit a transport interpretation inspired by the identity types of Voevodsky's C-systems. If a height factors through the underlying object, then a vertical comparison has equal endpoints, whatever its transport cost. We formulate an intensional Szpiro inequality for elliptic packages, where the discriminant height transports along a morphism with a controlled model defect, and we relate a bound of this form to the abc conjecture. The defect remains visible between non-minimal and normalized presentations, while conductor complexity stays attached to the underlying curve. We then treat the Birch and Swinnerton-Dyer conjecture in its pre-modularity form, using strong point-count asymptotics with an explicit convergence requirement. The refined constant retains separately typed arithmetic factors. A finite differential correction makes the period independent of the chosen differential and links its normalization to model transport. Within the declared finite-descent theory, Tate--Shafarevich finiteness reduces to finitely many closed prime columns of Selmer shadows and a uniform cutoff. For y^2=x^3−q^2x with q≡3pmod8 prime, two-isogeny descent closes the two-primary column uniformly. The statements admit expression in ACA_0 relative to certified descent and Mordell--Weil data, with certified Cauchy names for the period and regulator.
ER -