Peels of Equilateral Triangulated Surfaces
Abstract
This paper studies the modularity problem for elliptic curves from the elementary geometry of peels of decorated equilateral triangulations. A peel is a finite developing domain together with the boundary identifications that reconstruct the triangulated surface. In genus one it recovers the universal cover and its rank-two deck lattice. For a fixed prime ℓ, the finite unramified coverings [ℓn]:E→ E form a compatible tower whose deck groups are E[ℓn]; their inverse limit is the Tate module Tℓ(E), and over Q the compatible Galois action gives the usual ℓ-adic representation. Cyclic isogenies, by contrast, give the index-q lattice neighbors that form the local model for Hecke and Brandt correspondences. For a semistable elliptic curve E/Q of prime conductor p, the missing reciprocity is the construction, without using modularity, of a nonzero class cE in the degree-zero supersingular Brandt module such that BqcE=aq(E)cE at every good prime. I make this obstruction explicit and study oriented CM packets attached to the Frobenius discriminants Dq=aq(E)2−4q. New elementary results show that Dq is a nonsquare modulo p exactly when the residual Frobenius polynomial is irreducible, that any residual image containing such an element gives a positive-density set of primes for which p is inert in the quadratic field determined by Dq, and that the corresponding discriminants are necessarily unbounded.
Related
- Combinatorial Calabi flows on infinitely triangulated surfaces
- Combinatorial Yamabe flow on infinitely triangulated hyperbolic surfaces
- Bernstein Constraint Complexes for Multivariate Splines on Triangulated Surfaces
- Equilateral dimension of the Heisenberg group fHn
- Petty's equilateral-set conjecture in dimension five
BibTeX
@misc{verjovsky2026peels,
title = {{Peels of Equilateral Triangulated Surfaces}},
author = {Alberto Verjovsky},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26196},
eprint = {2609.26196},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26196},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Peels of Equilateral Triangulated Surfaces
AU - Alberto Verjovsky
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.26196
UR - https://arxiv.org/abs/2609.26196
AB - This paper studies the modularity problem for elliptic curves from the elementary geometry of peels of decorated equilateral triangulations. A peel is a finite developing domain together with the boundary identifications that reconstruct the triangulated surface. In genus one it recovers the universal cover and its rank-two deck lattice. For a fixed prime ℓ, the finite unramified coverings [ℓ^n]:E→ E form a compatible tower whose deck groups are E[ℓ^n]; their inverse limit is the Tate module T_ℓ(E), and over Q the compatible Galois action gives the usual ℓ-adic representation. Cyclic isogenies, by contrast, give the index-q lattice neighbors that form the local model for Hecke and Brandt correspondences. For a semistable elliptic curve E/Q of prime conductor p, the missing reciprocity is the construction, without using modularity, of a nonzero class c_E in the degree-zero supersingular Brandt module such that B_qc_E=a_q(E)c_E at every good prime. I make this obstruction explicit and study oriented CM packets attached to the Frobenius discriminants D_q=a_q(E)^2−4q. New elementary results show that D_q is a nonsquare modulo p exactly when the residual Frobenius polynomial is irreducible, that any residual image containing such an element gives a positive-density set of primes for which p is inert in the quadratic field determined by D_q, and that the corresponding discriminants are necessarily unbounded.
ER -