Short arithmetic terms for the cardinality of elliptic curves over rings of remainder classes and over finite fields
Year: 2026
Type: preprint
Abstract: Arithmetic terms are fixed finite compositions of addition, truncated subtraction, multiplication, integer division and exponentiation on natural numbers. We construct such terms for the number of affine solutions of $ y^2 = x^3 + Ax + B $. For arbitrary moduli $ n \geq 1 $, a specialization of Prunescu's general construction reduces the number of monomial contributions from fourty-nine to fifteen and the size of the packed integer from approximately $ 2n^{11} $ to approximately $ 2n^5 $ binary digits. For prime moduli $ p \geq 17 $, the Hasse invariant and the trace of Frobenius give a term of about thirty operations. For curves defined over $ \mathbb F_p $, a further term counts the points over $ \mathbb F_{p^k} $ with $ k $ as a variable. For primes $ p \geq 5 $, we also obtain arithmetic terms for the counts over $ \mathbb Z / p^k \mathbb Z $ with $ k $ variable, covering good reduction, nodes and cusps, including non-minimal equations and zero discriminant.
Source: arXiv (Cornell University)
Authors: Dumitru, Bogdan, Prunescu, Mihai
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7211905508