Short arithmetic terms express the cardinality of elliptic curves in Weierstra{\ss} normal form over finite fields

Abstract

Arithmetic terms are fixed finite compositions of additions, multiplications, subtractions, divisions with remainder and integer exponentiations. An arithmetic term in natural numbers $A$, $B$, $n$, obtained by refining the general method of the second author (arXiv:2608.22049) for elliptic curves in Weierstra{\ss} normal form, counts the solutions in $(\mathbb Z/n\mathbb Z)^2$ for every modulus $n \geq 1$, with intermediate integers of approximately $2n^5$ binary digits instead of approximately $2n^{11}$. For a prime modulus $p \geq 17$, a second construction, based on the Hasse invariant and on the trace of Frobenius, gives a term of about thirty operations, in place of the fifty-one products of generalized geometric progressions of the general method.