Hasse’s theorem

From Elliptic Curve Crypto

Helmut Hasse’s theorem is in German [1]. Hasse joined the Nazi Party (NSDAP) the year after it was published, and served as a concentration camp guard during WWII [2].

The number of points on an elliptic curve over a finite field GF(q) is within the range

q + 1 ± 2√q,

that is to say, of all q2 points (x,y) ∈ GF(q) ⨉ GF(q), the number of them that satisfy any given elliptic curve equation y2 = x3 + ax + b always falls in this range.

For “hyperelliptic” curves or other Abelian varieties of genus g > 1, Hasse’s theorem is still applicable when the permissible range is broadened by a factor of g:

q + 1 ± 2gq.

This result was proved by André Weil, and is known as the Hasse–Weil theorem [3][4].

  1. Helmut Hasse. „Zur Theorie der abstrakten elliptischen Funktionenkörper I, II, III.“ Journal für die reine und angewandte Mathematik, Band 175, 1936. https://www.digizeitschriften.de/id/243919689_0175 https://gdz.sub.uni-goettingen.de/id/PPN243919689_0175
  2. „Hasse, Helmut“, in: Hessische Biografie https://www.lagis-hessen.de/pnd/118708961 (Stand: 14.2.2024)
  3. André Weil “Numbers of solutions of equations in finite fields.” Bull. Amer. Math. Soc. 55 (1949), 497-508 https://www.ams.org/journals/bull/1949-55-05/S0002-9904-1949-09219-4/home.html PDF
  4. Mirjam Soeten. “Hasse's Theorem on Elliptic Curves with an extension to hyperelliptic curves of genus 2.” Master Thesis Mathematics, University of Groningen, June 24, 2013. https://fse.studenttheses.ub.rug.nl/10999/1/opzet.pdf