Hasse’s theorem: Difference between revisions
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The number of points on an elliptic curve over a [[finite field]] GF(''q'') is within the range | 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 ''q''<sup>2</sup> points | that is to say, of all ''q''<sup>2</sup> points (''x'',''y'') ∈ GF(''q'') ⨉ GF(''q''), the number of them that satisfy any given elliptic curve equation ''y''<sup>2</sup> = ''x''<sup>3</sup> + ''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'': | 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 ± 2''g''√''q''. | ||
This result was proved by André Weil, and is known as the '''Hasse–Weil theorem''' <ref>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 [https://www.ams.org/journals/bull/1949-55-05/S0002-9904-1949-09219-4/S0002-9904-1949-09219-4.pdf PDF]</ref><ref>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</ref>. | This result was proved by André Weil, and is known as the '''Hasse–Weil theorem''' <ref>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 [https://www.ams.org/journals/bull/1949-55-05/S0002-9904-1949-09219-4/S0002-9904-1949-09219-4.pdf PDF]</ref><ref>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</ref>. | ||
Latest revision as of 00:01, 12 February 2025
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 ± 2g√q.
This result was proved by André Weil, and is known as the Hasse–Weil theorem [3][4].
- ↑ 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
- ↑ „Hasse, Helmut“, in: Hessische Biografie https://www.lagis-hessen.de/pnd/118708961 (Stand: 14.2.2024)
- ↑ 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
- ↑ 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
