Hasse’s theorem: Difference between revisions
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'''Helmut Hasse’s theorem''' is in German <ref>Helmut Hasse. „Zur Theorie der abstrakten elliptischen Funktionenkörper I, II, III.“ ''Journal für die reine und angewandte Mathematik,'' Band 175, 1936. | '''Helmut Hasse’s theorem''' is in German <ref>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</ref>. | ||
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 | ||
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:<math>q+1\pm2g\sqrt q</math>. | :<math>q+1\pm2g\sqrt q</math>. | ||
This result was proved by André Weil, and is known as the '''Hasse–Weil theorem'''. | This result was proved by André Weil, and is known as the '''Hasse–Weil theorem''' <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>. | ||
Revision as of 09:18, 9 January 2025
Helmut Hasse’s theorem is in German [1].
The number of points on an elliptic curve over a finite field GF(q) is within the range
- ,
that is to say, of all q2 points in GF(q)⨉GF(q), the number of them that satisfy any given elliptic curve equation 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:
- .
This result was proved by André Weil, and is known as the Hasse–Weil theorem [2].
- ↑ 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
- ↑ 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
