Hasse’s theorem - Revision history

Rational Point: Go full Nazi, eliminate all use of math extension

2025-02-12T00:01:33Z

Go full Nazi, eliminate all use of math extension

← Older revision		Revision as of 00:01, 12 February 2025
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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

	:~~<math>~~q+1~~\pm2\sqrt~~ q~~</math>~~,		:''q'' + 1 &pm; 2√''q'',

	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.		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.
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	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'':

	:~~<math>~~q+1~~\pm2g\sqrt~~ q~~</math>~~.		:''q'' + 1 &pm; 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>.

Rational Point: make it more clear

2025-02-11T23:55:45Z

make it more clear

← Older revision		Revision as of 23:55, 11 February 2025
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	:<math>q+1\pm2\sqrt q</math>,		:<math>q+1\pm2\sqrt q</math>,

	that is to say, of all ''q''<sup>2</sup> points in GF(''q'')~~⨉GF~~(''q''), the number of them that satisfy any given elliptic curve equation always falls in this range.		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'':

Rational Point at 22:59, 11 February 2025

2025-02-11T22:59:03Z

← Older revision		Revision as of 22:59, 11 February 2025
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	that is to say, of all ''q''<sup>2</sup> points in GF(''q'')⨉GF(''q''), the number of them that satisfy any given elliptic curve equation always falls in this range.		that is to say, of all ''q''<sup>2</sup> 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'':		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'':

	:<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''' <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>.

Rational Point: Nazi

2025-01-11T15:03:06Z

Nazi

← Older revision		Revision as of 15:03, 11 January 2025
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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. https://www.digizeitschriften.de/id/243919689_0175 https://gdz.sub.uni-goettingen.de/id/PPN243919689_0175</ref>.		'''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>. Hasse joined the Nazi Party (NSDAP) the year after it was published, and served as a concentration camp guard during WWII <ref>„Hasse, Helmut“, in: Hessische Biografie https://www.lagis-hessen.de/pnd/118708961 (Stand: 14.2.2024)</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

Rational Point: ref

2025-01-10T19:45:32Z

ref

← Older revision		Revision as of 19:45, 10 January 2025
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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''' <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>.

Rational Point: masters thesis

2025-01-09T09:18:55Z

masters thesis

← Older revision		Revision as of 09:18, 9 January 2025
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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>.
	~~<br>~~https://www.digizeitschriften.de/id/243919689_0175 ~~<br>~~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
Line 12:		Line 11:
	:<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>.

Rational Point: refs

2025-01-09T08:01:22Z

refs

New page

'''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.
<br>https://www.digizeitschriften.de/id/243919689_0175 <br>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

:<math>q+1\pm2\sqrt q</math>,

that is to say, of all ''q''<sup>2</sup> 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'':

:<math>q+1\pm2g\sqrt q</math>.

This result was proved by André Weil, and is known as the '''Hasse–Weil theorem'''.