Conductor - Revision history

Rational Point: good and bad reduction: needs another article

2025-01-08T07:44:37Z

good and bad reduction: needs another article

← Older revision		Revision as of 07:44, 8 January 2025
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	with the '''exponent''' ''f''<sub>''p''</sub> of each prime ''p'' given by		with the '''exponent''' ''f''<sub>''p''</sub> of each prime ''p'' given by

	:<math>f_p=0</math> if ''E'' has ~~“[[good reduction]]”~~ at ''p'';<br><math>f_p=1</math> if ''E'' has ~~“[[multiplicative reduction]]”~~ at ''p'';<br><math>f_p=2</math> if ''E'' has ~~“[[additive reduction]]”~~ at ''p'' > 3;<br><math>f_p=2+\delta_p</math> if ''E'' has “additive reduction” at ''p'' = 2 or 3;		:<math>f_p=0</math> if ''E'' has “good reduction” at ''p'';<br><math>f_p=1</math> if ''E'' has “multiplicative reduction” at ''p'';<br><math>f_p=2</math> if ''E'' has “additive reduction” at ''p'' > 3;<br><math>f_p=2+\delta_p</math> if ''E'' has “additive reduction” at ''p'' = 2 or 3;

	where <math>\delta_p</math> is some <u>measure of “wild ramification” in the action of the inertia group</u> on <math>T_\ell(E)</math> for which Silverman <ref>Joseph H. Silverman. ''The Arithmetic of Elliptic Curves.'' Springer-Verlag, New York, 1986, corrected 2nd printing, p. 361.</ref> refers to Serre and Tate <ref>Jean-Pierre Serre and John Tate. “Good reduction of Abelian varieties.” ''The Annals of Mathematics,'' 2nd series, vol. 88, no. 3, Nov 1968, pp. 492-517.</ref> and Ogg <ref>A. P. Ogg. “Elliptic curves and wild ramification.” ''American Journal of Mathematics,'' vol. 89, no. 1, Jan 1967, pp. 1-21.</ref> who in turn rely on the minimal models of Néron <ref>André Néron. «Modèles ''p''-minimaux des variétés abéliennes.» ''Séminaire Bourbaki,'' vol. 7, no. 227, 1962, 16 pp. http://www.numdam.org/item/?id=SB_1961-1962__7__65_0</ref><ref>André Néron. «Modèles minimaux des variétés abéliennes sur les corps locaux et globaux.» ''Publications Mathématiques de l'IHÉS'', vol. 21 (1964), pp. 5-128. http://www.numdam.org/item/PMIHES_1964__21__5_0/ </ref> among other “monstrous moonshine” literature of the period for a “precise” definition.		where <math>\delta_p</math> is some <u>measure of “wild ramification” in the action of the inertia group</u> on <math>T_\ell(E)</math> for which Silverman <ref>Joseph H. Silverman. ''The Arithmetic of Elliptic Curves.'' Springer-Verlag, New York, 1986, corrected 2nd printing, p. 361.</ref> refers to Serre and Tate <ref>Jean-Pierre Serre and John Tate. “Good reduction of Abelian varieties.” ''The Annals of Mathematics,'' 2nd series, vol. 88, no. 3, Nov 1968, pp. 492-517.</ref> and Ogg <ref>A. P. Ogg. “Elliptic curves and wild ramification.” ''American Journal of Mathematics,'' vol. 89, no. 1, Jan 1967, pp. 1-21.</ref> who in turn rely on the minimal models of Néron <ref>André Néron. «Modèles ''p''-minimaux des variétés abéliennes.» ''Séminaire Bourbaki,'' vol. 7, no. 227, 1962, 16 pp. http://www.numdam.org/item/?id=SB_1961-1962__7__65_0</ref><ref>André Néron. «Modèles minimaux des variétés abéliennes sur les corps locaux et globaux.» ''Publications Mathématiques de l'IHÉS'', vol. 21 (1964), pp. 5-128. http://www.numdam.org/item/PMIHES_1964__21__5_0/ </ref> among other “monstrous moonshine” literature of the period for a “precise” definition.
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	[[Image:Conductor.png]]		[[Image:Conductor.png]]

	Only a finite number of ~~“[[bad reduction]]s”~~ are possible on any one elliptic curve ''E''. A particular algorithm of Tate <ref>J. Tate, Bryan J. Birch and Willem Kuyk. “Algorithm for determining the type of a singular fiber in an elliptic pencil.” ''Modular Functions of One Variable IV'', Springer, 1975. pp. 33-52.</ref> is supposed to find the minimal [[Weierstraß normal form\|Weierstraß form]] of the elliptic curve and calculate ''f''<sub>2</sub> and ''f''<sub>3</sub> for the last few special cases of bad reduction with wild ramification, in order to fully complete the definition of the '''conductor''', assuming an impartial outsider can ascertain exactly what is meant by vague terms such as ~~“good” or “bad~~ reduction.“		Only a finite number of “bad reductions” are possible on any one elliptic curve ''E''. A particular algorithm of Tate <ref>J. Tate, Bryan J. Birch and Willem Kuyk. “Algorithm for determining the type of a singular fiber in an elliptic pencil.” ''Modular Functions of One Variable IV'', Springer, 1975. pp. 33-52.</ref> is supposed to find the minimal [[Weierstraß normal form\|Weierstraß form]] of the elliptic curve and calculate ''f''<sub>2</sub> and ''f''<sub>3</sub> for the last few special cases of bad reduction with wild ramification, in order to fully complete the definition of the '''conductor''', assuming an impartial outsider can ascertain exactly what is meant by vague terms such as “[[good and bad reduction]].“

	This all needs to be simplified and brought down from any excessive abstraction. Cryptography is a martial art. References are inconsistent, disinformation exists, and it is unwise to accept mathematical results on face value which one does not prove oneself, or work through and satisfy oneself with the proofs. If anything is too difficult, another tack needs to be taken, a plan of attack from a different position. It is that strange feeling of invading a foreign library and opening up books one is not really welcome to.		This all needs to be simplified and brought down from any excessive abstraction. Cryptography is a martial art. References are inconsistent, disinformation exists, and it is unwise to accept mathematical results on face value which one does not prove oneself, or work through and satisfy oneself with the proofs. If anything is too difficult, another tack needs to be taken, a plan of attack from a different position. It is that strange feeling of invading a foreign library and opening up books one is not really welcome to.

Rational Point: Tate's algorithm

2025-01-07T11:56:14Z

Tate's algorithm

← Older revision		Revision as of 11:56, 7 January 2025
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	[[Image:Conductor.png]]		[[Image:Conductor.png]]

	Only a finite number of “[[bad reduction]]s” are possible on any one elliptic curve ''E''.		Only a finite number of “[[bad reduction]]s” are possible on any one elliptic curve ''E''. A particular algorithm of Tate <ref>J. Tate, Bryan J. Birch and Willem Kuyk. “Algorithm for determining the type of a singular fiber in an elliptic pencil.” ''Modular Functions of One Variable IV'', Springer, 1975. pp. 33-52.</ref> is supposed to find the minimal [[Weierstraß normal form\|Weierstraß form]] of the elliptic curve and calculate ''f''<sub>2</sub> and ''f''<sub>3</sub> for the last few special cases of bad reduction with wild ramification, in order to fully complete the definition of the '''conductor''', assuming an impartial outsider can ascertain exactly what is meant by vague terms such as “good” or “bad reduction.“

	This all needs to be simplified and brought down from any excessive abstraction. Cryptography is a martial art. References are inconsistent, disinformation exists, and it is unwise to accept mathematical results on face value which one does not prove oneself, or work through and satisfy oneself with the proofs. If anything is too difficult, another tack needs to be taken, a plan of attack from a different position. It is that strange feeling of invading a foreign library and opening up books one is not really welcome to.		This all needs to be simplified and brought down from any excessive abstraction. Cryptography is a martial art. References are inconsistent, disinformation exists, and it is unwise to accept mathematical results on face value which one does not prove oneself, or work through and satisfy oneself with the proofs. If anything is too difficult, another tack needs to be taken, a plan of attack from a different position. It is that strange feeling of invading a foreign library and opening up books one is not really welcome to.

Rational Point: more minimal models

2025-01-05T16:55:25Z

more minimal models

← Older revision		Revision as of 16:55, 5 January 2025
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	:<math>f_p=0</math> if ''E'' has “[[good reduction]]” at ''p'';<br><math>f_p=1</math> if ''E'' has “[[multiplicative reduction]]” at ''p'';<br><math>f_p=2</math> if ''E'' has “[[additive reduction]]” at ''p'' > 3;<br><math>f_p=2+\delta_p</math> if ''E'' has “additive reduction” at ''p'' = 2 or 3;		:<math>f_p=0</math> if ''E'' has “[[good reduction]]” at ''p'';<br><math>f_p=1</math> if ''E'' has “[[multiplicative reduction]]” at ''p'';<br><math>f_p=2</math> if ''E'' has “[[additive reduction]]” at ''p'' > 3;<br><math>f_p=2+\delta_p</math> if ''E'' has “additive reduction” at ''p'' = 2 or 3;

	where <math>\delta_p</math> is some <u>measure of “wild ramification” in the action of the inertia group</u> on <math>T_\ell(E)</math> for which Silverman <ref>Joseph H. Silverman. ''The Arithmetic of Elliptic Curves.'' Springer-Verlag, New York, 1986, corrected 2nd printing, p. 361.</ref> refers to Serre and Tate <ref>Jean-Pierre Serre and John Tate. “Good reduction of Abelian varieties.” ''The Annals of Mathematics,'' 2nd series, vol. 88, no. 3, Nov 1968, pp. 492-517.</ref> and Ogg <ref>A. P. Ogg. “Elliptic curves and wild ramification.” ''American Journal of Mathematics,'' vol. 89, no. 1, Jan 1967, pp. 1-21.</ref> who in turn rely on the minimal models of Néron <ref>André Néron. «Modèles minimaux des variétés abéliennes sur les corps locaux et globaux.» ''Publications Mathématiques de l'IHÉS'', vol. 21 (1964), pp. 5-128. http://www.numdam.org/item/PMIHES_1964__21__5_0/ </ref> among other “monstrous moonshine” literature of the period for a “precise” definition.		where <math>\delta_p</math> is some <u>measure of “wild ramification” in the action of the inertia group</u> on <math>T_\ell(E)</math> for which Silverman <ref>Joseph H. Silverman. ''The Arithmetic of Elliptic Curves.'' Springer-Verlag, New York, 1986, corrected 2nd printing, p. 361.</ref> refers to Serre and Tate <ref>Jean-Pierre Serre and John Tate. “Good reduction of Abelian varieties.” ''The Annals of Mathematics,'' 2nd series, vol. 88, no. 3, Nov 1968, pp. 492-517.</ref> and Ogg <ref>A. P. Ogg. “Elliptic curves and wild ramification.” ''American Journal of Mathematics,'' vol. 89, no. 1, Jan 1967, pp. 1-21.</ref> who in turn rely on the minimal models of Néron <ref>André Néron. «Modèles ''p''-minimaux des variétés abéliennes.» ''Séminaire Bourbaki,'' vol. 7, no. 227, 1962, 16 pp. http://www.numdam.org/item/?id=SB_1961-1962__7__65_0</ref><ref>André Néron. «Modèles minimaux des variétés abéliennes sur les corps locaux et globaux.» ''Publications Mathématiques de l'IHÉS'', vol. 21 (1964), pp. 5-128. http://www.numdam.org/item/PMIHES_1964__21__5_0/ </ref> among other “monstrous moonshine” literature of the period for a “precise” definition.

	[[Image:Conductor.png]]		[[Image:Conductor.png]]

Rational Point: soft math

2025-01-05T09:05:38Z

soft math

← Older revision		Revision as of 09:05, 5 January 2025
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	[[Category:Properties]]		[[Category:Properties]]
			:'' '''Caveat emptor: ''' This is an incomplete definition which has never been fully elicited or nailed down outside of a certain highly insular “[[monstrous moonshine]] groupie” community of “[[soft math]]” researchers. The working goal here would be to develop a simple, self-contained and rigorous definition suitable for outside work. ''

	The '''conductor''' <ref>PlanetMath: conductor of an elliptic curve. https://planetmath.org/conductorofanellipticcurve</ref><ref>Wikipedia: Conductor of an elliptic curve. https://en.wikipedia.org/wiki/Conductor_of_an_elliptic_curve		The '''conductor''' <ref>PlanetMath: conductor of an elliptic curve. https://planetmath.org/conductorofanellipticcurve</ref><ref>Wikipedia: Conductor of an elliptic curve. https://en.wikipedia.org/wiki/Conductor_of_an_elliptic_curve
	</ref><ref>LMFDB: Knowledge → ec → Conductor of an elliptic curve (reviewed). https://www.lmfdb.org/knowledge/show/ec.conductor</ref><ref>Silverman, Joseph H., and Brumer, Armand. "The number of elliptic curves over Q with conductor N.." Manuscripta mathematica 91.1 (1996): 95-102. http://eudml.org/doc/156217.</ref> of an elliptic curve ''E'' over a field ''K'' is an integer, (or an ~~“ideal”~~ of the ring of integers in any algebraic field,) defined by its prime factorization:		</ref><ref>LMFDB: Knowledge → ec → Conductor of an elliptic curve (reviewed). https://www.lmfdb.org/knowledge/show/ec.conductor</ref><ref>Silverman, Joseph H., and Brumer, Armand. "The number of elliptic curves over Q with conductor N.." Manuscripta mathematica 91.1 (1996): 95-102. http://eudml.org/doc/156217.</ref> of an elliptic curve ''E'' over a field ''K'' is an integer, (or an “[[ideal]]” of the [[ring of integers]] in any algebraic field,) defined by its prime factorization:

	:<math>N=\prod_{\mathrm{all\;primes\;} p}p^{f_p}</math>		:<math>N=\prod_{\mathrm{all\;primes\;} p}p^{f_p}</math>

	~~where~~ the '''exponent''' ''f''<sub>''p''</sub> of each prime ''p'' is given by		with the '''exponent''' ''f''<sub>''p''</sub> of each prime ''p'' given by

	:<math>f_p=0</math> if ''E'' has ~~“good”~~ reduction at ''p'';<br><math>f_p=1</math> if ''E'' has ~~“multiplicative”~~ reduction at ''p'';<br><math>f_p=2</math> if ''E'' has ~~“additive”~~ reduction at ''p'' > 3;<br><math>f_p=2+\delta_p</math> if ''E'' has ~~“additive” reduction~~ at ''p'' = 2 or 3;		:<math>f_p=0</math> if ''E'' has “[[good reduction]]” at ''p'';<br><math>f_p=1</math> if ''E'' has “[[multiplicative reduction]]” at ''p'';<br><math>f_p=2</math> if ''E'' has “[[additive reduction]]” at ''p'' > 3;<br><math>f_p=2+\delta_p</math> if ''E'' has “additive reduction” at ''p'' = 2 or 3;

	where <math>\delta_p</math> is some <u>measure of “wild ramification” in the action of the inertia group</u> on <math>T_\ell(E)</math> for which Silverman <ref>Joseph H. Silverman. ''The Arithmetic of Elliptic Curves.'' Springer-Verlag, New York, 1986, corrected 2nd printing, p. 361.</ref> refers to Serre and Tate <ref>Jean-Pierre Serre and John Tate. “Good reduction of Abelian varieties.” ''The Annals of Mathematics,'' 2nd series, vol. 88, no. 3, Nov 1968, pp. 492-517.</ref> and Ogg <ref>A. P. Ogg. “Elliptic curves and wild ramification.” ''American Journal of Mathematics,'' vol. 89, no. 1, Jan 1967, pp. 1-21.</ref> who in turn rely on the minimal models of Néron <ref>André Néron. «Modèles minimaux des variétés abéliennes sur les corps locaux et globaux.» ''Publications Mathématiques de l'IHÉS'', vol. 21 (1964), pp. 5-128. http://www.numdam.org/item/PMIHES_1964__21__5_0/ </ref> among other “monstrous moonshine” literature of the period for a “precise” definition.		where <math>\delta_p</math> is some <u>measure of “wild ramification” in the action of the inertia group</u> on <math>T_\ell(E)</math> for which Silverman <ref>Joseph H. Silverman. ''The Arithmetic of Elliptic Curves.'' Springer-Verlag, New York, 1986, corrected 2nd printing, p. 361.</ref> refers to Serre and Tate <ref>Jean-Pierre Serre and John Tate. “Good reduction of Abelian varieties.” ''The Annals of Mathematics,'' 2nd series, vol. 88, no. 3, Nov 1968, pp. 492-517.</ref> and Ogg <ref>A. P. Ogg. “Elliptic curves and wild ramification.” ''American Journal of Mathematics,'' vol. 89, no. 1, Jan 1967, pp. 1-21.</ref> who in turn rely on the minimal models of Néron <ref>André Néron. «Modèles minimaux des variétés abéliennes sur les corps locaux et globaux.» ''Publications Mathématiques de l'IHÉS'', vol. 21 (1964), pp. 5-128. http://www.numdam.org/item/PMIHES_1964__21__5_0/ </ref> among other “monstrous moonshine” literature of the period for a “precise” definition.
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	[[Image:Conductor.png]]		[[Image:Conductor.png]]

	Only a finite number of ~~“bad” reductions~~ are possible on an elliptic curve ''E''.		Only a finite number of “[[bad reduction]]s” are possible on any one elliptic curve ''E''.

	This all needs to be simplified and brought down from any excessive abstraction. Cryptography is a martial art. References are inconsistent, disinformation exists, and it is unwise to accept mathematical results on face value which one does not prove oneself, or work through and satisfy oneself with the proofs. If anything is too difficult, another tack needs to be taken, a plan of attack from a different position. It is that strange feeling of invading a foreign library and opening up books one is not really welcome to.		This all needs to be simplified and brought down from any excessive abstraction. Cryptography is a martial art. References are inconsistent, disinformation exists, and it is unwise to accept mathematical results on face value which one does not prove oneself, or work through and satisfy oneself with the proofs. If anything is too difficult, another tack needs to be taken, a plan of attack from a different position. It is that strange feeling of invading a foreign library and opening up books one is not really welcome to.

Rational Point: monstrous moonshine

2025-01-05T04:21:51Z

monstrous moonshine

← Older revision		Revision as of 04:21, 5 January 2025
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	:<math>f_p=0</math> if ''E'' has “good” reduction at ''p'';<br><math>f_p=1</math> if ''E'' has “multiplicative” reduction at ''p'';<br><math>f_p=2</math> if ''E'' has “additive” reduction at ''p'' > 3;<br><math>f_p=2+\delta_p</math> if ''E'' has “additive” reduction at ''p'' = 2 or 3;		:<math>f_p=0</math> if ''E'' has “good” reduction at ''p'';<br><math>f_p=1</math> if ''E'' has “multiplicative” reduction at ''p'';<br><math>f_p=2</math> if ''E'' has “additive” reduction at ''p'' > 3;<br><math>f_p=2+\delta_p</math> if ''E'' has “additive” reduction at ''p'' = 2 or 3;

	where <math>\delta_p</math> is some <u>measure of “wild ramification” in the action of the inertia group</u> on <math>T_\ell(E)</math> for which Silverman <ref>Joseph H. Silverman. ''The Arithmetic of Elliptic Curves.'' Springer-Verlag, New York, 1986, corrected 2nd printing, p. 361.</ref> refers to Serre and Tate <ref>Jean-Pierre Serre and John Tate. “Good reduction of Abelian varieties.” ''The Annals of Mathematics,'' 2nd series, vol. 88, no. 3, Nov 1968, pp. 492-517.</ref> and Ogg <ref>A. P. Ogg. “Elliptic curves and wild ramification.” ''American Journal of Mathematics,'' vol. 89, no. 1, Jan 1967, pp. 1-21.</ref> who in turn rely on the minimal models of Néron <ref>André Néron. «Modèles minimaux des variétés abéliennes sur les corps locaux et globaux.» ''Publications Mathématiques de l'IHÉS'', vol. 21 (1964), pp. 5-128. http://www.numdam.org/item/PMIHES_1964__21__5_0/ </ref> for a ~~precise~~ definition.		where <math>\delta_p</math> is some <u>measure of “wild ramification” in the action of the inertia group</u> on <math>T_\ell(E)</math> for which Silverman <ref>Joseph H. Silverman. ''The Arithmetic of Elliptic Curves.'' Springer-Verlag, New York, 1986, corrected 2nd printing, p. 361.</ref> refers to Serre and Tate <ref>Jean-Pierre Serre and John Tate. “Good reduction of Abelian varieties.” ''The Annals of Mathematics,'' 2nd series, vol. 88, no. 3, Nov 1968, pp. 492-517.</ref> and Ogg <ref>A. P. Ogg. “Elliptic curves and wild ramification.” ''American Journal of Mathematics,'' vol. 89, no. 1, Jan 1967, pp. 1-21.</ref> who in turn rely on the minimal models of Néron <ref>André Néron. «Modèles minimaux des variétés abéliennes sur les corps locaux et globaux.» ''Publications Mathématiques de l'IHÉS'', vol. 21 (1964), pp. 5-128. http://www.numdam.org/item/PMIHES_1964__21__5_0/ </ref> among other “monstrous moonshine” literature of the period for a “precise” definition.

	[[Image:Conductor.png]]		[[Image:Conductor.png]]

Rational Point: another primary reference

2025-01-05T04:09:30Z

another primary reference

← Older revision		Revision as of 04:09, 5 January 2025
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	:<math>f_p=0</math> if ''E'' has “good” reduction at ''p'';<br><math>f_p=1</math> if ''E'' has “multiplicative” reduction at ''p'';<br><math>f_p=2</math> if ''E'' has “additive” reduction at ''p'' > 3;<br><math>f_p=2+\delta_p</math> if ''E'' has “additive” reduction at ''p'' = 2 or 3;		:<math>f_p=0</math> if ''E'' has “good” reduction at ''p'';<br><math>f_p=1</math> if ''E'' has “multiplicative” reduction at ''p'';<br><math>f_p=2</math> if ''E'' has “additive” reduction at ''p'' > 3;<br><math>f_p=2+\delta_p</math> if ''E'' has “additive” reduction at ''p'' = 2 or 3;

	where <math>\delta_p</math> is some <u>measure of “wild ramification” in the action of the inertia group</u> on <math>T_\ell(E)</math> for which Silverman <ref>Joseph H. Silverman. ''The Arithmetic of Elliptic Curves.'' Springer-Verlag, New York, 1986, corrected 2nd printing, p. 361.</ref> refers to Serre and Tate <ref>Jean-Pierre Serre and John Tate. “Good reduction of Abelian varieties.” ''The Annals of Mathematics,'' 2nd series, vol. 88, no. 3, Nov 1968, pp. 492-517.</ref> and Ogg <ref>A. P. Ogg. “Elliptic curves and wild ramification.” ''American Journal of Mathematics,'' vol. 89, no. 1, Jan 1967, pp. 1-21.</ref> on the precise definition.		where <math>\delta_p</math> is some <u>measure of “wild ramification” in the action of the inertia group</u> on <math>T_\ell(E)</math> for which Silverman <ref>Joseph H. Silverman. ''The Arithmetic of Elliptic Curves.'' Springer-Verlag, New York, 1986, corrected 2nd printing, p. 361.</ref> refers to Serre and Tate <ref>Jean-Pierre Serre and John Tate. “Good reduction of Abelian varieties.” ''The Annals of Mathematics,'' 2nd series, vol. 88, no. 3, Nov 1968, pp. 492-517.</ref> and Ogg <ref>A. P. Ogg. “Elliptic curves and wild ramification.” ''American Journal of Mathematics,'' vol. 89, no. 1, Jan 1967, pp. 1-21.</ref> who in turn rely on the minimal models of Néron <ref>André Néron. «Modèles minimaux des variétés abéliennes sur les corps locaux et globaux.» ''Publications Mathématiques de l'IHÉS'', vol. 21 (1964), pp. 5-128. http://www.numdam.org/item/PMIHES_1964__21__5_0/ </ref> for a precise definition.

	[[Image:Conductor.png]]		[[Image:Conductor.png]]

Rational Point: fix graphic

2025-01-04T23:46:33Z

fix graphic

← Older revision		Revision as of 23:46, 4 January 2025
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	where <math>\delta_p</math> is some <u>measure of “wild ramification” in the action of the inertia group</u> on <math>T_\ell(E)</math> for which Silverman <ref>Joseph H. Silverman. ''The Arithmetic of Elliptic Curves.'' Springer-Verlag, New York, 1986, corrected 2nd printing, p. 361.</ref> refers to Serre and Tate <ref>Jean-Pierre Serre and John Tate. “Good reduction of Abelian varieties.” ''The Annals of Mathematics,'' 2nd series, vol. 88, no. 3, Nov 1968, pp. 492-517.</ref> and Ogg <ref>A. P. Ogg. “Elliptic curves and wild ramification.” ''American Journal of Mathematics,'' vol. 89, no. 1, Jan 1967, pp. 1-21.</ref> on the precise definition.		where <math>\delta_p</math> is some <u>measure of “wild ramification” in the action of the inertia group</u> on <math>T_\ell(E)</math> for which Silverman <ref>Joseph H. Silverman. ''The Arithmetic of Elliptic Curves.'' Springer-Verlag, New York, 1986, corrected 2nd printing, p. 361.</ref> refers to Serre and Tate <ref>Jean-Pierre Serre and John Tate. “Good reduction of Abelian varieties.” ''The Annals of Mathematics,'' 2nd series, vol. 88, no. 3, Nov 1968, pp. 492-517.</ref> and Ogg <ref>A. P. Ogg. “Elliptic curves and wild ramification.” ''American Journal of Mathematics,'' vol. 89, no. 1, Jan 1967, pp. 1-21.</ref> on the precise definition.

	[[Image:Conductor.~~svg~~]]		[[Image:Conductor.png]]

	Only a finite number of “bad” reductions are possible.		Only a finite number of “bad” reductions are possible on an elliptic curve ''E''.

	This all needs to be simplified and brought down from any excessive abstraction. Cryptography is a martial art. References are inconsistent, disinformation exists, and it is unwise to accept mathematical results on face value which one does not prove oneself, or work through and satisfy oneself with the proofs. If anything is too difficult, another tack needs to be taken, a plan of attack from a different position. It is that strange feeling of invading a foreign library and opening up books one is not really welcome to.		This all needs to be simplified and brought down from any excessive abstraction. Cryptography is a martial art. References are inconsistent, disinformation exists, and it is unwise to accept mathematical results on face value which one does not prove oneself, or work through and satisfy oneself with the proofs. If anything is too difficult, another tack needs to be taken, a plan of attack from a different position. It is that strange feeling of invading a foreign library and opening up books one is not really welcome to.

Rational Point: authoritative references

2025-01-04T23:36:27Z

authoritative references

← Older revision		Revision as of 23:36, 4 January 2025
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	where the '''exponent''' ''f''<sub>''p''</sub> of each prime ''p'' is given by		where the '''exponent''' ''f''<sub>''p''</sub> of each prime ''p'' is given by

	:<math>f_p=0</math> if ''E'' has “good” reduction at ''p'';<br><math>f_p=1</math> if ''E'' has “multiplicative” reduction at ''p'';<br><math>f_p=2~~+??~~</math> if ''E'' has “additive” reduction at ''p'' ~~= 2 or~~ 3;<br><math>f_p=2+~~???~~</math> if ''E'' has “additive” reduction at ''p'' ~~&gt~~; 3;		:<math>f_p=0</math> if ''E'' has “good” reduction at ''p'';<br><math>f_p=1</math> if ''E'' has “multiplicative” reduction at ''p'';<br><math>f_p=2</math> if ''E'' has “additive” reduction at ''p'' > 3;<br><math>f_p=2+\delta_p</math> if ''E'' has “additive” reduction at ''p'' = 2 or 3;

			where <math>\delta_p</math> is some <u>measure of “wild ramification” in the action of the inertia group</u> on <math>T_\ell(E)</math> for which Silverman <ref>Joseph H. Silverman. ''The Arithmetic of Elliptic Curves.'' Springer-Verlag, New York, 1986, corrected 2nd printing, p. 361.</ref> refers to Serre and Tate <ref>Jean-Pierre Serre and John Tate. “Good reduction of Abelian varieties.” ''The Annals of Mathematics,'' 2nd series, vol. 88, no. 3, Nov 1968, pp. 492-517.</ref> and Ogg <ref>A. P. Ogg. “Elliptic curves and wild ramification.” ''American Journal of Mathematics,'' vol. 89, no. 1, Jan 1967, pp. 1-21.</ref> on the precise definition.

	[[Image:Conductor.svg]]		[[Image:Conductor.svg]]

	''' ''N.B.'':''' The definition here is incomplete and more authoritative references on the advanced abstract algebra should be consulted. All types of reduction at each prime ''p'' that are not “good” are said to be “bad,” and there are several special cases; only a finite number of “bad” reductions are possible.		Only a finite number of “bad” reductions are possible.

	This all needs to be simplified and brought down from any excessive abstraction. Cryptography is a martial art. References are inconsistent, disinformation exists, and it is unwise to accept mathematical results on face value which one does not prove oneself, or work through and satisfy oneself with the proofs. If anything is too difficult, another tack needs to be taken, a plan of attack from a different position. It is that strange feeling of invading a foreign library and opening up books one is not really welcome to.		This all needs to be simplified and brought down from any excessive abstraction. Cryptography is a martial art. References are inconsistent, disinformation exists, and it is unwise to accept mathematical results on face value which one does not prove oneself, or work through and satisfy oneself with the proofs. If anything is too difficult, another tack needs to be taken, a plan of attack from a different position. It is that strange feeling of invading a foreign library and opening up books one is not really welcome to.

Rational Point: delineation graph

2025-01-04T22:10:45Z

delineation graph

← Older revision		Revision as of 22:10, 4 January 2025
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	:<math>f_p=0</math> if ''E'' has “good” reduction at ''p'';<br><math>f_p=1</math> if ''E'' has “multiplicative” reduction at ''p'';<br><math>f_p=2+??</math> if ''E'' has “additive” reduction at ''p'' = 2 or 3;<br><math>f_p=2+???</math> if ''E'' has “additive” reduction at ''p'' > 3;		:<math>f_p=0</math> if ''E'' has “good” reduction at ''p'';<br><math>f_p=1</math> if ''E'' has “multiplicative” reduction at ''p'';<br><math>f_p=2+??</math> if ''E'' has “additive” reduction at ''p'' = 2 or 3;<br><math>f_p=2+???</math> if ''E'' has “additive” reduction at ''p'' > 3;

			[[Image:Conductor.svg]]

	''' ''N.B.'':''' The definition here is incomplete and more authoritative references on the advanced abstract algebra should be consulted. All types of reduction at each prime ''p'' that are not “good” are said to be “bad,” and there are several special cases; only a finite number of “bad” reductions are possible.		''' ''N.B.'':''' The definition here is incomplete and more authoritative references on the advanced abstract algebra should be consulted. All types of reduction at each prime ''p'' that are not “good” are said to be “bad,” and there are several special cases; only a finite number of “bad” reductions are possible.

	This all needs to be simplified and brought down from any excessive abstraction. Cryptography is a martial art. References are inconsistent, disinformation exists, and it is unwise to accept mathematical results on face value which one does not prove oneself, or work through and satisfy oneself with the proofs. If anything is too difficult, another tack needs to be taken, a plan of attack from a different position. It is that strange feeling of invading a foreign library and opening up books one is not really welcome to.		This all needs to be simplified and brought down from any excessive abstraction. Cryptography is a martial art. References are inconsistent, disinformation exists, and it is unwise to accept mathematical results on face value which one does not prove oneself, or work through and satisfy oneself with the proofs. If anything is too difficult, another tack needs to be taken, a plan of attack from a different position. It is that strange feeling of invading a foreign library and opening up books one is not really welcome to.

Rational Point: fw

2025-01-04T14:12:23Z

← Older revision		Revision as of 14:12, 4 January 2025
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			[[Category:Properties]]
	The '''conductor''' <ref>PlanetMath: conductor of an elliptic curve. https://planetmath.org/conductorofanellipticcurve</ref><ref>Wikipedia: Conductor of an elliptic curve. https://en.wikipedia.org/wiki/Conductor_of_an_elliptic_curve		The '''conductor''' <ref>PlanetMath: conductor of an elliptic curve. https://planetmath.org/conductorofanellipticcurve</ref><ref>Wikipedia: Conductor of an elliptic curve. https://en.wikipedia.org/wiki/Conductor_of_an_elliptic_curve
	</ref><ref>LMFDB: Knowledge → ec → Conductor of an elliptic curve (reviewed). https://www.lmfdb.org/knowledge/show/ec.conductor</ref><ref>Silverman, Joseph H., and Brumer, Armand. "The number of elliptic curves over Q with conductor N.." Manuscripta mathematica 91.1 (1996): 95-102. http://eudml.org/doc/156217.</ref> of an elliptic curve ''E'' over a field ''K'' is an integer, (or an “ideal” of the ring of integers in any algebraic field,) defined by its prime factorization:		</ref><ref>LMFDB: Knowledge → ec → Conductor of an elliptic curve (reviewed). https://www.lmfdb.org/knowledge/show/ec.conductor</ref><ref>Silverman, Joseph H., and Brumer, Armand. "The number of elliptic curves over Q with conductor N.." Manuscripta mathematica 91.1 (1996): 95-102. http://eudml.org/doc/156217.</ref> of an elliptic curve ''E'' over a field ''K'' is an integer, (or an “ideal” of the ring of integers in any algebraic field,) defined by its prime factorization:
Line 8:		Line 9:
	:<math>f_p=0</math> if ''E'' has “good” reduction at ''p'';<br><math>f_p=1</math> if ''E'' has “multiplicative” reduction at ''p'';<br><math>f_p=2+??</math> if ''E'' has “additive” reduction at ''p'' = 2 or 3;<br><math>f_p=2+???</math> if ''E'' has “additive” reduction at ''p'' > 3;		:<math>f_p=0</math> if ''E'' has “good” reduction at ''p'';<br><math>f_p=1</math> if ''E'' has “multiplicative” reduction at ''p'';<br><math>f_p=2+??</math> if ''E'' has “additive” reduction at ''p'' = 2 or 3;<br><math>f_p=2+???</math> if ''E'' has “additive” reduction at ''p'' > 3;

	''' ''N.B.'':''' The definition here is incomplete and more authoritative references on the abstract algebra should be consulted. All types of reduction at each prime ''p'' that are not “good” are said to be “bad,” and there are several special cases; only a finite number of “bad” reductions are possible.		''' ''N.B.'':''' The definition here is incomplete and more authoritative references on the advanced abstract algebra should be consulted. All types of reduction at each prime ''p'' that are not “good” are said to be “bad,” and there are several special cases; only a finite number of “bad” reductions are possible.

			This all needs to be simplified and brought down from any excessive abstraction. Cryptography is a martial art. References are inconsistent, disinformation exists, and it is unwise to accept mathematical results on face value which one does not prove oneself, or work through and satisfy oneself with the proofs. If anything is too difficult, another tack needs to be taken, a plan of attack from a different position. It is that strange feeling of invading a foreign library and opening up books one is not really welcome to.