Classical modular curve: Difference between revisions

Revision as of 22:42, 6 June 2015

Revision as of 15:57, 7 August 2015

Line 20:

is (up to the constant term) the [[McKay–Thompson series]] for the class 2B of the [[Monster group|Monster]], and {{mvar|η}} is the [[Dedekind eta function]], then

:<math>x = \frac{(j_2+256)^3}{j_2^2},</math>

:<math>y = \frac{(j_2+16)^3}{j_2}</math>

Line 36:

If in the curve {{math|''y''² + ''y'' {{=}} ''x''³ − ''x''² − 10''x'' − 20}}, isomorphic to {{math|''X''₀(11)}} we substitute

:<math>x \mapsto \frac{x^5-2x^4+3x^3-2x+1}{x^2(x-1)^2}</math>

:<math>y \mapsto y-\frac{(2y+1)(x^4+x^3-3x^2+3x-1)}{x^3(x-1)^3}</math>

Line 63:

*Erich Hecke, ''Die eindeutige Bestimmung der Modulfunktionen q-ter Stufe durch algebraische Eigenschaften'', Math. Ann. '''111''' (1935), 293-301, reprinted in ''Mathematische Werke'', third edition, Vandenhoeck & Ruprecht, Göttingen, 1983, 568-576 [http://dz-srv1.sub.uni-goettingen.de/sub/digbib/pdftermsconditions?did=D37958&p=297]

*Anthony Knapp, ''Elliptic Curves'', Princeton, 1992

*[[Serge Lang]], ''Elliptic Functions'', Addison-Wesley, 1973

*Goro Shimura, ''Introduction to the Arithmetic Theory of Automorphic Functions'', Princeton, 1972