By Willem Kuyk (Editor), J.-P. Serre (Editor)
Read or Download Modular Functions of One Variable III: Proceedings International Summer School, University of Antwerp, RUCA, July 17 - August 3, 1972 PDF
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Extra resources for Modular Functions of One Variable III: Proceedings International Summer School, University of Antwerp, RUCA, July 17 - August 3, 1972
Now for v = 3,5 or 7 denote by G 1v the subgroup of G consisting of those a for which f1 - 1 or v mod 8, and denote by G1 the subgroup of G for which t::. - 1 mod 8. The argument that proved G :J H only used the como mutators of elements of G ; so it certainly proves [G ,G ] :J H ' and 17 o 17 17 it now follows easily from (41) that (G ,G ] consists of those ele17 17 ments of [G,G] for which Band C are divisible by 4. It is convenient next to consider the commutator subgroup of G • It is 1 easily verified that if a and a are in G then the congruences (41) 1 2 1 hold mod 2 8 • Now f1 = t::.
19 ( 1 +A) ) mod 32 if t. - 3 mod (38) - Be II.. -1) - II.. ( 11 ( 1 +t. ) ) mod 64 if t. - mod 8 if A Note that D and t. are linked by the congruence D - 1/2 (t. - 1) mod 64 5 mod 47 which is a weak form of (31). SwD-47 It is also convenient at this point to record some formulae for the product of two matrices in A,B,C,D form; all mod 2 7 • LEMMA 10. Each a in G satisfies B + C~ == 0 mod 8 and the congruence con- ditions stated in the following table : ± 1 6 mod 8 A mod 16 Y. B and C (a6 - ± 3 1) - 2C 2 1/.
Of ~ Finally e,~,~ will be the characters mod 8 whose values are given by the following table : I ~ mod 8 1 3 5 7 e 1 1 -1 -1 ~ 1 -1 1 -1 ~ 1 -1 -1 1 If we have to consider several a simultaneously, we shall distinguish them by subscripts and we shall attach the corresponding sUbscripts to the associated letters a,b,c,d,A,B,C,D,S,~,e,~,~. Let Go be the set of elements a of GL 2 GZ ) which satisfy the conditions 2 44 SwD-44 Band C are both even if ~ =± ~ =± 1 mod 8 and both odd if 3 mod 8, (3 3) A - 2C 2 mod 64.
Modular Functions of One Variable III: Proceedings International Summer School, University of Antwerp, RUCA, July 17 - August 3, 1972 by Willem Kuyk (Editor), J.-P. Serre (Editor)