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Extra resources for A First Course in Modular Forms (Graduate Texts in Mathematics)

Sample text

9. Verify the Fourier expansions of G2,2 and G2,4 given in the section. 10. (a) Show that the series S(τ ) = n=1 log(1−q n ) converges absolutely and uniformly on compact subsets of H. ) (b) Show the logarithmic diﬀerentiation formula (log( fn )) = fn /fn for ﬁnite products. The analogous result for inﬁnite products applies to the Dedekind eta function. 11. Let GL+ 2 (Q) be the group of 2-by-2 matrices with positive determinant and rational entries. The elements of this group act as fractional linear transformations on the upper half plane since Im(γ(τ )) = det γ·Im(τ )/|cτ +d|2 for γ ∈ GL+ 2 (Q).

A) Let p be a prime and let e be a positive integer. Show that |SL2 (Z/pe Z)| = p3e (1 − 1/p2 ). ) (b) Show that |SL2 (Z/N Z)| = N 3 p|N (1 − 1/p2 ), so this is also the index [SL2 (Z) : Γ (N )]. (c) Show that the map Γ1 (N ) −→ Z/N Z given by ac db → b (mod N ) surjects and has kernel Γ (N ). (d) Show that the map Γ0 (N ) −→ (Z/N Z)∗ given by ac db → d (mod N ) surjects and has kernel Γ1 (N ). (e) Show that [SL2 (Z) : Γ0 (N )] = N p|N (1 + 1/p). 4. Let Γθ be the group generated by the matrices ± [ 10 11 ] and ± [ 14 01 ].

10. (a) Show that the series S(τ ) = n=1 log(1−q n ) converges absolutely and uniformly on compact subsets of H. ) (b) Show the logarithmic diﬀerentiation formula (log( fn )) = fn /fn for ﬁnite products. The analogous result for inﬁnite products applies to the Dedekind eta function. 11. Let GL+ 2 (Q) be the group of 2-by-2 matrices with positive determinant and rational entries. The elements of this group act as fractional linear transformations on the upper half plane since Im(γ(τ )) = det γ·Im(τ )/|cτ +d|2 for γ ∈ GL+ 2 (Q).