By Fred Diamond, Jerry Shurman

This booklet introduces the idea of modular kinds, from which all rational elliptic curves come up, with a watch towards the Modularity Theorem. dialogue covers elliptic curves as advanced tori and as algebraic curves; modular curves as Riemann surfaces and as algebraic curves; Hecke operators and Atkin-Lehner conception; Hecke eigenforms and their mathematics houses; the Jacobians of modular curves and the Abelian kinds linked to Hecke eigenforms. because it offers those principles, the booklet states the Modularity Theorem in a number of varieties, bearing on them to one another and relating their functions to quantity concept. The authors suppose no history in algebraic quantity idea and algebraic geometry. routines are incorporated.

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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).

### A First Course in Modular Forms (Graduate Texts in Mathematics) by Fred Diamond, Jerry Shurman

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