# New PDF release: A First Course in Modular Forms (Graduate Texts in

By Fred Diamond, Jerry Shurman

This booklet introduces the idea of modular types, from which all rational elliptic curves come up, with an eye fixed 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 concept; Hecke eigenforms and their mathematics houses; the Jacobians of modular curves and the Abelian kinds linked to Hecke eigenforms. because it offers those rules, the booklet states the Modularity Theorem in a variety of varieties, touching on them to one another and pertaining to their functions to quantity thought. The authors suppose no historical past in algebraic quantity conception and algebraic geometry. routines are integrated.

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

**Sample text**

This section shows that there is a bijection from the ﬁrst quotient set to the second. That is, the equivalence classes of points in the upper half plane under the action of the modular group are described by the isomorphism classes of complex elliptic curves. 2 are described by the sets of equivalence classes of elliptic curves enhanced by corresponding torsion data. We begin by describing the relevant torsion data for the congruence subgroups. Let N be a positive integer. An enhanced elliptic curve for Γ0 (N ) is an ordered pair (E, C) where E is a complex elliptic curve and C is a cyclic subgroup of E of order N .

Let Γθ be the group generated by the matrices ± [ 10 11 ] and ± [ 14 01 ]. This exercise shows that Γθ = Γ0 (4). The containment “⊂” is clear. For the other containment, let α = ac db be a matrix in Γ0 (4). 1, the identity ab 1n a b = cd 01 c nc + d shows that unless c = 0, some matrix αγ with γ ∈ Γθ has bottom row (c , d ) with |d | < |c |/2. ) Use the identity ab cd 1 0 a b = 4n 1 c + 4nd d 22 1 Modular Forms, Elliptic Curves, and Modular Curves to show that unless d = 0, some matrix αγ with γ ∈ Γθ has bottom row (c , d ) with |c | < 2|d |.

3) and m(1/N + Λτ ) = 1/N + Λτ . 10) so in particular m = cτ + d. Now the second condition becomes 1 cτ + d + Λτ = + Λτ , N N showing that (c, d) ≡ (0, 1) (mod N ) and γ ∈ Γ1 (N ). 10), it follows that Γ1 (N )τ = Γ1 (N )τ . 3 and at the beginning of this section. This lets us associate a complex number to each isomorphism class. 1, an SL2 (Z)-invariant function on H. Each isomorphism class of complex elliptic curves has an associated orbit SL2 (Z)τ ∈ SL2 (Z)\H and thus has a well deﬁned invariant j(SL2 (Z)τ ).

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

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