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Using Diamond, I have read and understand the definitions of modular curves as quotient groups of the upper half plane, modded out by congruence sub groups of $SL_2(Z)$, I understand that they can be interpreted as Riemann surfaces, then compactified, and as such, have an algebraic representation. The algebraic modular polynomial when equated to 0, has solutions that are the j-invariants, and so the solutions represent isomorphism classes of elliptic curves.

My question is this, I'm trying to present the above ideas and I did not really get a good understanding of why it is the case that we can go from working with quotient groups of the upper half plane, to parameterizing l-isogenous elliptic curves for example. I would greatly benefit from an explanation of the intuition behind all of the technical details and definitions.

Thanks!

s3binator
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    Well, a short answer is just that elliptic curves come from lattices, isomorphisms of elliptic curves are isomorphisms of lattices and the action of $\text{SL}_2(\mathbb Z)$ on the upper half plane corresponds to mapping a lattice into an isomorphic one. Thus it is not completely surprising that say $\text{SL}_2(\mathbb Z)\backslash \mathbb H$ parametrizes isomorphism classes of elliptic curves, and analogously for $X_0(N)$. What is much more surprising (to me) is the fact that you can descend to $\mathbb Q$ these curves! – Ferra Oct 12 '16 at 11:49

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