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I am studying modular forms and I have two questions on modular forms over finite index subgroups of the modular group.

  1. What is the role of the enhanced elliptic curve on number theory?

    Modular form of weight $2k$ over $\text{SL}_2(\mathbb{Z})$ is holomorphic differential form of weight $k$ on modular curve $\text{SL}_2(\mathbb{Z})\setminus\mathbb{H}$, and this is classifying space of elliptic curves over $\mathbb{C}$. I agree this is important because the elliptic curve is important in number theory. For example, we can use the elliptic curve to describe class field theory of imaginary quadratic fields. For some important subgroups of $\text{SL}_2(\mathbb{Z})$ such as $\Gamma(N)$, the modular curve $\Gamma(N)\setminus\mathbb{H}$ is a classifying space of 'enhanced elliptic curve', in this case, $(E,B)$ where $B$ is a basis of $E[N]$. We can say modular form over these subgroups is a holomorphic differential form on modular curve. However, I haven't seen yet the usage of enhanced elliptic curves in number theory. So I'm wondering why the enhanced elliptic curve is important in number theory.

  2. What is the meaning of odd weight modular forms?

    If $\Gamma\le \text{SL}_2(\mathbb{Z})$ doesn't contain $-I$, then we may have odd weight modular forms over $\Gamma$. However, these modular forms cannot be interpreted as differential forms on modular curve. Then what is the geometric meaning of modular forms of odd weight? Is there an half-integral weight differential forms? Also, I told that there exist modular forms of half-integral weight. Then what is the geometric meaning of that?

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There is a potentially justification from the moduli space point of view, but I can give a partial response to your first question from the point of view of the BSD conjecture (which overlaps quite a bit), and I'll leave the second to someone else more knowledgeable about modular forms of non-even weight.

The group of rational points on an elliptic curve $E$ defined over $\mathbf{Q}$ is finitely generated, and the Birch and Swinnerton-Dyer (BSD) conjecture predicts $$ \mathrm{rank}(E(\mathbf{Q})) = \mathrm{ord}_{s=1}L(E,s). $$ As a consequence, if $L(E,1) = 0$ then BSD predicts that there must be a rational point of infinite order.

But how do you construct a rational point on this (arbitrary) elliptic curve? One idea is to take advantage of the modular parametrization $\pi: X_0(N) \to E$ that exists as a consequence of modularity. Since $\pi$ takes rational points to rational points, one can reduce the question to constructing rational points on $X_0(N)$ that give rise to points of infinite order on $E$.

This maybe doesn't seem any simpler at first; I've simply suggested that constructing points directly on an elliptic curve is hard, and I've claimed that I can potentially solve that problem by reducing it to constructing rational points on a different curve. But, $X_0(N)$ is a moduli space (of generalized enhanced elliptic curves), which allows us to construct certain points and show that they are actually rational.

For example, non-cuspidal points of $X_0(N)$ parametrize cyclic isogenies $E \to E'$ of degree $N$ between elliptic curves. Let $K$ be an imaginary quadratic field and suppose that all primes dividing $N$ split in $K$. Then there is a ideal $\mathfrak{N}$ of $\mathcal{O}_K$ such that $\mathcal{O}_K/\mathfrak{N} \cong \mathbf{Z}/N\mathbf{Z}$. Then $$ \mathbf{C}/\mathcal{O}_K \to \mathbf{C}/\mathfrak{N}^{-1} $$ is a cylic $N$-isogeny and so corresponds to a point $P$ on $X_0(N)$. Using the theory of complex multiplication, one can show that $P$ is actually defined over the Hilbert class field, $H$, of $K$. Then $$ x_K := \sum_{\sigma \in \mathrm{Gal}(H/K)} P^\sigma $$ is a divisor on $X_0(N)$ defined over $K$, sometimes referred to as a Heegner point. Then $\pi(x_K)$ is a $K$-rational point on $E$, and work of Gross-Zagier and Kolyvagin actually shows that if $\mathrm{ord}_{s=1} L(E,s) = 1$, then $\pi(x_K)$ is actually defined over $\mathbf{Q}$ and is a non-torsion point of $E(\mathbf{Q})$.

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