# Is there a characterization of the cuspidal automorphic representations that arise from elliptic curves?

Is there a way to characterize (e.g. by their local components) the cuspidal automorphic representations that arise from elliptic curves? There is a procedure to go from elliptic curves to cuspidal automorphic representations of $$GL_2$$ but can the image be characterized?

## 1 Answer

I'll assume that you're talking about elliptic curves over $$\mathbf{Q}$$ - much of what I'm going to say should generalize to totally real fields, replacing modular forms by Hilbert modular forms. I think the question is much less well-understood outside of that setting.

So your question is a little bit backwards: there's a procedure to attach an elliptic curve to certain automorphic representations of $$\mathrm{GL}_2/\mathbf{Q}$$, and the modularity theorem says that this is surjective. The automorphic representations in question are exactly those spanned by a Hecke eigenform of weight 2 with Hecke eigenvalues contained in $$\mathbf{Q}$$.

Let $$f = \sum_n a_n q^n$$, $$a_n \in \mathbf{Q}$$ be such an eigenform, normalized with $$a_1 = 1$$. We can think of $$f$$ as a function on $$\mathrm{GL}_2(\mathbf{Q}) \backslash \mathrm{GL}_2(\mathbf{A}_\mathbf{Q})$$, and it spans an irreducible cuspidal automorphic representation $$\pi_f$$ of $$\mathrm{GL}_2$$. (This transformation is described in many places, such as here or here.)

We can decompose $$\pi_f = \bigotimes'_p \pi_p$$. The fact that $$f$$ is a modular form of weight $$2$$ corresponds to $$\pi_\infty$$ being the discrete series representation of $$\mathrm{GL}_2(\mathbf{R})$$ of weight $$2$$.

For $$p$$ not dividing the level of $$f$$, we have that $$\pi_p$$ is the unramified principal series representation of $$\mathrm{GL}_2(\mathbf{Q}_p)$$ with Satake parameters equal to the two roots of the polynomial $$x^2 - a_p x + p$$.

It's a bit harder to characterize $$\pi_p$$ for $$p$$ dividing the level - these will be ramified representations of $$\mathrm{GL}_2(\mathbf{Q}_p)$$. However, the strong multiplicity one theorem says that the automorphic representation is determined by all but finitely many of its components anyway.

Now, given such a modular form $$f$$, we can construct the corresponding elliptic curve over $$\mathbf{Q}$$ as follows. We regard $$f$$ as a one-form on the modular curve $$X(N)$$. The Hecke operators away from $$N$$ act by correspondences on $$X(N)$$, so they determine a legitimate commuting action on its Jacobian $$J(N)$$. The Jacobian $$J(N)$$ breaks up into a product of "Hecke eigenspaces", and the elliptic curve $$E_f$$ is the quotient of $$J(N)$$ on which the Hecke operators act by the $$a_p$$'s.

What happens when we relax some of the conditions we put on $$f$$? If we do not require the Hecke eigenvalues $$a_p$$ to be in $$\mathbf{Q}$$, the same construction produces an abelian variety of dimension greater than $$1$$. The corresponding Galois representation will be a $$2$$-dimensional representation over the field $$K$$ generated by the $$a_p$$'s, and the abelian variety will have "real multiplication" over $$K$$.

If we replace $$f$$ by a Hecke eigenform of higher (even) weight, the corresponding motive looks like a symmetric product of elliptic curves. On the level of Galois representations, a weight $$2k$$ Hecke eigenform with rational eigenvalues corresponds to a $$2$$-dimensional representation over $$\mathbf{Q}$$ with "Hodge-Tate weights" $$(k, 0)$$. Roughly speaking, this means it looks like $$\mathrm{H}^k(X, \underline{\mathbf{Q}_\ell})$$ for a smooth proper variety $$X$$.

• Over other number fields, there will be a similar conjectural description of which ones come from elliptic curves, but it won’t be probable. – Will Sawin Jul 21 at 9:26