Since 2008 we have the following remarkable correspondence:

Odd irreducible 2-dim Galois repn $\longleftrightarrow$ weight 1 newforms

note: all Galois representations in this question are ment to be continuous complex linear representations.

That this is a bijection is the consequence of three deep results.

  • Serre-Deligne: for each weight 1 newform there is an irreducible 2-dim Galois representation.

  • Weil-Langlands: the converse is true if the Artin conjecture holds for those representations.

  • Khare-Wintenberger: the Artin conjecture holds for odd irreducible 2-dim Galois representation (from Serre's modularity conjecture).

Now, we also famously have (Wiles-Breuil-Conrad-Diamond-Taylor):

Elliptic curves over $\mathbb{Q}$ $\longrightarrow$ weight 2 newforms

My question is, what is known/conjectured on the converse direction? This is, does every newform of weight 2 arises from an elliptic curve over $\mathbb{Q}$? (Note that this correspondence can only hold up to isogeny) Are there any other source for newforms of this weight? How can we tell apart those that came from elliptic curves?

  • $\begingroup$ If your conjecture holds only up to isogeny, then it most likely relies on the deep properties of L-functions of rational elliptic curves. $\endgroup$ Sep 18 '15 at 17:48
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    $\begingroup$ Weight 2 newforms that correspond to elliptic curves are exactly those that have integral coefficients. This follows from basic results of Shimura (and is much easier than the modularity of elliptic curves). $\endgroup$ Sep 18 '15 at 18:11
  • 2
    $\begingroup$ You mixed up Deligne-Serre and Weil-Langlands: Deligne-Serre associated to weight 1 newforms 2-dimensional odd Galois representations, and it follows from the work of Weil-Langlands (and also Hecke) that exactly those odd 2-dimensional Galois representations arise this way whose twists by 1-dimensional representations satisfy the Artin conjecture. $\endgroup$
    – GH from MO
    Sep 18 '15 at 19:44

This should be a comment, but is getting too long.

First, unless you are using the strange convention that Galois representations have by definition complex coefficients, odd, irreducible 2-dimensional Galois representations do not correspond to weight 1 newforms. Only a very small subset of the former set corresponds to the latter one, namely the one with values in $\operatorname{GL}_2(\mathbb C)$ or with finite image.

Second, you exchanged the contributions of Weil-Langlands and Deligne-Serre.

Third, the newforms whose $L$-function is the $L$-function of an elliptic curve are exactly those with integral coefficients. This is elementary (at least if one disregards Euler factors at places of bad reduction, in order to conclude for all Euler factors, one needs Faltings's theorem): just take the quotient of the Jacobian of the modular curve given by the Hecke action on $f$. The other newforms of weight 2 are attached to modular abelian varieties (the dimension of the abelian variety being the degree of the field generated by the coefficients of $f$ over $\mathbb Q$). This is a classical result of G.Shimura.

  • $\begingroup$ You're absolutely right about 1 and 2 (thank to Denis and GH too). Also, your third point is precisely what I was looking for with the first question: I was missing Shimura's results. I'm tempted to accept this answer and take the Maass part to another question. $\endgroup$
    – Myshkin
    Sep 18 '15 at 21:18

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