Analytic continuation for $L$-functions of elliptic curves

Question

Let $E$ be an elliptic curve over a number field.

When $E$ has no CM and is a $\mathbb Q$-curve (i.e. it is $\overline{\mathbb Q}$-isogenous to all of its conjugates), it is nowadays known that $E$ is a factor up to isogeny of $J_1(N)$. In a nice paper X. Guitart and J. Quer define a subclass of $\mathbb Q$-curves, called strongly modular curves, which enjoy the property that their $L$-function is a product of $L$-functions of newforms. Thus, the $L$-function of these curves can be analitically continued to $\mathbb C$.

If $E$ has CM, the $L$-function of $E$ is known to admit an analytic continuation to $\mathbb C$ because it coincides with a product of $L$-functions of Hecke characters.

My first question is: are $L$-functions of elliptic curves with CM also products of $L$-functions of (classical) newforms? It is not clear at all to me what is the connection between $L$-functions of cuspforms and $L$-functions of Hecke characters.

My second question is: what is the state of art of the problem? Namely, is there any other class of elliptic curves whose $L$-function is known to have an analytic continuation?

Answer 1

As said in the comment above, the answer to your first question is no. Hecke L-functions are expected to factor as a product of irreducible cuspidal automorphic representations of $\mathrm{GL}_n(\mathbb{A})$ (they are known to be a product of Artin L-functions, but we don't know those are entire in general). In the case of an elliptic curve with CM (using Deuring's theorem) this means

$$L_E(s)=L(s,\chi)L(s,\bar\chi)=\prod_{i=1}^r L(s,\pi_i)^{e_i}$$

But in general those representations $\pi$ are $n$-dimensional.

To see this in the simplest case, consider a Hecke L-function with trivial character, that is, $L(s,1)=\zeta_K(s)$, the Dedekind zeta function of the number field. We know from Aramata-Brauer (in black) and conditionally on the strong Artin conjecture (in red) that

$$\zeta_K(s)=\zeta(s)\prod_{\rho \neq 1} L(s,\rho)=\color{red}{\zeta(s)\prod_{\pi \neq 1} L(s,\pi)}$$

where the product goes over all the nontrivial irreducible representations of $\mathrm{Gal}(K/\mathbb{Q})$.

Now, in order for $\zeta_K$ to be expressible in terms of classical newforms, all (or some, depending on how one interprets your question) those representations of that Galois groups have to be of degree $2$. But that's definitely not true. Number fields with $\mathrm{Gal}(K/\mathbb{Q}) \cong S_n$, $n>4$ are alredy a counterexample.

Of course classical newforms also show up, but I'm not sure that there's a nice characterization of when that happens, depending on $E$, $K$ and $\mathrm{End}_K(E)\otimes\mathbb{Q}$.

Regarding the second question, analytic continuation (a special case of the Hasse-Weil conjecture) is only known in general when $E$ has CM (Deuring), when $E$ is defined over $\mathbb{Q}$ (Wiles-Breuil-Conrad-Diamond-Taylor) and when $K$ is real quadratic (Freitas-Hung-Siksek).