I have recently attempted to read some number-theoretic texts. Here is an excerpt from a paper by Breuil-Conrad-Diamond-Taylor:

Now consider an elliptic curve $E/\mathbb{Q}$. Let $\rho_{E, l}$ (resp. $\overline{\rho_{E, l}}$) denote the representation of $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ on the $l$-adic Tate module (resp. the $l$-torsion) of $E(\mathbb{Q})$. Let $N(E)$ denote the conductor of $E$. It is known that the following conditions are equivalent:

(1) The $L$-function $L(E,s)$ of $E$ equals the $L$-function $L(f,s)$ for some eigenform $f$.

(2) The $L$-function $L(E,s)$ of $E$ equals the $L$-function $L(f,s)$ for some eigenform $f$ of weight $2$ and level $N(E)$.

(3) For some prime $l$, the representation $\rho_{E,l}$ is modular.

(4) For all primes $l$, the representation $\rho_{E,l}$ is modular.

(5) There is a non-constant holomorphic map $X_1(N)(\mathbb{C}) \rightarrow E(\mathbb{C})$ for some positive integer $N$.

(6) There is a non-constant morphism $X_1(N(E)) \rightarrow E$ which is defined over $\mathbb{Q}$.

The implications (2) ⇒ (1), (4) ⇒ (3) and (6) ⇒ (5) are tautological. The implication (1) ⇒ (4) follows from the characterisation of $L(E,s)$ in terms of $\rho_{E,l}$. The implication (3) ⇒ (2) follows from a theorem of Carayol [Ca1].

The first question (very naive): what does the weight and the level of a modular form mean in the automorphic terms? Do these notions somehow generalize to other groups?

The second question (possibly less naive): if my understanding is correct, the Langlands philosophy tells us that given an elliptic curve, there is some random modular form which has the same $L$-function. Does the Langlands philosophy make any specific predictions about weight and level of the modular form?

The third, loosely related question: does the Langlands philosophy make any predictions about the strong Serre conjecture? I am not even sure that it makes any predictions about the weak Serre conjecture, just asking.

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