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.

  • $\begingroup$ See mathoverflow.net/questions/296682/… for your first question. $\endgroup$ – dhy Jun 1 '19 at 18:37
  • $\begingroup$ And the second (and third) questions are pretty vague --- what counts as "the Langlands philosophy"? $\endgroup$ – Pound Sterling Jun 1 '19 at 20:59
  • $\begingroup$ @PoundSterling I agree they are vague, but there were some pretty well-received questions about that on this site so I thought it was OK. No idea how to define that formally, give any argument you want and then we will judge (I think there is some set of statements which by community consensus are considered Langlands philosophy, but I am not able to define it completely rigorously, and I don't think anyone else can). $\endgroup$ – user140765 Jun 1 '19 at 21:01
  • $\begingroup$ I would argue that those questions are of a qualitatively different nature. But leaving that aside, if you want to include "global reciprocity conjectures for motives" as part of the Langlands philosophy, then I don't see why you wouldn't also include as part of the Langlands philosophy that "global reciprocity is compatible with local Langlands", in which case the answer to the second question is yes, the level and weight are pinned down exactly. $\endgroup$ – Pound Sterling Jun 1 '19 at 22:29
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    $\begingroup$ The answer to your first question ("what does the weight and the level of a modular form mean in the automorphic terms") can be found in textbooks (e.g. Bump or Goldfeld-Hundley). Briefly, the weight and the level (and the nebentypus) tell us how an automorphic form transforms under the right action of a certain compact subgroup of the underlying adelic group. $\endgroup$ – GH from MO Jun 11 '19 at 9:45

Modular forms are in some sense not the exactly right object to look at, because there are many modular forms of different levels that give rise to L-functions which are basically the same. Namely any oldform arising from a newform has the same L-function up to some factor. So in Langlands philosophy only newforms matter I would say. Modular forms are also in automorphic side, so I do not know what you meant by your first question. From the automorphic representation corresponding to a newform, level=conductor, weight=Blattner parameter of infinity part of aut. rep.

An elliptic curve supposed to correspond to newforms of weight 2 and level N where N is the conductor of the elliptic curve.

The original Langlands philosophy does not have a say about congruences, so the Serre conjecture is a little different story (although very related).

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    $\begingroup$ Newform theory is only available for a few groups such as $\mathrm{GL}_n$. In general, one needs to consider irreducible admissible representations (with special attention to the cuspidal ones). $\endgroup$ – GH from MO Jun 11 '19 at 9:49

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