Are there any general conjectures/properties (in the Langlands Program) for automorphic representations of $GL_n$ which are still open for $n=1$?

  • $\begingroup$ Maybe you should explicate what you mean by "general conjectures." The existence of the Langlands group is still unknown, which applies to all $n$. $\endgroup$
    – Kimball
    Sep 2 '16 at 9:47
  • $\begingroup$ I guess Langlands defined automorphic $L$-functions for automorphic representations of $GL(n)$ and proved that the product converged for $Re(s)$ sufficiently large? So now one can conjecture that they have meromorphic continuation (which is true) and that a suitable form of the Riemann Hypothesis holds (which is unknown even for $GL_1$). Is that part of the Langlands Program? $\endgroup$
    – znt
    Sep 2 '16 at 17:24

The question is subject to interpretation, because the Langlands program was never a clearly delimited set of conjectures to begin with, and moreover many things were added during the following decades, and the answer depends if you want to consider these innovations as part of the Langlands program or not.

I believe nevertheless that essentially everything in the original Langlands program (of the 70's) is known for $GL_1$, and actually was known long before Langlands began working on his program. Every instance of Langlands functoriality involving only $GL_1$ results easily from Class field theory. As for reciprocity, if the existence of the Langlands group is still unknown, only its abelianization matters for $GL_1$, and we know, again by CFT, what this abelianization is.

If we add to the Langlands program the correspondance of algebraic automorphic forms with Galois representations, as for example formulated in Clozel's 1988 paper "Motifs et Formes Automorphes" in Clozel-Milne, then again the case of $GL_1$ was well-known long before, and due to André Weil (that's his theory of algebraic Grössencharacters).

But if you are willing to consider the theory of $p$-adic automorphic forms and their families as part of the Langlands program, then there is one thing that is still unknown: the dimension of the eigenvariety of automorphic forms for $GL_1$ over a number field. Determining this dimension is equivalent to proving Leopoldt's conjecture.


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