50

Very briefly: until work of Hans Maass c. 1949, "modular" or "automorphic" both referred to holomorphic functions invariant-up-to-cocycle (that is, invariant holomorphic sections of a bundle) on a quotient $\Gamma\backslash X$. For $X$ the upper half-plane, these were ellipic modular forms, visible since the 19th century. For $X$ a product of upper half-...


29

Knowing the organizers well and working in the field, I can try an answer, but this is nothing more than an educated guess. First, the breakthroughs in question include (i) The construction and study of Galois representations attached to self-dual cohomological automorphic forms for $Gl_n$ (satisfying local-global compatibility, etc.) This is the work of ...


28

I can tell you the complete answer for $g \leq 2$: When $g=0$ and $n \geq 3$ no nontrivial automorphic forms appear. When $g=1$ and $n \geq 1$ the class of automorphic forms which appear are exactly the cusp forms for $\mathrm{SL}(2,\mathbf Z)$. (Deligne, Birch) When $g=2$ and $n \geq 0$ the automorphic forms are precisely the cusp forms for $\mathrm{SL}(...


25

A brief update: in his amazing talk yesterday at MSRI (available here), Laurent Fargues explained (building on work of Peter Scholze, see his phenomenal talk two weeks ago here and his historic Berkeley lectures here) a conjectural picture for the local Langlands correspondence as a geometric Langlands correspondence over the "Fargues-Fontaine curve". [...


25

This question is a bit like saying "what's the point of the theory of bases for vector spaces -- this just gives you an isomorphism of your space with $\mathbb{R}^n$. What is the point of defining this isomorphism if the thing we defined is just isomorphic to our original representation? Why would it be important to regard $V$ as a set of vectors in $\mathbb{...


24

Now that our paper Geometrization of the local Langlands correspondence with Fargues is finally out (ooufff!!), it may be worth giving an update to Ben-Zvi's answer above. In brief: we give a formulation of Local Langlands over a $p$-adic field $F$ so that it is finally an actual conjecture, in the sense that it asks for properties of a given construction, ...


24

To keep things simple, let $G$ be a finite group and $K$ a subgroup of it. The simplest definition of the Hecke algebra associated to this pair $(G, K)$ is that it is the algebra of $G$-endomorphisms $\text{End}_G(\mathbb{C}[G/K])$ of the permutation representation of $G$ on $G/K$. The significance of this algebra is that $\mathbb{C}[G/K]$ represents the ...


21

The only CAS's that have built-in support for modular and automorphic forms, as far as I know, are Sage and Magma. [Edit: I had forgotten Pari/GP, which will introduce substantial modular forms functionality as of version 2.10 which is currently in alpha testing; see Aurel's comment below]. Both Sage and Magma offer roughly comparable functionality. In Sage ...


19

To complement Joel's wonderful and (as far as I understand) very much on point answer, let me quote from the proposal for the parallel program on Geometric Representation Theory, which touches on several related themes: Representation theory is the study of the basic symmetries of mathematics and physics. The primary aim of the subject is to understand ...


19

Excellent question indeed. The quick answer is that $E_2(z)$ is an almost holomorphic modular form of weight $2$ and level $1$, so the automorphic representation generated by it is not irreducible. For more details (and my thought process), read below. Consider the Maass raising operator in weight $0$, $$ R:=y\left(i\frac{\partial}{\partial x}+\frac{\...


18

Now I think the answer is yes, Arthur's work is now unconditional for quasi-split special orthogonal and symplectic groups. Wee Teck Gan kindly directed me to the relevant papers of Waldspurger and Moeglin addressing the assumptions 2-4 above, though I was not able to verify with certainty that their stated results precisely cover what Arthur requires. ...


17

I first disclaim being up-to-date on the precise issue in the question! Given that: The only truly interesting example I know to have been definitely worked out, that exactly fits the question is Ramakrishnan's result from 2000 (Annals) which proves that the Rankin-Selberg convolutions for $GL_2\times GL_2$ (when there's no pole!) are standard cuspidal $L$-...


16

I am not sure it makes sense to ask "what is the p-adic local Langlands conjecture for $\mathrm{GL}_1$". Nobody has succeeded in even formulating a reasonable candidate for a p-adic LLC for $\mathrm{GL}_n$, so you can't just plug $n = 1$ in and see what it says. Morally, the conjecture should be something like this: for $K$ a p-adic field, there is a ...


16

If $D$ has degree $\leq 3$ there won't be any cusp forms. By the Langlands correspondence these correspond to irreducible Galois representations into $GL_2$, unramified away from a degree $3$ divisor, with unipotent local monodromy at $3$ points. Such representations would have Euler characteristic $1$, contradicting the claim that they are irreducible. One ...


15

This is not exactly an answer, but should illustrate how tentative the opinion of experts on L-functions is, when it comes to explain what L-function really are. Two quotes from the first volume on number theory by Kazuya Kato, Nobushige Kurokawa and Takeshi Saito: It seems as if the homeland where $\zeta$ functions originally come form is an unknown ...


15

I think few people would argue claims that Langlands' SLN 544 was written "in a different time" (e.g., pre-rigidity-theorems), and never really edited carefully (although the retyping is a relief), nor that Moeglin-Waldspurger aim for maximum generality/encyclopedic-ness, ... Selberg's sketches were of an even more different time. My colleague D. Hejhal ...


15

Here are two applications that I know of which involve direct crossover. In section 5 of the Hopkins ICM address referenced in Drew's answer to the other question, he gives a topological proof of the following congruence originally due to Borcherds. Suppose that $L$ is a positive definite, even unimodular lattice of dimension $24k$. The theta function $\...


15

If $K$ is a local field and $\mathcal O$ is its ring of integers, we say an irreducible representation of $G(K)$ is unramified if it contains a vector invariant under $G(\mathcal O)$. It is known that such a representation naturally has a unique vector invariant under $G(\mathcal O)$. The Hecke eigenvalue of a double coset in $G(\mathcal O)\backslash G(K)/G(\...


14

To give a brief answer, which I think applies to all audiences, and I hope is not too "elementary" for you (I'm not attempting to give details, which of course need to be specialized for the intended audience, and I'm not entirely sure what you're looking for): Automorphic forms provide analytic ways to study solutions to equations in integers or rational ...


14

Sure, and you can even do this for $\mathrm{SL}_n$; I believe this goes back to Siegel. A good hands-on reference is Chapter 1 of Automorphic Forms and $L$-Functions for the group $\mathrm{GL}(n,\mathbb{R})$ by Dorian Goldfeld. More precisely, we have the Iwasawa decomposition $z = xy$ for $z \in \mathrm{SL}_n(\mathbb{R}) / \mathrm{SO}_n(\mathbb{R})$, where ...


13

The Paris lectures, along with others he gave later, were spliced together into a publication: Introduction aux groupes arithmetiques (softcover, Hermann, Paris, 1969). As his nominal assistant at IAS in 1968-69, I tried to help with the splicing process but didn't manage to clean up all the inconsistent notation and typos (especially on pages 90-94) ...


13

EDIT. A colleague wrote to me to point out that my original answer to this question was actually completely wrong: I had confused "isobaric" representations with "pure" representations (which are not at all the same thing). The correct answer, according to my colleague, is this. Consider the one-dimensional representation $\sigma = |\cdot|^{1/2} \boxtimes|\...


13

We have that \[-\frac{L'}{L}(s,\mathrm{sym}^2 f) = \sum_{n = 1}^{\infty} \frac{\Lambda_{\mathrm{sym}^2 f}(n)}{n^s},\] where $\Lambda_{\mathrm{sym}^2 f}(n)$ is equal to $\lambda_f(p^2) \log p$ if $n = p$ with $p \nmid N$, is essentially a bounded multiple of $\log p$ if $n = p^k$, and vanishes otherwise. (There are some minor issues at $p \mid N$ that are no ...


13

Up to a normalisation factor for the measure you fixed on $G_\gamma(\mathbb A)$, this is the Tamagawa number of $G_\gamma$. There is indeed a formula for this. It is $$ \frac{ \left| \pi_0\left( Z(\hat{G})^\Gamma\right) \right|}{ \left|\operatorname{ker}^1\left(F, Z(\hat{G}) \right)\right|}$$ where $\hat{G}$ is the Langlands dual group, $Z$ is its centre, $...


13

The right person to answer this question is probably Dinakar Rimakrishnan (a good working reference is his paper "Modularity of the Rankin–Selberg $L$-series, and multiplicity one for $\mathrm{SL}(2)$"), but since he doesn't seem to use MathOverflow, here is my understanding of this. In each case, I will first discuss the local theory, then the ...


13

The Bessel functions $J_\ell$ for $\ell\geq 1$ odd are pairwise orthogonal on the positive axis with respect to the measure $dx/x$. They correspond to the holomorphic spectrum (of various even weights $\ell+1$) of $L^2(\Gamma\backslash H)$. The orthogonal complement of the span of these $J_\ell$'s is continuously (and orthogonally) spanned by the functions $...


13

This is a question which has no "right" answer. A posh interpretation of the choice of exponent is that a Hecke eigenform $f$ determines an equivalence class of irreducible representations $\Pi = \bigotimes'_v \Pi_v$ of $GL_2(\mathbb{A}_\mathbb{Q})$, differing by twists by powers of the character $g \mapsto \|\det(g) \|$, and the power of $\det$ ...


12

A natural generalization of the geometric modularity conjecture which is compatible with your formulation Do you expect some form of modularity to correspond to the existence of a map from some space (like a Shimura variety) to the variety in question? is to ask whether a variety always appears as a quotient of the Picard variety of (the smooth ...


12

This is a special case of the shifted convolution problem for modular forms. For example, see Chapter 12 of Iwaniec's book "Spectral methods for automorphic forms" (AMS Grad studies in Math 53). Theorem 12.5 there gives (for $n$ odd) $$ \sum_{m\le x} r(m) r(m+n) = 8 \Big(\sum_{d|n} d^{-1} \Big) x + O(x^{2/3} n^{1/3+\epsilon}), $$ where $r(m)$ denotes ...


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