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13
votes
3answers
995 views

Constructing coherent sheaves on Shimura varieties.

Let me first run through the setting of my question in an example I understand well; that of modular curves. If $Y_1(N)$ denotes the usual modular curve over the complexes, the quotient of the upper ...
4
votes
1answer
604 views

Rallis inner product formula for U(2,2) and U(3)

Victor Tan has a couple of papers on a regularized Siegel-Weil formula for U(2,2) and U(3). The papers I'm talking about are: "A Regularized Siegel-Weil Formula on U(2,2) and U(3)", Duke, 1998. "An ...
6
votes
2answers
799 views

modularity of algebraic varieties

Hello, Are there any examples of varieties which are not Shimura varieties or abelian varieties and whose L-functions have been shown to be a product of automorphic L-functions? Thanks. N
9
votes
2answers
2k views

p-adic L-functions

For modular forms, it is known that you can construct p-adic L-functions by interpolating (p-power conductor) twists of their associated L-functions at special values. Similarly, Kubota-Leopoldt's ...
6
votes
3answers
1k views

Terminology occuring in automorphic representation and relationship between them

When one tries to read about automorphic representation few terms come up more than others namely, 1.Cuspidal 2.Square Integrable 3.Absolutely Cuspidal 4.Super Cuspidal My understanding about ...
9
votes
2answers
508 views

Is there a canonical notion of “mod-l automorphic representation”?

As the title says. In particular, I am interested in the story for a general reductive group $G$, say defined over $\mathbb{Q}$. I can imagine that mod-$\ell$ (algebraic) automorphic representation ...
4
votes
2answers
433 views

Conductor of monomial forms with trivial nebentypus

Is it true that the conductor of a holomoprhic or a Maass cusp form with trivial nebentypus corresponding to a two-dimensional dihedral representation (over $\mathbb{Q}$ )is non-square-free?
5
votes
1answer
406 views

extending cusp forms

Let $E/F$ be a quadratic extension of number fields, and let $V$ be an $n$-dimensional Hermitian space over $E$. Let $\tilde{G} := GU(V)$ and $G := U(V)$. Suppose that $(\pi, V_{\pi})$ is an ...
41
votes
6answers
3k views

What are the local Langlands conjectures nowadays, for connected reductive groups over a $p$-adic field?

Let me stress that I am only interested in $p$-adic fields in this question, for reasons that will become clear later. Let me also stress that in some sense I am basically assuming that the reader ...
14
votes
2answers
1k views

Overview of automorphic representations for $SL(2)/{\mathbf{Q}}$?

In short: what does Labesse-Langlands say? Slightly more precise: what are the cuspidal automorphic representations of $SL_2(\mathbf{A}_{\mathbf{Q}})$, together with multiplicities? Let's say that I ...
22
votes
5answers
4k views

Where stands functoriality in 2009?

Robert Langlands is famous in number theory for making famous and deep conjectures about very abstract things called automorphic forms, somewhere in the 60s. There's a very interesting article by ...