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Results tagged with automorphic-forms
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user 2284
An automorphic form is a well-behaved function from a topological group $G$ to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup $\Gamma \subset G$ of the topological group. Automorphic forms are a generalization of the idea of periodic functions in Euclidean space to general topological groups.
4
votes
Are there motives which do not, or should not, show up in the cohomology of any Shimura vari...
In their 2017 preprint
On subquotients of the étale cohomology of Shimura varieties
C.Johannsson and J.Thorne have made substantial progresses towards question 4. In particular, they show that under …
- 10.9k
8
votes
Accepted
Proving automorphy of the Galois representations of number fields without considering the re...
The canonical answer to that question is certainly the world of so called converse theorems, whose basic ideas go back to Hecke's remark that an holomorphic $L$-function satisfying a suitable function …
- 13k
12
votes
To what extent are modular parametrizations expected to generalize?
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 sp …
- 10.9k
5
votes
Accepted
Matsushima-Murakami Isomorphism for $L^2$-cohomology
Morally yes, by
Franke, Jens Harmonic analysis in weighted $L_{2}$-spaces. Ann. Sci. École Norm. Sup. (4) 31 (1998), no. 2, 181--279
See especially theorem 4 (and note that if you want an equivarian …
- 10.9k
2
votes
The historical development of automorphic geometry
A common answer to question 1 is to mention the entries of Gauss's diary from 1814, including famously (but not restricted to) the last one, in which he studies some properties of biquadratic reciproc …
- 10.9k
4
votes
Accepted
To which automorphic forms/rep's over a function field can we associate a Galois representat...
Let $X$ be a smooth projective geometrically irreducible curve over $\mathbb F_{q}$ a finite field and $F$ its global field. Let $\mathbf G/F$ be a split connected reductive group and $\widehat{\mathb …
- 10.9k
7
votes
modular form Fourier coefficients and associated automorphic representation
No.
A supercuspidal representation, a Steinberg twisted by a ramified character and a principal series ramified at both characters at $p$ will all have zero $a_{p}$. A reference for this is Jacquet- …
- 10.9k
5
votes
Langlands in dimension 2: the Yoshida conjecture
And so it turns out that I was in the audience of a seminar talk just today on this very subject. The opinion I expressed in comments is apparently not too far from the truth: V.Pilloni and B.Stroh no …
- 10.9k
4
votes
Are there motives which do not, or should not, show up in the cohomology of any Shimura vari...
Let me expand and hopefully clarify my first comment about the more specific question of whether the cohomology of a modular elliptic curve with everywhere good reduction shows up in the cohomology of …
- 10.9k
7
votes
p-adic L-functions
The following is more a long comment than an answer per se.
One thing to keep in mind when discussing $p$-adic $L$-functions is that to a given algebraic automorphic representation $\pi$ or Galois re …
- 10.9k