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Results tagged with automorphic-forms
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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.
5
votes
Langlands' original observation about Ramanujan conjecture
Aurel has already given a good answer to this, and as I note in my comment to his answer one needs only the holomorphy of $L(s,\pi, \text{sym}^k)$ in the region Re$(s)>1$ to obtain the Ramanujan conje …
- 43.7k
9
votes
Classical Lower Bound of L(1) assuming GRH
I'll assume that the degree of the $L$-function is fixed, and that the conductor is going to infinity. (If the degree is growing, then it is not clear what the right notion of analytic conductor is, …
- 43.7k
15
votes
Using Eichler-Selberg trace formula to compute dimension of modular forms?
Yes; here's an elaboration of my comments.
The Eichler-Selberg formula (see Theorem 2 of Zagier's http://people.mpim-bonn.mpg.de/zagier/files/google/es-trace-sl2z/fulltext.pdf which appears in Lang's …
- 43.7k
13
votes
How many integer solutions of $a^2+b^2=c^2+d^2+n$ are there?
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). Th …
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