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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.

6 votes
1 answer
253 views

Divergence of integrals in the trace formula

I am trying to understand the following situation for $G=GL(2)$, when going from the compact trace formula to the non-compact case. The integral over $G(\mathbb{A})^1_\gamma \backslash G(\mathbb{A})^1 …
TheStudent's user avatar
8 votes
1 answer
586 views

Does the symmetric square L-function vanish at one?

Take a cuspidal automorphic representation $\pi$ of $GL(3)$ over a number field. My question is quite straightforward and can be related to this one : Can $L(1, \pi, \mathrm{sym}^2)$ be zero? If y …
TheStudent's user avatar
5 votes
0 answers
104 views

Archimedean L-factors for symplectic group

Let $\pi$ be an automorphic representation of $GSp(4)$. Provided a representation $r$ of the Langlands dual group of $GSp(4)$ (namely, the standard or the spinor one), it is possible to define a parti …
TheStudent's user avatar
6 votes
1 answer
540 views

Analogues of Hecke relations for Maass forms

If a (suitably normalised) holomorphic cusp newform has q-expansion $$f(z) = \sum_n \lambda_f(n) e(nz),$$ then we know the Hecke relations for $(mn,q)=1$, $$(\star) \qquad \lambda_f(m)\lambda_f(n) = …
TheStudent's user avatar
8 votes
1 answer
565 views

Functional equation for general number fields

When it comes to general number fields beyond $\mathbb{Q}$, the litterature is not so abundant in analytic number theory. For instance over $\mathbb{Q}$, for primitve Dirichlet characters modulo $q$, …
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4 votes
0 answers
133 views

Plancherel measure and dimension

I would like to understand what the Plancherel measure is on the dual of a compact group, in more explicit terms than with an implicit definition. Indeed, to the extent of my knowledge it is defined a …
TheStudent's user avatar
7 votes
0 answers
127 views

Bound for orbital integrals

Let $F$ be a number field, and $G$ be the group of units of a quaternion algebra $D$ over $F$. At a certain ramified place $v$, for $\gamma_v \in G(F_v)$, could we bound the orbital integral $$\mathca …
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2 votes
0 answers
100 views

Function equation over general number fields

Let $\chi$ be a Hecke character on a number field $k$, where could I find a precise reference for the function equation of the $GL(1)$ L-functions $$L(s, \chi)?$$ I only find references for the case …
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1 vote
0 answers
50 views

Mean value estimates for general number fields

Results are known in many different cases to bound powers of L-functions on average over a wide enough family. I am interested in results for general number fields, not only for the rationals, for pow …
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