## Tesi etd-09012017-175323 |

Thesis type

Tesi di laurea magistrale

Author

PENNETTA, GIORDANO

URN

etd-09012017-175323

Title

Automorphic representations and associated L-functions

Struttura

MATEMATICA

Corso di studi

MATEMATICA

Supervisors

**relatore**Prof. Maffei, Andrea

Parole chiave

- automorphic representation
- automorphic form

Data inizio appello

22/09/2017;

Consultabilità

Completa

Riassunto analitico

This thesis gives an introduction to the theory of automorphic forms and automorphic representations. Automorphic forms are smooth functions defined on the space of invertibles 2x2 matrices with coefficients in the adele ring A of a global field F, and automorphic representations are irreducible admissible representations of GL(2, A) on the space of automorphic forms. In particular in the first part we will see how to express an admissible representation of GL(2, A) as a product of local admissible representation, where with local we mean a representation of GL(2) over a completion of F.

Then we will use the existence and the uniqueness of the Whittaker model of an automorphic cuspidal representation to obtain the Multiplicity one Theorem: this asserts that two automorphic cuspidal representations of GL(2, A) such that almost all local factors are isomorphic, actually are equal.

The importance of the Whittaker model will be clear also in the last part, where we will associate a L-function to an automorphic cuspidal representation, and we will see that this function satisfies a functional equation.

Then we will use the existence and the uniqueness of the Whittaker model of an automorphic cuspidal representation to obtain the Multiplicity one Theorem: this asserts that two automorphic cuspidal representations of GL(2, A) such that almost all local factors are isomorphic, actually are equal.

The importance of the Whittaker model will be clear also in the last part, where we will associate a L-function to an automorphic cuspidal representation, and we will see that this function satisfies a functional equation.

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