An $L$-function is a function defined additively by a Dirichlet series with a multiplicative Euler product.
The initial work of Dirichlet was generalized to number fields by Hecke and given an adelic interpretation by Tate
which paved the way to move from $GL(1)$ to higher degree $L$ functions associated to automorphic forms of $GL(n)$ for a general n .
Automorphic $L$ functions are essentially analytic objects and allow translations between arithmetic and analysis.
We recall the legendary correspondence in the case of the
Riemann-$zeta$ function, the prototype of higher $L$-functions both analytic and arithmetic-algebraic,
between an arithmetic statement on the distribution of primes and an analytic one on the distribution of its zeros.
Now there are many more parameters to explore.
For example, bounding twisted automorphic forms in terms of conductors help obtain the asymptotics of representations
of totally positive integers by ternary quadratic forms.
The aim of this school is to prepare the participants to appreciate the contemporary theory of automorphic representations, their $L$-functions
and their surprising links with diverse fields like Fourier analysis, algebraic geometry and theoretical physics.
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Scientific program is available on the local website of the school:
https://www.mathconf.org/alfiitr2024
Official language of the school: english