In this seminar, Dirichlet L-series are investigated from an adelic point of view. We will review L-series and some applications, locally compact (abelian) groups and Haar measures, Pontryagin duality, the Fourier transform, p-adic numbers, the ring of adeles, and finally construct Dirichlet L-series in the adelic setting.

The study of polynomial equations has deep historical roots, exemplified by Fermat's Last Theorem, which highlights the difficulty of determining solvability. For homogeneous quadratic polynomials, however, a 'local-global principle' provides a clear criterion for identifying non-trivial rational roots. This seminar will explore this principle and some implications in number theory.