Davide Lombardo is a number theorist based at the University of Pisa, Italy. He studied at the Scuola Normale Superiore (Italy) and the Université Paris-Saclay (France), and obtained his PhD from the latter in 2016. After a short post-doctoral position in Hannover (Germany), he joined the Department of Mathematics in Pisa in 2017, first as a researcher and then – since 2022 – as an associate professor. His work focuses mainly on the geometric and arithmetic interplay between Galois representations and abelian varieties. L-functions are a fundamental tool in this area, providing a bridge between arithmetic and analytic objects.

• - Part I Classical 𝑳-functions and applications.- 1. What is an 𝑳-function?.- 2. The Prime Number Theorem.- 3. Review of Algebraic Number Theory.- 4. A Primer of Representation Theory.- 5. The 𝑳-Function of a Complex Galois Representation.- 6. Dirichlet’s Theorem on Arithmetic Progressions.- 7. The Chebotarev Density Theorem.- Part II Prerequisites for Tate’s Thesis.- 8. The Haar Measure.- 9. Abstract Fourier Analysis.- 10. Review of Local Fields.- 11. Restricted Direct Products.- Part III Tate’s Thesis.- 12. The Local Theory.- 13. The Global Theory.- 14. Hecke 𝑳-Functions.- 15. Recovering the Classical Theory.- 16. An Extended Example: the 𝑳-Function of a CM Elliptic Curve.

Davide Lombardo is a number theorist based at the University of Pisa, Italy. He studied at the Scuola Normale Superiore (Italy) and the Université Paris-Saclay (France), and obtained his PhD from the latter in 2016. After a short post-doctoral position in Hannover (Germany), he joined the Department of Mathematics in Pisa in 2017, first as a researcher and then – since 2022 – as an associate professor. His work focuses mainly on the geometric and arithmetic interplay between Galois representations and abelian varieties. L-functions are a fundamental tool in this area, providing a bridge between arithmetic and analytic objects.

• - Part I Classical 𝑳-functions and applications.- 1. What is an 𝑳-function?.- 2. The Prime Number Theorem.- 3. Review of Algebraic Number Theory.- 4. A Primer of Representation Theory.- 5. The 𝑳-Function of a Complex Galois Representation.- 6. Dirichlet’s Theorem on Arithmetic Progressions.- 7. The Chebotarev Density Theorem.- Part II Prerequisites for Tate’s Thesis.- 8. The Haar Measure.- 9. Abstract Fourier Analysis.- 10. Review of Local Fields.- 11. Restricted Direct Products.- Part III Tate’s Thesis.- 12. The Local Theory.- 13. The Global Theory.- 14. Hecke 𝑳-Functions.- 15. Recovering the Classical Theory.- 16. An Extended Example: the 𝑳-Function of a CM Elliptic Curve.

Davide Lombardo is a number theorist based at the University of Pisa, Italy. He studied at the Scuola Normale Superiore (Italy) and the Université Paris-Saclay (France), and obtained his PhD from the latter in 2016. After a short post-doctoral position in Hannover (Germany), he joined the Department of Mathematics in Pisa in 2017, first as a researcher and then – since 2022 – as an associate professor. His work focuses mainly on the geometric and arithmetic interplay between Galois representations and abelian varieties. L-functions are a fundamental tool in this area, providing a bridge between arithmetic and analytic objects.

• - Part I Classical 𝑳-functions and applications.- 1. What is an 𝑳-function?.- 2. The Prime Number Theorem.- 3. Review of Algebraic Number Theory.- 4. A Primer of Representation Theory.- 5. The 𝑳-Function of a Complex Galois Representation.- 6. Dirichlet’s Theorem on Arithmetic Progressions.- 7. The Chebotarev Density Theorem.- Part II Prerequisites for Tate’s Thesis.- 8. The Haar Measure.- 9. Abstract Fourier Analysis.- 10. Review of Local Fields.- 11. Restricted Direct Products.- Part III Tate’s Thesis.- 12. The Local Theory.- 13. The Global Theory.- 14. Hecke 𝑳-Functions.- 15. Recovering the Classical Theory.- 16. An Extended Example: the 𝑳-Function of a CM Elliptic Curve.