In this course we will describe in numerous examples how methods from l-adic cohomology as developed by Grothendieck, Deligne, Katz (and others) interact with methods from analytic number theory (the study of the distribution of prime numbers, size methods or the analytic theory of modular forms and associated L-functions).

 

Although the major results from  l-adic cohomology will be treated as a black box (for instance  Deligne's main theorem from « Weil II ») we explain in concrete details how to get ones hands on the «  trace function » attached to an l-adic sheaf on the affine line (which is simply the Frobenius trace function of the underlying Galois representation) and how the geometric properties of the sheaf impact its behavior as a function.

 

Basic properties on traces functions and their associated sheaves:

 

Sheaves as Galois representations over function fields.
The Frobenius trace functions.
The geometric representation and its invariants: local and global monodromy groups.
The Grothendieck-Ogg-Shafarevich and the Grothendieck-Lefshetz trace formula.
Deligne main theorem from "Weil ll" and some of its consequences:
- Correlation of trace functions
- Diophantine criterion for irreducibility

 

Examples of trace functions and their associated sheaves

 

Artin-Schreier and Kummer sheaves.
Kloosterman sheaves.
Pull-backs, tensor products, Fourier transform.

 

Trace functions in everyday life:

 

-Trace functions and codes.
-Sato-Tate equidistribution laws.
-Trace functions over intervals: the Polya-Vinogradov method and beyond.
-Primes and divisor functions in large arithmetic progressions.
-Trace functions vs. Modular forms.
-Trace functions vs. the primes.