# Drinfeld modules

The seminar was held during the winter semester 2015. Its aim was to understand the results of Lenny Taelman on special values of Goss L-functions attached to Drinfeld modules.

## Talks

23-09-15

Organizational meeting

30-09-15

Class number formula for global fields

07-10-15

Introduction to Drinfeld modules

14-10-15

Goss L-functions I

21-10-15

Goss L-functions II
(notes for both talks: pdf)

28-10-15

Explicit class field theory

18-11-15

Irrationality and transcendence of special values

25-11-15

Class number formula I

16-12-15

Class number formula II

## Organizers

Pasha Solomatin <p.solomatin@math.leidenuniv.nl>
Maxim Mornev <m.mornev@math.leidenuniv.nl>

## Approximate program

Introduction: class number formula for global fields, special values of classical L-functions and their irrationality.

Drinfeld modules and their analytic uniformization.

Goss L-functions.

Explicit class field theory.

Taelman's class number formula.

Irrationality of special values in positive characetristic.

Uniformization via Drinfeld shtukas.

## References

G. Anderson. $t$-Motives. Duke Math. J. 53 (1986), no. 2, 457–502.

P. Deligne, D. Husemöller. Survey of Drinfeld modules, in: Current Trends in Arithmetical Algebraic Geometry (Arcata, Calif., 1985). Contemporary mathematics 67 (1987), 25–91.

V. Drinfeld. Elliptic modules. Math. USSR-Sb. 23 (1974), no. 4, 561–592.

V. Drinfeld. Elliptic modules, II. Math. USSR-Sb. 31 (1977), no. 2, 159–170.

D. Goss. Basic structures of function field arithmetic. Springer (1998)

D. Goss. $v$-adic zeta functions, $L$-series and measures for function fields. Invent. Math. 55 (1979), no. 2, 107–119.

D. Goss. On a new type of $L$-function for algebraic curves over finite fields. Pacific J. Math. 105 (1983), no. 1, 143–181.

B. Lutes. Special values of the Goss $L$-function and special polynomials. PhD thesis. Texas A&M (2010)

R. Perkins. What is Anderson's log-algebraicity? (2013)

R. Perkins. On special values of Pellarin's $L$-series. PhD thesis. OSU (2013)

L. Taelman. A Dirichlet unit theorem for Drinfeld modules. Math. Ann. 348 (2010), no. 4, 899–907.

L. Taelman. Special $L$-values of Drinfeld modules. Ann. Math. 175 (2012), no. 1, 369–391.

L. Taelman. The Carlitz shtuka. J. Number Theory 131 (2011), no. 3, 410–418.

L. Taelman. Sheaves and functions modulo $p$, lectures on the Woods-Hole trace formula. LMS Lecture Note Series, volume 429 (2015)

D. Thakur. On characteristic $p$ zeta functions. Compositio Math. 99 (1995), no. 3, 231–247.

D. Zywina. Explicit class field theory for global function fields. (2011)