Drinfeld modules

The Carlitz module

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.



Organizational meeting


Class number formula for global fields


Introduction to Drinfeld modules


Goss L-functions I


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


Explicit class field theory


Irrationality and transcendence of special values


Class number formula I


Class number formula II


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

The organizers of the seminar

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.


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

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

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. external link

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

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

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

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

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

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

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

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

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

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