Registration Deadline: January 24, 2027

Instructor: Professor Kumar Murty, University of Toronto

Course Dates: 
Mid-Semester Break: 
Lecture Time: 
Office Hours: 

Registration Fee:

• Students from our Principal Sponsoring & Affiliate Universities: Free
• Other Students: CAD$500

Capacity Limit: 

Format: 

Course Description

The transcendence (or even irrationality) of special values of $L$-functions is a central theme in number theory. Euler’s theorem that the Riemann zeta function at positive even integers is a rational multiple of a power of a single transcendental number, namely $\pi$, is an early success story. But Euler also tried to find a similar theorem about the values of the zeta function at positive odd integers and was unable to do so.

In this course, we will describe the modern perspective on this theme, namely to interpret these values as periods of certain motives (or mixed motives). We will give an exposition of the work of Kontsevich and Zagier on periods, some of the work of Francis Brown on multiple zeta values and motives, and Grothendieck's period conjecture.

Finally, we will give an exposition of recent work on some special values of Dirichlet $L$-functions, including the (apparently non-motivic) irrationality result by Calegari, Dimitrov and Tang, as well as the (motivic) work of Eskandari, Murty and Nemoto on Catalan’s constant.

Prerequisites and Evaluation

We will assume basic number theory and algebraic geometry, but will attempt to make the content as self-contained as possible. The field is just opening up now and the aim of the course will really be to get more students up to speed on these ideas. Evaluation will be based on attendance and class participation.

References

1. F. Brown, Irrationality proofs of zeta values, moduli spaces and dinner parties, Moscow Journal of Combinatorics and Number Theory, 6(2016), 102-165.
2. F. Calegari, V. Dimitrov and Y. Tang, The linear independence of $1$, $\zeta(2)$, and $L(2,_{X_{-3}})$, arXiv:2408.15403 [math.NT], 2024.
3. P. Eskandari and V. Kumar Murty, On unipotent radicals of motivic Galois groups, Algebra and Number Theory, 17(2023), 165-214.
4. P. Eskandari, V. Kumar Murty and Y. Nemoto, Mixed motives and the Catalan constant, arXiv:2510.20648 [math.NT], 2025.
5. M. Kontsevich and D. Zagier, Periods, IHES Publication, 2001.