What is the Selberg trace formula?

The Selberg trace formula is associated with the work of Atle Selberg, who introduced the Rankin-Selberg method in 1940 for constructing and analytically continuing automorphic L-functions. Trace formulas of this type connect to various generalizations in modern algebraic geometry and number theory. Pierre Deligne notably found an example showing that Behrend's trace formula can be interpreted as a sort of the Selberg trace formula.

What we know

• The Rankin-Selberg method was introduced by Rankin in 1939 and Selberg in 1940 as a technique for constructing and analytically continuing several important examples of automorphic L-functions. [4]
• Pierre Deligne found an example that shows Behrend's trace formula may be interpreted as a sort of the Selberg trace formula. [2]
• Behrend's trace formula is a generalization of the Grothendieck-Lefschetz trace formula to a smooth algebraic stack over a finite field. [2]

Where to go next

• whoWho was Atle Selberg and what did he discover?
• how it workedHow does the Rankin-Selberg method work?
• compared withWhat is Behrend's trace formula?
• the bigger pictureWhat is the Poisson summation formula?
• how we knowHow do trace formulas relate to the Riemann hypothesis?
• an unexpected connectionHow does stacky point counting connect to geometric traces?Pierre Deligne discovered that Behrend's algebraic stack trace formula can be interpreted as a version of the geometric Selberg trace formula.

Sources

• [1]Riemann hypothesis · CC BY-SA 4.0
• [2]Behrend's trace formula · CC BY-SA 4.0
• [3]Poisson summation formula · CC BY-SA 4.0
• [4]Rankin–Selberg method · CC BY-SA 4.0

Your path

1. What is rock mass classification?
2. /What is the Selberg trace formula?