An analytic class number type formula for PSL2(Z)

Apparaît dans la collection : Summer School 2017 - Arakelov Geometry and diophantine applications

For any Fuchsian subgroup Γ⊂PSL2(R) of the first kind, Selberg introduced the Selberg zeta function in analogy to the Riemann zeta function using the lengths of simple closed geodesics on Γ∖H instead of prime numbers. In this talk, we report on a formula that determines the special value at s=1 of the derivative of the Selberg zeta function for Γ=PSL2(Z). This formula is obtained as an application of a generalized Riemann-Roch isometry for the trivial sheaf on ¯¯¯¯¯¯¯¯¯¯¯Γ∖H, equipped with the Poincaré metric. This is joint work with Gerard Freixas.