On the local Bump–Friedberg L-function II

Abstract

Let F be a p-adic field with residue field of cardinality q. To each irreducible representation of GL(n, F ), we attach a local Euler factor L BF (q −s , q −t , π) via the Rankin-Selberg method, and show that it is equal to the expected factor L(

A class number formula for Picard modular surfaces

“…Thus the right-hand side of ( 51) is E(g, Φ N , s) by Proposition 5.4, which yields (50). Claim (1) is just the Kronecker limit formula given in [37, §20.4] combined with the functional equation E(g, s) = E(g, 1 − s) of the Eisenstein series.…”

“…When π is unramified at p, we can show that the GCD of the local integral is equal to L p (π p × ν 1,p , Std, s), which was computed explicitly in Section 3.4. We will deduce this from special cases of far more general results of Miyauchi [51] if p is inert and Matringe [50] if p is split. We remark that the construction of Matringe differs from ours, since our group GU(J)(Q p ) is isomorphic to GL 3 ×G m rather than GL 3 .…”

“…If p is split, we may deduce this from the work of Matringe [50] as follows. Suppose that π 0 is any irreducible admissible generic representation of G 0 (Q p ) = GL 3 (Q p ) and let W 0 (g v ) be any Whittaker function for π 0 on G 0 .…”

“…However, many essential properties have been established by Baruch [4] and Miyauchi [53,54,52,51], which we leverage in Sections 7.1 and 8.3 below. It turns out that although the Bump-Friedberg integral [13] yields a different local L-function for GL 3 than ours at split places, the detailed work of Matringe [49,50] and a computation of Miyauchi-Yamauchi [56] may be used to deduce facts about our construction as well.…”

Test vectors for non‐Archimedean Godement–Jacquet zeta integrals

“…Alternatively, one can write this as an integral over (N n (F ) ∩ H m,m (F ))\H m,m (F ) for n = 2m and as an integral over (N n (F ) ∩ H m+1,m (F ))\H m+1,m (F ) for n = 2m + 1 (cf. [Mat15,Mat17]). The Bump-Friedberg integrals converges absolutely for (s) sufficiently large and extends meromorphically to the entire complex plane.…”