Rankin–Selberg and relative-character nonvanishing in the mixed conductor case

Solve the remaining case in which the tempered hermitian irreducible representation \(\Pi=\Pi_n\otimes\Pi_{n+1}\) of \(G'\) has trivial central character, \(\Pi_n\) has conductor one, and \(\Pi_{n+1}\) is unramified, in order to establish the corresponding nonvanishing result for the local relative characters.

Background

The paper evaluates the Rankin–Selberg integral of opposite Whittaker newforms when both Πn\Pi_n and Πn+1\Pi_{n+1} are tempered representations with conductor one and trivial central character. This yields nonvanishing of the relevant local Rankin–Selberg period and, via a transfer identity, proves a case of the Disegni–Zhang conjecture on nonvanishing of local relative characters.

The authors explain that, among tempered hermitian representations Π=Πn⊗Πn+1\Pi=\Pi_n\otimes\Pi_{n+1} with both factors of conductor at most one, the only case not covered by the cited results and the theorem proved in the paper is the mixed-conductor situation where Πn\Pi_n has conductor one and Πn+1\Pi_{n+1} is unramified. They state that this case still involves unresolved problems.

References

In general, assume that \Pi=\Pi_n\otimes \Pi_{n+1} is a tempered hermitian irreducible representation of G' with trivial central character, such that \Pi_r (r=n, n+1) has conductor at most one. In view of Conjecture 3.10, Remark 3.11, Theorem 1.2 and Theorem \ref{DZ}, the remaining case is that \Pi_n has conductor one and \Pi_{n+1} is unramified, for which some challenging problems remain to be solved.

— Rankin--Selberg integrals of opposite conductor--one newforms  (2609.11045 - Liu et al., 10 Sep 2026) in Remark following Theorem 6.1, Section 4 (Section “Nonvanishing of local relative characters”)