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arXiv · 2602.14845

Relative Character Asymptotics Beyond Stability for $\mathrm{PGL}_2 \times \mathrm{GL}_1$

Trajan Hammonds
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Abstract

The asymptotics of relative characters for real Lie groups were studied for representations $(\pi, \sigma)$ arising from Gan-Gross-Prasad pairs $(G,H)$ by Nelson and Venkatesh. They successfully compute the asymptotics of relative characters whenever the conductor of the associated Rankin-Selberg $L$-function $L(\pi \boxtimes \sigma^\vee)$ lies in a stable locus, i.e. away from conductor dropping. In this paper, we express asymptotics for relative characters in the non-archimedean setting for $(\mathrm{PGL}_2, \mathrm{GL}_1)$. The key new innovation is that our method overcomes the stability hypothesis and allows for significant conductor dropping.

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Trajan Hammonds. 2026-02-16. Relative Character Asymptotics Beyond Stability for $\mathrm{PGL}_2 \times \mathrm{GL}_1$. https://arxiv.org/abs/2602.14845

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