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Nonvanishing of special L-values of cusp forms on GL(2) with totally split prime power twists: without the class number hypothesis
Abstract
Let F be a number field and f a normalized cuspidal Hecke eigenform on GL(2) over F, with k/2 as the central critical point of its L-function L(f,s). Let p be a totally split prime ideal of F lying above an odd prime p and coprime to the level ideal of f. In this article, we show that L(f⊗ϕ, k/2) does not vanish for sufficiently ramified Hecke characters of F of p-power order and p-power conductor. Our result improves previous works of Jaesung Kwon and Hae-Sang Sun, in which the additional hypothesis that p does not divide the class number of F is required.
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BibTeX
@misc{xu2026nonvanishing,
title = {{Nonvanishing of special $L$-values of cusp forms on $\mathrmGL(2)$ with totally split prime power twists: without the class number hypothesis}},
author = {Ruichen Xu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03976},
eprint = {2610.03976},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03976},
}
RIS
TY - UNPB
TI - Nonvanishing of special L-values of cusp forms on GL(2) with totally split prime power twists: without the class number hypothesis
AU - Ruichen Xu
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03976
UR - https://arxiv.org/abs/2610.03976
AB - Let F be a number field and f a normalized cuspidal Hecke eigenform on GL(2) over F, with k/2 as the central critical point of its L-function L(f,s). Let p be a totally split prime ideal of F lying above an odd prime p and coprime to the level ideal of f. In this article, we show that L(f⊗ϕ, k/2) does not vanish for sufficiently ramified Hecke characters of F of p-power order and p-power conductor. Our result improves previous works of Jaesung Kwon and Hae-Sang Sun, in which the additional hypothesis that p does not divide the class number of F is required.
ER -