Automorphic symbols and automorphic \emph{L}-values of GL(2) over imaginary quadratic fields
Abstract
We study non-vanishing modulo primes of critical values of L-functions over imaginary quadratic fields twisted by Coates--Wiles characters. For a parallel weight two Hecke eigenform over an imaginary quadratic field, we prove that a positive proportion of Coates--Wiles characters have nonzero integral L-values modulo each prime in a positive-density set. The argument constructs automorphic symbols in parabolic homology with integral coefficients and expresses the critical values through their pairings with parabolic cohomology classes. A vertical family of additive averages of these pairings recovers Fourier coefficients and force the module generated by symbols to have full rank. We transfer this full-rank property to non-vanishing modulo primes. This extends the homological strategy of Kim--Sun from classical modular curves to arithmetic orbifolds, while addressing the unit obstructions specific to the imaginary quadratic setting.
Related
- Maximal unramified 3-extensions of imaginary quadratic fields and \textrm SL\sb 2(\mathbb Z\sb 3)
- p-class groups in the cyclotomic ℤp-extension of imaginary quadratic fields
- The enhanced polylogarithm conjecture in weight two for imaginary quadratic fields
- Zeta elements for p-ordinary modular forms over imaginary quadratic fields and applications
- Imaginary quadratic fields of class number at most 2000
BibTeX
@misc{kwon2026automorphic,
title = {{Automorphic symbols and automorphic \emphL-values of GL(2) over imaginary quadratic fields}},
author = {Jaesung Kwon},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11210},
eprint = {2610.11210},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11210},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Automorphic symbols and automorphic \emph{L}-values of GL(2) over imaginary quadratic fields
AU - Jaesung Kwon
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11210
UR - https://arxiv.org/abs/2610.11210
AB - We study non-vanishing modulo primes of critical values of L-functions over imaginary quadratic fields twisted by Coates--Wiles characters. For a parallel weight two Hecke eigenform over an imaginary quadratic field, we prove that a positive proportion of Coates--Wiles characters have nonzero integral L-values modulo each prime in a positive-density set. The argument constructs automorphic symbols in parabolic homology with integral coefficients and expresses the critical values through their pairings with parabolic cohomology classes. A vertical family of additive averages of these pairings recovers Fourier coefficients and force the module generated by symbols to have full rank. We transfer this full-rank property to non-vanishing modulo primes. This extends the homological strategy of Kim--Sun from classical modular curves to arithmetic orbifolds, while addressing the unit obstructions specific to the imaginary quadratic setting.
ER -