Automorphic symbols and automorphic \emph{L}-values of GL(2) over imaginary quadratic fields

Jaesung Kwon

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arXiv · 2610.11210

Automorphic symbols and automorphic \emph{L}-values of GL(2) over imaginary quadratic fields

Jaesung Kwon

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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.

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BibTeX RIS

Jaesung Kwon. 2026-10-08. Automorphic symbols and automorphic \emph{L}-values of GL(2) over imaginary quadratic fields. https://arxiv.org/abs/2610.11210

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