## Kyoto Journal of Mathematics

### On mod $p$ nonvanishing of special values of $L$-functions associated with cusp forms on $\operatorname{GL}_{2}$ over imaginary quadratic fields

Kenichi Namikawa

#### Abstract

Let $f$ be a cusp form on $\operatorname{GL}_{2}$ over an imaginary quadratic field $F$ of class number 1, and let $p$ be an odd prime which satisfies some mild conditions. Then we show the existence of a finite-order Hecke character $\varphi$ of $F^{\times}_{\mathbf{A}}$ such that the algebraic part of the special value of $L$-functions of $f\otimes \varphi$ at $s=1$ is a $p$-adic unit. This is an analogous result to the result of A. Ash and G. Stevens for $\operatorname{GL}_{2}$ over the field of rationals obtained in [AS].

#### Article information

Source
Kyoto J. Math., Volume 52, Number 1 (2012), 117-140.

Dates
First available in Project Euclid: 19 February 2012

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1329684745

Digital Object Identifier
doi:10.1215/21562261-1503782

Mathematical Reviews number (MathSciNet)
MR2892770

Zentralblatt MATH identifier
1284.11087

#### Citation

Namikawa, Kenichi. On mod $p$ nonvanishing of special values of $L$ -functions associated with cusp forms on $\operatorname{GL}_{2}$ over imaginary quadratic fields. Kyoto J. Math. 52 (2012), no. 1, 117--140. doi:10.1215/21562261-1503782. https://projecteuclid.org/euclid.kjm/1329684745

#### References

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