Base-change Bianchi modular forms

For every classical elliptic newform $f$ and every imaginary quadratic field $K$ one can define the base-change $F$ of $f$ to $K$.

Usually the base-change of an elliptic newform $f$ is a cuspidal Bianchi newform, but there is one exceptional situation, when $f$ is a CM form whose character is that of the extension $K/\Q$. In this case the base-change still exists but is not cuspidal. One explanation for this is that the automorphic representation attached to $f$ is always irreducible, but after base-change to $K$ the representation splits as a sum of two Hecke characters over $K$, whereas the representations attached to Bianchi (cuspidal) newforms are irreducible.

The Hecke eigenvalues of the base-change $F$ of $f$ to $K$ may be determined easily from those of the original elliptic newform $f$. For every prime $p$ not dividing the level of $f$, if the eigenvalue of $T_p$ on $f$ is $a_p$ then the eigenvalue(s) $a_{\frak{p}}$ for ${\frak{p}}\mid p$ of its base-change are given by the following (where $k$ is the weight of $f$):

• if $p$ splits in $K$ as $p\mathcal{O}_K=\frak{p}\overline{\frak{p}}$ then $a_{\frak{p}}=a_{\overline{\frak{p}}}=a_p$;
• if $p$ ramifies in $K$ as $p\mathcal{O}_K=\frak{p}^2$ then $a_{\frak{p}}=a_p$;
• if $p$ is inert in $K$ with $p\mathcal{O}_K=\frak{p}$ then $a_{\frak{p}}=a_p^2-2p^{k-1}$.

For an explanation of the last formula, note that $a_p=\alpha+\beta$ where $\alpha,\beta$ are the associated eigenvalues of Frobenius, so that $\alpha\beta=p^{k-1}$, while $a_{\frak{p}}=\alpha^2+\beta^2$.

One consequence of this formula is the following fact: it is possible for the Hecke field of the base-change of $f$ to be of smaller degree than that of $f$ itself. For example, it is possible to have a classical newform with quadratic Hecke field whose base-change has $\Q$ as its Hecke field.

Base-change as defined here is a special case of a more general definition of base change for GL(2) modular forms.