Let $f \in S_2(\Gamma_1(N))$ be an eigenform. By a theorem of Shimura, there are associated "periods" $\Omega_f^\pm$ such that, after normalizing by these periods, the L-function associated to $f$ takes algebraic values.

If $\chi$ is a Dirichlet character, one can form the twist $f_\chi$ of $f$.

How are the periods $\Omega^\pm_f$ and $\Omega^\pm_{f_\chi}$ related? In particular, are they algebraic multiples of each other?


By a famous theorem of Manin, one can define $\Omega^{\pm}$ such that $L(f\otimes\chi,j)\in \Omega^{\epsilon}_{f}\mathbb Q$ with $\chi(-1)(-1)^{j}=\epsilon$. So the period depends on $\chi$ only insofar as you need to know $\chi(-1)$ to determine if you should choose $\Omega_f^{+}$ or $\Omega_f^{-}$.

This result is the key property allowing one to construct the $p$-adic $L$-function of a modular form.

  • $\begingroup$ Or maybe it's already in Shimura, I don't remember now. $\endgroup$ – Olivier Sep 7 '13 at 19:21

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.