Mazur-Tate-Teitelbaum p-adic L-function

Question

I start to read the paper "On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer" by Mazur, Tate and Teitelbaum (referred as [MTT]) to learn how we can associate p-adic L-function to certain eigenforms.

For an eigenform $f$ (with certain conditions) and a choice of the root $\alpha$, the authors define the $V_f$-valued measure $\mu_{f,\alpha}$ in page 13. Here $V_f=C_p\otimes_\bar{Q}L_f\bar{Q}$ is a finite dimensional $C_p$-vector space defined in line 5 of page 13. And $L_f\bar{Q}$ (I think) is the $\bar{Q}$-vector subspace of $C$ generated by elements of $L_f$, where $L_f$ is defined right before the proposition in page 6.

Finally, they define the p-adic L-function $$L_p(f,\alpha,\chi,s):=L_p(f,\alpha,\chi\chi_s)$$ as in page 19.

Apparently, this function has values in the $C_p$-vector space $V_f$. But in the rest of the paper, it seems to me that they treat this as a $C_p$-valued function. On the other hand, other references, such as Greenberg-Stevens' paper in Inventiones 1993, and the earlier paper ``Arithmetic of Weil Curvess'' by Mazur--Swinnerton-Dyer use the two periods $\Omega_f^{\pm}$ to get out two measures $\mu_f^{\pm}$. Then use them to define a $C_p$-valued function.

Question: do we need to change the functions in [MTT] into $C_p$-valued functions, say, by using the above periods as in Greenberg-Stevens?

Answer 1

In order to get $\mathbb{C}_p$-valued functions, you need to choose a basis for the vector space $V_f$. If $f$ corresponds to an elliptic curve, there is a reasonably canonical way of doing this (using the periods of a Neron differential), as in the paper by Mazur and Swinnerton-Dyer. If $f$ is a more general modular form it is much less clear what the "right" normalisation is for a basis of $V_f$. This is an important issue, though, because one needs to fix such a normalisation to make sense of whether or not two L-values are congruent modulo a prime. The question has been studied in detail by Vinayak Vatsal in his paper "Canonical periods and congruence formulae" (Duke Math Journal 98 no. 2, 1999), which determines a canonical normalisation up to p-adic units.