Suppose $f$ is a modular form of weight $k \ge 2$.

It's "well-known" that there are "periods" $\Omega_-$ and $\Omega_+ \in \mathbb{C}$, such that the $L$-values $L(f, \chi, j)$, for $\chi$ a Dirichlet character and $1 \le j \le k-1$, are $(2\pi i)^j$ times an algebraic multiple of one of $\Omega_{\pm}$.

What is the correct statement of this "well known" result? Colmez claims in his Bourbaki seminar on $p$-adic BSD (here, Section 3.1.3) that

$$L(f, \chi, j) \in \begin{cases} \overline{\mathbb{Q}} \cdot (2\pi i)^j \Omega_+ & \text{if $\chi(-1) = (-1)^j$} \\ \overline{\mathbb{Q}} \cdot (2 \pi i)^j\Omega_- & \text{if $\chi(-1) =-(-1)^{j}$.}\end{cases} $$

On the other hand, Vatsal claims in Theorem 0.1 of this paper (in which he defines canonical choices for the periods $\Omega_{\pm}$ up to $p$-adic units for a given prime $p$) that the conditions should be

$$L(f, \chi, j) \in \begin{cases} \overline{\mathbb{Q}} \cdot (2\pi i)^j \Omega_+ & \text{if $\chi(-1) = 1$} \\ \overline{\mathbb{Q}} \cdot (2 \pi i)^j\Omega_- & \text{if $\chi(-1) =-1$.}\end{cases} $$

Vatsal's statement is repeated verbatim in the MathSciNet review of the paper (here if you have an institutional subscription).

These can't both be right, surely? If $k = 2$ then the only possibility for $j$ is $1$, so the two claims can be reconciled by simply switching the labelling of the two periods; but for $k \ge 3$ then the only way they can both hold is if $\Omega_+ / \Omega_-$ is algebraic, which I gather isn't expected to be the case unless $f$ is CM.

Colmez doesn't give a reference, while Vatsal cites a 1976 paper of Shimura, which I haven't been able to get hold of to check the exact statement.

Which of these two statements are correct? Or are they both correct but I've misunderstood them?


Check out Manin's paper Periods of parabolic forms and $p$-adic Hecke series. He only deals with level 1 and even weight, but this is enough to know that Vatsal's statement is not the right one (though not enough to fully confirm Colmez's statement, though it's true as given in Theorem 1 of Shimura's paper you can't get your hands on; you should also be able to dig it out of Mazur–Tate–Teitelbaum or Emerton–Pollack–Weston, which generalizes Vatsal). In particular, see section 1.2: The Periods Theorem of Manin's paper.

Update: In fact, the statement in the form you're looking for is written down in Theorem 1(ii) of Shimura's On the periods of modular forms (doi:10.1007/BF01391466), where he even tells you what you can take as $\Omega_\pm$ (and the result in the weight 2 case in Shimura's aforementioned 1976 paper was still conditional on a result proved in this 1977 paper).

  • $\begingroup$ Re. update: Those periods don't seem very canonical, depending as they do on an arbitrary choice of twisting character. $\endgroup$ – David Hansen Aug 24 '11 at 6:39
  • $\begingroup$ They certainly are not! That's a big problem with the periods of modular forms as opposed to those of elliptic curves. There has however been progress in writing down periods that canonically vary in $p$-adic families, so that one can take about congruences and $\mu$-invariants of $p$-adic $L$-functions. This is what the papers of Vatsal and Emerton–Pollack–Weston discuss. $\endgroup$ – Rob Harron Aug 24 '11 at 14:42

Vatsal actually would write $(-2\pi i)^j$, so both his and Colmez's are the same correct statement.

  • $\begingroup$ I don't follow: $\Omega_+$ is not the negative of $\Omega_-$. $\endgroup$ – David Loeffler Jun 18 '11 at 16:28

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