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It is often noted in the literature that there are certain complex periods that allow one to normalize the modular symbol associated to a modular form in such a way that its coefficients are algebraic. This process of normalization by complex periods is regularly attributed to Shimura, though I can't seem to find a concrete reference explaining this result.

More precisely, let $ \Gamma=\Gamma_0(N)$ and fix an eigenform $f\in S_k(\Gamma)$. The modular symbol $\xi_f\in \operatorname{Hom}_{\Gamma}(\operatorname{Div}^0(\mathbb{P}^1(\mathbb{Q})),V_{k-2}(\mathbb{C}))$, where $V_{k-2}(\mathbb{C})$ is the space of homogeneous polynomials with complex coefficients of degree $k-2$, attached to $f$ is defined by $$ \xi_f(\{r\}-\{s\})=2\pi i \int_s^r f(z)(zX+Y)^{k-2}dz. $$ One can expand this into a homogeneous polynomial $\sum_{j=0}^{k-2} c_jX^jY^{k-2-j}$ where $c_j=\binom{k-2}{j}2\pi i \int_s^rf(z)z^jdz$. The matrix $\begin{pmatrix} -1 &0\\ 0&1\end{pmatrix}$ normalizes $\Gamma$, so the modular symbols come equipped with an involution, and hence there is a unique eigenspace decomposition $\xi_f=\xi_f^++\xi_f^-$, with $\xi^\pm$ in the $\pm 1$-eigenspace.

The following theorem is stated in the literature (see, for example, [Greenberg-Stevens, 3.5.4], [Bertolini-Darmon,1.1], or [Pollack-Weston,page 7]).

Theorem. There exists complex numbers $\Omega_f^\pm$ such that $\xi_f^\pm/\Omega_f^\pm$ takes values in $V_{k-2}(K_f)$, where $K_f$ is the number field generated by the Fourier coefficients of $f$.

Greenberg-Stevens cite this 1977 paper of Shimura, Pollack-Weston cite Shimura's book on automorphic functions, and the Bertolini-Darmon does not give a reference. I could not find anything helpful in Shimura's automorphic function book, but I think theorem 1 from the 1977 paper is probably what we want. For simplicity, I state it below in the case where $f$ has rational coefficients.

Theorem. (Shimura, Theorem 1) Fix a primitive Dirichlet character $\chi$. There exist complex numbers $u_f^\pm$ such that $$ \frac{L(f_\chi,j)}{u_f^\epsilon\tau(\chi)(2\pi i)^j}\in K_fK_\chi $$ where $0< j< k$, $\epsilon$ is the sign of $\chi(-1)(-1)^j$, $\tau(\chi)$ is the classical Gauss sum, and $L(f_\chi,s)=\sum\chi(n)a_nn^{-s}$ is the $L$-function of $f$ twisted by $\chi$.

In fact, Shimura gives precise (though rather noncanonical) descriptions of these periods $u_f^\pm$: they are essentially the value of the $L$-function at $k-1$.

I would like to know how the first theorem stated above follows from this theorem 1 of Shimura.

It seems like a nontrivial exercise, or perhaps I am just having some trouble connecting the dots. I would also be content to see a reference which outlines a proof of the first theorem above.

My thoughts are roughly the following. With the notation as above, let $m$ be the conductor of $\chi$. I know that (see [Mazur-Tate-Teitelbaum, 8.6], for example) one has the following connection between coefficients of the modular symbols and special values of $L$-functions $$ \frac{j!}{(-2\pi i)^{j+1}}\frac{m^{j+1}}{\tau(\bar \chi)}L(f_{\bar\chi},j+1) =\sum_{a\in (\mathbb{Z}/m\mathbb{Z})^\times}\chi(a)\int_{-a/m}^{i\infty}f(z)(mz+a)^j dz, $$ for $0\leq j \leq k-2$. This tells us, for example, that certain weighted sums of the coefficients of $\xi_f(\{\infty\}-\{-a/m\})$ can be scaled to be algebraic. Even more, after writing down the symbols $\xi_f^\pm$, I can find periods $\Omega_f^\pm$ such that, roughly speaking, $$ \frac{1}{\Omega_f^\pm}\sum\chi(a)(\text{$j$th coefficient of $\xi_f^\pm(\{\infty\}-\{a/m\}$}) ) $$ is algebraic, but again, this only tells me that (a) certain weighted sums of the coefficients are algebraic, and (b) only gives information about the modular symbol evaluated at $\{\infty\}-\{a/m\}$, which as far as I can tell, is not the generality needed for the first theorem above.

(I posted this question on MSE a few days ago, but did not have much luck there. I hope re-posting it here is not too much of a faux pas.)

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    $\begingroup$ For (b): The path from $r$ to $s$ can be split up into the path from $r$ to $i\infty$ and the path from $i\infty$ to $s$. $\endgroup$ – Chris Wuthrich Jul 27 '20 at 14:26
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    $\begingroup$ For (a): If $\sum_a \chi(a) X_a = Y_\chi$ then $X_a = 1/d \sum_{\chi} \bar{\chi}(a) Y_{\chi}$. But maybe I misunderstand the actual question $\endgroup$ – Chris Wuthrich Jul 27 '20 at 14:29
  • $\begingroup$ Having ploughed through a paper of Shimura many years ago I don't know if MSE was really the right place for this question! $\endgroup$ – Yaakov Baruch Nov 5 '20 at 15:20
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You can find a proof of this theorem (the first in the OP) in Section 5.3 of the following paper by Pasol and Popa: https://arxiv.org/abs/1202.5802

The idea is to use the action of Hecke operators. More precisely, the map $f \mapsto \xi_f^{\pm}$ is Hecke-equivariant, the Hecke operators preserve the rational structures of both sides, and the eigenspaces are 1-dimensional.

This theorem could also, in principle, be deduced from Shimura's theorem (Theorem 1 in the OP), but the proof I have in mind would be very technical. The idea is to start from the formula expressing the values $L(f,\chi,j+1)$ in terms of modular symbols and then take the inverse Fourier transform. But there are many technical problems due to the fact that the Dirichlet characters are not necessarily primitive, and Shimura's formula is a priori only for primitive characters. Nevertheless, in the case of weight 2, Merel has proved a completely general formula expressing modular symbols in terms of twisted $L$-values, see the article Symboles de Manin et valeurs de fonctions $L$.

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