# Generalized Jacobians and modular units

Let $$X$$ be a proper algebraic smooth curve over a characteristic zero field $$k$$ and let $$J$$ be the Jacobian variety of $$X$$. Let $$K$$ be the function field of $$X$$. Assume that we are given $$n$$ distinct points $$P_1$$, ..., $$P_n$$ in $$X(k)$$ such that the image of the group of degree zero divisors supported on the $$P_i$$'s has finite image in $$J(k)$$. Let $$\tilde{J}$$ be the generalized Jacobian of $$X$$ with respect to the divisor $$D:=P_1+...+P_n$$.

Typically, I'm thinking about the case where $$X$$ is a modular curve, and the $$P_i$$'s are the cusps of $$X$$. But maybe my question has a general pure thought answer.

There is a rather non-standard description of $$\tilde{J}(k)$$, as follows (I found it in a paper of Yamazaki and Yang, Documenta Mathematica 21 (2016) 1669-1690, Rational torsion on the generalized Jacobian of a modular curve with cuspidal modulus''). Let $$Y = X \backslash \{P_1, ..., P_n\}$$, $$\text{Div}(Y)$$ be the set of $$k$$-rational divisors with support on $$Y$$. For each $$i$$ let $$K_{P_i}$$ be the completed local ring at $$P_i$$ and $$U_{P_i}$$ be the principal units of $$K_{P_i}$$. Let $$\text{Div}(X,D) = \text{Div}(Y) \oplus \bigoplus_{i=1}^n K_{P_i}^{\times}/U_{P_i}$$. There is a surjective map $$\text{Div}(X,D) \rightarrow \text{Div}(X)$$ induced by the inclusion $$\text{Div}(Y) \hookrightarrow \text{Div}(X)$$ and by the map $$K_{P_i}^{\times}/U_{P_i} \rightarrow \text{Div}(X)$$ given by $$f \mapsto \text{val}_{P_i}(f)\cdot (P_i)$$. Let $$\text{Div}^0(X,D)$$ be the kernel of the composition $$\text{Div}^0(X,D)\rightarrow \text{Div}(X) \rightarrow \mathbf{Z}$$ where the last map is the degree. There is an obvious map $$K^{\times} \rightarrow \text{Div}(X,D)$$. Then we have a canonical isomorphism $$\text{Div}^0(X,D)/K^{\times} \simeq \tilde{J}(k)$$.

Suppose now that $$k = \mathbf{C}$$. There is an Abel-Jacobi style description of $$\tilde{J}(k)$$, given as $$\text{Hom}_{\mathbf{C}}(\Omega^1(-D), \mathbf{C})/H_1(Y, \mathbf{Z})$$ (think of $$\text{Hom}_{\mathbf{C}}(\Omega^1(-D), \mathbf{C})$$ as the whole space of modular forms, that is cusp forms $$\textbf{and}$$ Eisenstein series).

Question: how to relate explicitly the two descriptions that I gave of $$\tilde{J}(k)$$? More specifically I have the following question. Fix a uniformizer $$t_i$$ at each $$P_i$$. Let $$m_1$$, ..., $$m_n$$ be integers such that $$\sum_{i=1}^n m_i=0$$. Consider the point $$M:=0 \bigoplus (t_i^{m_i})_{i=1}^n$$ in $$\tilde{J}(k)$$ with the description above. The image of this point in $$J(k)$$ is $$\sum_{i=1}^n m_i\cdot (P_i)$$, which by assumption has finite order, denoted by $$m$$. The element $$m\cdot M \in \text{Ker}(\tilde{J}(k) \rightarrow J(k))$$ corresponds to a morphism $$\varphi \in \text{Hom}_{\mathbf{C}}(\Omega^1(-D), \mathbf{C})$$, well-defined modulo the periods $$H_1(Y, \mathbf{Z})$$. On the other hand, let $$u \in K^{\times}$$ be such that its poles and zeros are supported on the $$P_i$$'s (think of $$u$$ as a modular unit). Consider the element $$\omega := d\log(u) \in \Omega^1(-D)$$ (think of it as an Eisenstein series). Its periods along $$H_1(Y, \mathbf{Z})$$ are in $$2\pi i \mathbf{Z}$$. Therefore, the number $$\varphi(\omega)$$ is well-defined in $$\mathbf{C}/2\pi i \mathbf{Z} \simeq \mathbf{C}^{\times}$$. Is there a relation between $$\varphi(\omega) \in \mathbf{C}^{\times}$$ and the quantity $$\prod_{i=1}^n \left((\frac{u}{t_i^{\text{ord}_{P_i}(u)}})(P_i)\right)^{m\cdot n_i} \in \mathbf{C}^{\times}$$? Or at least is there a relation between the collection of the $$\varphi(\omega)$$'s and the collection of $$\prod_{i=1}^n \left((\frac{u}{t_i^{\text{ord}_{P_i}(u)}})(P_i)\right)^{m\cdot n_i} \in \mathbf{C}^{\times}$$'s when $$u$$ varies through a basis of modular units'' (such a basis has $$n-1$$ elements)?