This is linked to my question on math.Stackexchange for which I had no answer.

I have found nowhere a historical account on connections (or rather called logarithmic connections?) that would help me fully grasp this object.(except maybe Katz's paper) that links the notion of connection on a vector bundle to:

solvability of homogeneous systems of first-order differential equations(a question first raised by Grothendieck).

Trying to clear my confusion about connections, I tried to build a connection on the Legendre smooth projective elliptic curve over a char $p>0$ field k: $(X): x_0 x_2^2- x_1(x_1-x_0)(x_1+x_0)=0$ living in $\Bbb P^1_{ k}$.

1. Construction of $\mathcal E$ rank $2$ degree $0$ vector bundle.

The covering $\Bbb P^1= D_+(x_0)\cup D_+(x_1)\cup D_+(x_2) $ gives a covering $X=U_0 \cup U_2$, with $U_0= D_+(x_0) \cap X$ and $U_2= D_+(x_2) \cap X$

In summary we have $\mathcal O_X(U_0)= \frac{k[x,y]}{y^2-x(x-1)(x+1)}$ and $\mathcal O_X(U_2)= \frac{k[xy^{-1},y^{-1}]}{y^{-1}-xy^{-1}(xy^{-1}-y^{-1})(xy^{-1}+y^{-1})}$

$\mathcal E$ is given by a transition matrices, $(M_{02}, M_{20}=M_{02}^{-1})$, say $M_{02}=\begin{pmatrix} y & 0 \\ 0 & y^2 \end{pmatrix}$. The compatibility to verify is: $A_0=M_{02}A_2$

2. Building a connection on $\mathcal E$

Here is where I have few confusions:$\nabla|_{U_i} = d_{X} + A_i$ where $A_i$ is a $2\times 2$ matrix with entries $1$-form in $\Omega^1_X$.

• Few computations show that $\Omega^1$ is spanned by $ (3x^2-2)dx-\frac{9}{2}xy dy$
• Taking $A_2= \begin{pmatrix} dx & 0 \\ 0 & dy \end{pmatrix} $, $A_0$ must verify the compatibility condition $A_0=M_{02}A_2$.

I am confused by:

1. How can we make sense of $\nabla|_{U_i}$, in other words, how can we add a scalar with a matrix. Do we add the scalar to each entry in $A_i$?
2. I've seen in some paper a confusing version of compatibility condition: $A_0=M_{02} A_2 M_{02}^{-1} + d M_{02}M_{02}^{-1} $ where each $A_i$ is a $r × r$ matrix of meromorphic $1$-forms having at most simple poles. I don't know why it should be this way?
3. I would be happy if you have any references on this subject.

Mnay thanks for your help.

This is linked to my question on math.Stackexchange for which I had no answer.

I have found nowhere a historical account on connections (or rather called logarithmic connections?) that would help me fully grasp this object.(except maybe Katz's paper) that links the notion of connection on a vector bundle to:

solvability of homogeneous systems of first-order differential equations(a question first raised by Grothendieck).

Let $X$ be an elliptic curve and let $(U_i)_i$ be an open covering of $X$.

A connection on a vector bundle $\nabla|_{U_i} = d_{X} + A_i$ where $A_i$ is a $2\times 2$ matrix with entries $1$-form in $\Omega^1_X$.

1. How can we make sense of $\nabla|_{U_i}$, in other words, how can we add a scalar ($\Omega^1$ is of dimension $1$ as $X$ is smooth) with a matrix?
2. I would be happy if you have any references on this subject.

Mnay thanks for your help.

This is linked to my question on math.Stackexchange for which I had no answer.

I have found nowhere a historical account on connections (or rather called logarithmic connections?) that would help me fully grasp this object.(except maybe Katz's paper) that links the notion of connection on a vector bundle to:

solvability of homogeneous systems of first-order differential equations(a question first raised by Grothendieck).

Trying to clear my confusion about connections, I tried to build a connection on the Legendre smooth projective elliptic curve over a char $p>0$ field k: $(X): x_0 x_2^2- x_1(x_1-x_0)(x_1+x_0)=0$ living in $\Bbb P^1_{ k}$.

1. Construction of $\mathcal E$ rank $2$ degree $0$ vector bundle.

The covering $\Bbb P^1= D_+(x_0)\cup D_+(x_1)\cup D_+(x_2) $ gives a covering $X=U_0 \cup U_2$, with $U_0= D_+(x_0) \cap X$ and $U_2= D_+(x_2) \cap X$

In summary we have $\mathcal O_X(U_0)= \frac{k[x,y]}{y^2-x(x-1)(x+1)}$ and $\mathcal O_X(U_2)= \frac{k[xy^{-1},y^{-1}]}{y^{-1}-xy^{-1}(xy^{-1}-y^{-1})(xy^{-1}+y^{-1})}$

$\mathcal E$ is given by a transition matrices, $(M_{02}, M_{20}=M_{02}^{-1})$, say $M_{02}=\begin{pmatrix} y & 0 \\ 0 & y^2 \end{pmatrix}$. The compatibility to verify is: $A_0=M_{02}A_2$

2. Building a connection on $\mathcal E$

Here is where I have few confusions:$\nabla|_{U_i} = d_{X} + A_i$ where $A_i$ is a $2\times 2$ matrix with entries $1$-form in $\Omega^1_X$.

• Few computations show that $\Omega^1$ is spanned by $ (3x^2-2)dx-\frac{9}{2}xy dy$
• Taking $A_2= \begin{pmatrix} dx & 0 \\ 0 & dy \end{pmatrix} $, $A_0$ must verify the compatibility condition $A_0=M_{02}A_2$.

I am confused by:

1. How can we make sense of $\nabla|_{U_i}$ for $\lambda \in \mathcal O_X(U_i)$, in other words, how can we add a scalar with a matrix. Do we add the scalar to each entry in $A_i$?
2. I've seen in some paper a confusing version of compatibility condition: $A_0=M_{02} A_2 M_{02}^{-1} + d M_{02}M_{02}^{-1} $ where each $A_i$ is a $r × r$ matrix of meromorphic $1$-forms having at most simple poles. I don't know why it should be this way?
3. I would be happy if you have any references on this subject.

Mnay thanks for your help.

This is linked to my question on math.Stackexchange for which I had no answer.

I have found nowhere a historical account on connections (or rather called logarithmic connections?) that would help me fully grasp this object.(except maybe Katz's paper) that links the notion of connection on a vector bundle to:

solvability of homogeneous systems of first-order differential equations(a question first raised by Grothendieck).

Trying to clear my confusion about connections, I tried to build a connection on the Legendre smooth projective elliptic curve over a char $p>0$ field k: $(X): x_0 x_2^2- x_1(x_1-x_0)(x_1+x_0)=0$ living in $\Bbb P^1_{ k}$.

1. Construction of $\mathcal E$ rank $2$ degree $0$ vector bundle.

The covering $\Bbb P^1= D_+(x_0)\cup D_+(x_1)\cup D_+(x_2) $ gives a covering $X=U_0 \cup U_2$, with $U_0= D_+(x_0) \cap X$ and $U_2= D_+(x_2) \cap X$

In summary we have $\mathcal O_X(U_0)= \frac{k[x,y]}{y^2-x(x-1)(x+1)}$ and $\mathcal O_X(U_2)= \frac{k[xy^{-1},y^{-1}]}{y^{-1}-xy^{-1}(xy^{-1}-y^{-1})(xy^{-1}+y^{-1})}$

$\mathcal E$ is given by a transition matrices, $(M_{02}, M_{20}=M_{02}^{-1})$, say $M_{02}=\begin{pmatrix} y & 0 \\ 0 & y^2 \end{pmatrix}$. The compatibility to verify is: $A_0=M_{02}A_2$

2. Building a connection on $\mathcal E$

Here is where I have few confusions:$\nabla|_{U_i} = d_{X} + A_i$ where $A_i$ is a $2\times 2$ matrix with entries $1$-form in $\Omega^1_X$.

• Few computations show that $\Omega^1$ is spanned by $ (3x^2-2)dx-\frac{9}{2}xy dy$
• Taking $A_2= \begin{pmatrix} dx & 0 \\ 0 & dy \end{pmatrix} $, $A_0$ must verify the compatibility condition $A_0=M_{02}A_2$.

I am confused by:

1. How can we make sense of $\nabla|_{U_i}$, in other words, how can we add a scalar with a matrix. Do we add the scalar to each entry in $A_i$?
2. I've seen in some paper a confusing version of compatibility condition: $A_0=M_{02} A_2 M_{02}^{-1} + d M_{02}M_{02}^{-1} $ where each $A_i$ is a $r × r$ matrix of meromorphic $1$-forms having at most simple poles. I don't know why it should be this way?
3. I would be happy if you have any references on this subject.

Mnay thanks for your help.