Let $A$ be an abelian variety over a field $k$ of characteristic $0$. How do I prove, without using transcendental methods, that if $\nabla$ is an integrable connection on a vector bundle $L$ on $A$ then $(L, \nabla)$ comes from a $G$-bundle with connection for some abelian algebraic subgroup $G \subset GL(n)$? And the corollary there exists a $\nabla$-stable flag$$L = L\_n \supset L\_{n-1} \supset \dots \supset L\_1 \supset 0$$with $\text{rank }L\_i = i$? It is easy to prove for $k = \mathbb{C}$ by looking at the monodromy of $(L, \nabla)$, and the general case follows by the Lefschetz principle, but I want to find a different proof which does not use $\mathbb{C}$ at all.


Any help would be greatly appreciated!



$(L, \nabla)$ comes from a $G$-bundle with connection for some abelian algebraic subgroup $G \subset GL(n)$?
Here is one approach using the Fourier transform for $D$-modules on an abelian variety due to Laumon. 


Let $A^\flat$ be the moduli space of rank one local systems on $A$, it is the universal extension of the dual abelian variety $A^\vee$ by a vector space. Note that L is a D-module such that the underlying O-module is coherent. This means that its Fourier transform $\hat{L}\in D^b(A^\flat)$ has the property that its pushforward to $A^\vee$ is coherent (or more precisely all of its cohomology are coherent). This is equivalent to the $supp(\hat{L})$ being finite over $A^\vee$. Thus, $supp(\hat{L})$ is complete (i.e., proper over $k$).


The claim we are trying to prove is equivalent to finiteness of $supp(\hat{L})$. Thus, we are trying to prove that $A^\flat$ contains no non-trivial complete sub-varieties. This is very easy analytically (it is analytically equivalent to an affine variety), but there is also an algebraic proof. 


Indeed, it suffices to check that any map from a projective curve $X$ to $A^\flat$ is constant. Such a map is simply a line bundle on $X\times A$ together with a flat connection along $A$. Now looking at it as an $A$-family of line bundles on $X$, we see that it is an exterior product of two line bundles, and it is easy to see that the connection must be constant as well.


