pullback of poincare bundle Let $X$ be a smooth curve and ${x,y}$ are two smooth points. Let $J_X$ be the jacobian of X i.e, the variety which parametrizes the degree 0 line bundles on X and let $\mathcal{P}$ is the poincare bundle over $X\times J_X$. We denote by $\mathcal{P}_x$ and $\mathcal{P}_y$ the restrictions of $\mathcal{P}$ along $x\times J_X$ and $y\times J_X$ respectively. There is an isomorphism $\phi_{y-x}:J_X\rightarrow J_X$ defined by $L\mapsto L\otimes \mathcal{O}_X(y-x)$. My question is: Is $\phi_{y-x}^*\mathcal{P}_x\cong \mathcal{P}_y$?
 A: I am just writing the comments as an answer.  First of all, the essential point made by abx is that for an invertible sheaf $\mathcal{L}$ on an Abelian variety, if $\mathcal{L}$ has degree $0$, then the pullback of $\mathcal{L}$ by each translation is isomorphic to $\mathcal{L}$.  For one reference, this follows from Facts (i) and (iv) on pp. 75-76 of the following.
MR2514037 (2010e:14040) 
Mumford, David 
Abelian varieties. 
With appendices by C. P. Ramanujam and Yuri Manin.  
Corrected reprint of the second (1974) edition.  
Tata Institute of Fundamental Research Studies in Mathematics, 5.  Published for the Tata Institute of Fundamental Research, Bombay;  
by Hindustan Book Agency, New Delhi, 2008. xii+263 pp. 
Mumford uses a definition of $\text{Pic}^0$ that might not agree with every definition.  By (iv), for any other definition of the degree of an invertible sheaf, so long as it is compatible with pullback by the "multiplication-by-$n$" maps on the Abelian variety, it follows that "degree zero" implies that the invertible sheaf is in $\text{Pic}^0$ with Mumford's definition.  Then (i) implies that the invertible sheaf is stable for translations.
It is a matter of convention how to define a Poincare sheaf, and different authors follow different conventions.  In the sources that I know, every Poincare sheaf on $X \times \text{Pic}^0(X)$ is normalized so that its restrictions to fibers $\{x\}\times \text{Pic}^0(X)$ are each invertible sheaves of degree $0$, i.e., parameterized by $\text{Pic}^0$.  
