# When $\phi_{\mathcal L}=0$ for $\mathcal L$ a line bundle over an abelian scheme $X/S$

Let $$X\rightarrow S$$ be a projective abelian scheme. To a line bundle $$\mathcal L$$ on $$X$$, we associate its Mumford line bundle $$\Lambda(\mathcal L):= \mu^{\star}\mathcal L\otimes p_1^{\star}\mathcal L^{-1}\otimes p_2^{\star}\mathcal L^{-1}$$ on $$X\times_S X$$ where $$\mu$$ is the group law, and $$p_i$$ the projection morphisms. This defines a homomorphism $$\phi_{\mathcal L}:X\rightarrow \hat{X}$$ of $$X$$ into its dual. Eventually, denote by $$\mathcal P$$ the Poincaré sheaf on $$X\times_S \hat{X}$$ trivialized along $$\epsilon\times id$$, where $$\epsilon$$ is the unit section of $$X$$.

Claim: Assume that $$\mathcal L\cong (id\times\hat{x})^{\star}\mathcal P$$ for some section $$\hat{x}:S\rightarrow \hat{X}$$. Then $$\phi_{\mathcal L}=0$$.

The hypothesis $$\mathcal L\cong (id\times\hat{x})^{\star}\mathcal P$$ means that the $$S$$-point of $$\mathcal{Pic}_{X/S}$$ defined by the line bundle $$\mathcal L$$ on $$X=X\times_S S$$ factors through $$\hat{X}$$, and it is exactly $$\hat{x}$$.

The paper I am reading (Genestier and Ngô's lecture on Shimura varieties) gives as a justification that the statement is trivial for $$\mathcal L\cong \mathcal O_X$$, and that we can continously deform a general $$\mathcal L$$ to $$\mathcal O_X$$ using rigidity lemma to obtain the desired result.

I have trouble writing down a rigorous proof for this. Given an $$S$$-scheme $$T$$ and a $$T$$-valued point $$z$$ of $$X$$, at the level of points, $$\phi_{\mathcal L}(z)$$ is the element of $$Pic_{X/S}(T)$$ determined by the pullbak of $$\Lambda(\mathcal L)$$ by $$id\times z$$. I must somehow prove that this pullback on $$X\times_S T$$ is actually the pullback of some line bundle on $$T$$. It would be true if I could prove that $$\Lambda(\mathcal L)$$ is the pullback by $$p_2$$ of some line bundle on $$X$$. Thus I am looking at $$\mu^{\star}\mathcal L\otimes p_1^{\star}\mathcal L^{-1}\cong \mu^{\star}(id\times\hat{x})^{\star}\mathcal P\otimes p_1^{\star}(id\times\hat{x})^{\star}\mathcal P^{-1}$$.

I naturally considered the morphism $$(id\times\hat{x})\mu - (id\times\hat{x})p_1: X\times_S X\rightarrow X\times_S \hat{X}$$, which we can look as a map between two abelian schemes over $$X$$ (with structure morphisms the second projection at the start, the first projection at the target). This map sends the unit section on the unit section, hence by rigidity it is a group homomorphism. But I am stuck there, I do not see how to progress further.

Would somebody be able to give a hand there ?

Edit : I actually just noticed that my last argument is not true, the map $$(id\times\hat{x})\mu - (id\times\hat{x})p_1$$ isn't sending the unit to the unit after all. I left the wrong argument, but I fear I should find another way to use rigidity.