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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.

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