Symmetric homomorphisms are Mumford maps

Work first over an algebraic closure $\bar{k}$ of $k$ and write $\lambda$ again for ${\lambda }_{\bar{k}}$. Put $M=(\mathrm{id},\lambda {)}^{\ast }\mathcal{P}$ and $N=M^2$. The Poincare biextension identities expand $P(x+y,\lambda (x+y))$ into $P(x,\lambda x)\otimes P(y,\lambda y)\otimes P(x,\lambda y)\otimes P(y,\lambda x)$. Thus the normalized square family of $M$ is the product of the two cross families. The first represents $\lambda$, and the switched second represents ${\lambda }^{\vee }\circ {\kappa }_A$ by the defining dual-pullback and biduality identities in [F2]. Hence ${\phi }_M=\lambda +{\lambda }^{\vee }\circ {\kappa }_A=2\lambda$. Tensor multiplicativity gives ${\phi }_N=4\lambda$, so $A[4]\subseteq K(N)$.

Now let $k$ be arbitrary and consider the fppf sheaf $Z$ on $k$-schemes whose $T$-points are the relative Picard classes of invertible sheaves $\mathcal{L}$ on $A_T$, normalized along the identity with ${\phi }_{\mathcal{L}}={\lambda }_T$. If $Z(T)\mathrm{\ne }\mathrm{\varnothing }$ and $\mathcal{L}\in Z(T)$, then ${\mathcal{L}}^{\mathrm{\prime }}\mapsto {\mathcal{L}}^{\mathrm{\prime }}\otimes {\mathcal{L}}^{-1}$ identifies $Z(T)$ with $\ker (\phi )(T)={\mathrm{Pic}}^0(A_T)=A^{\vee }(T)$ by [F4], so $Z$ is an $A^{\vee }$-torsor for the fppf topology.

Restrict the alternating bilinear commutator pairing $e_N$ to $A[4]\times A[4]$, which is legitimate because $A[4]\subseteq K(N)$ by step 1.1. Its values lie in ${\mu }_4$: bilinearity gives $e_N(x^{\mathrm{\prime }},y^{\mathrm{\prime }}{)}^4=e_N(4x^{\mathrm{\prime }},y^{\mathrm{\prime }})=1$. For $x,y\in A[2]$, lift fppf-locally to $x^{\mathrm{\prime }},y^{\mathrm{\prime }}\in A[4]$ with $x=2x^{\mathrm{\prime }}$ and $y=2y^{\mathrm{\prime }}$ by [F6]. Then $e_N(x,y)=e_N(x^{\mathrm{\prime }},y^{\mathrm{\prime }}{)}^4=1$. Descent of equality proves isotropy on the full group scheme $A[2]\times A[2]$, including characteristic two. This uses $e_N$ on its actual domain and does not extend $e_M$ outside $K(M)$.

Apply the isotropic descent of [F3] with $H=A[2]$ and the quotient morphism $[2]:A\to A$: since $e_N$ is trivial on $A[2]\times A[2]$, the line bundle $N$ descends through $[2]$, so there is an invertible sheaf $L$ on $A$ with $N\cong [2{]}^{\ast }L$. Then $4\lambda ={\phi }_N=[2{]}^{\ast }{\phi }_L=4{\phi }_L$, using ${\phi }_{[2{]}^{\ast }L}=[2{]}^{\ast }\circ {\phi }_L\circ [2]=4{\phi }_L$. Therefore $\lambda -{\phi }_L$ is a homomorphism whose image lies in $\ker [4]=A^{\vee }[4]$, a finite group scheme; since $A$ is proper and geometrically integral, every map from $A$ to a finite affine scheme factors through $\mathrm{\Gamma }(A,{\mathcal{O}}_A)=k$, so a pointed homomorphism to that finite scheme is zero, so $\lambda ={\phi }_L$.

The realization over $\bar{k}$ in step 3.1 descends to a finite extension $K\mathrm{/}k$: an invertible sheaf is described by finitely many generators and transition functions on a finite affine cover, and the equality of its Mumford morphism with $\lambda$ is described by finitely many equations on affine covers; all their algebraic coefficients lie in a finite extension. Call this realizing bundle $L_0$ on $A_K$. Tensoring $L_0$ with the universal Poincare bundle identifies $Z_K$ with $A_K^{\vee }$ on all tests by the kernel equality [F4]. In particular $Z$ is fppf-locally nonempty, as required in step 1.2, and $Z_K$ carries the canonical descent datum obtained from equality of its classifying functor over $K{\otimes }_{k}K$. This datum satisfies the cocycle by uniqueness of the functor identification. The scheme $Z_K$ is projective, since it is an abelian variety. Its finite descent orbits lie in affine opens: choose closed specializations of the finitely many orbit points and a sufficiently high very ample power; Serre vanishing and eventual generation in [F5] supply a section nonzero at each specialization, whose nonvanishing affine open contains the whole orbit. The finite-field descent theorem [F5] therefore descends $Z_K$ to a finite-type $k$-scheme representing $Z$, including inseparable $K\mathrm{/}k$. This argument requires only the represented dual and its universal bundle and does not presume that the full Picard scheme has already been constructed over $k$.

The scheme $Z$ is smooth over $k$: it is an $A^{\vee }$-torsor by step 1.2, $A^{\vee }$ is smooth over $k$ by [F1], and smoothness is fppf-local. It is nonempty because after extending scalars to $\bar{k}$ the realization $\lambda ={\phi }_L$ of step 3.1 gives a $\bar{k}$-point of $Z$. Since $Z$ is a nonempty smooth finite-type $k$-scheme, the finite-separable-point theorem [F6] provides a closed point $z\in Z$ whose residue field $k^{\mathrm{\prime }}$ is finite and separable over $k$; the tautological bundle at $z$ is an invertible sheaf $\mathcal{L}$ on $A_{k^{\mathrm{\prime }}}$ with ${\phi }_{\mathcal{L}}={\lambda }_{k^{\mathrm{\prime }}}$, as required.