Suppose we have two sections for our abelian fibration $$0,\Sigma:B \ra \cA.$$ After restricting to a suitable open subset of $B$ (also denoted $B$), we may assume that \begin{enumerate} \item{ the image of $0$ and $\Sigma$ lie in the smooth locus of $\eta$;} \item{ the group law on $\cA$, with identity $0$, is well-defined wherever $\eta$ is smooth.} \end{enumerate} \begin{prop}\label{prop:dense} Retain the notations and assumptions above, and assume that all the relevant varieties and morphism are defined over a number field $K$. Assume furthermore that \begin{enumerate} \item{ the union $\cup_{n\in \bN}(n\Sigma)$ is dense;} \item{$K$-rational points of $\Sigma$ are dense.} \end{enumerate} Then $K$-rational points of $\cA$ are also dense. \end{prop} {\em proof:} Each $n\Sigma\simeq_K \Sigma$ and therefore has dense rational points. Thus rational points in $\cA$ are dense as well. $\square$ \begin{defn} A section $\Sigma$ of an abelian fibration is {\em nondegenerate} if $\cup_{n\in \bN}(n\Sigma)$ is dense, and {\em nontorsion} if $\cup_{n\in \bN}(n\Sigma)$ is infinite. \end{defn} These two notions coincide for an elliptic fibration. Restrict $B$ further so that $J(\cA)_B$, the relative Jacobian group-scheme of $\cA$ over $B$ is defined, i.e., all cycles of relative degree zero. After basechange to $M$, $\sigma_M$ induces an identification between $J(\cA)_M:=J(\cA)_B\times_B M$ and the smooth locus of $\cA\times_B M$. The image of $-\Tr_M(M\times_B M)$ in $J(\cA)_B$ is a multisection, denoted $\tau_M$. If $m\in M$ and $\eta^{-1}(\eta(m))=\{m,m_2,\ldots,m_{\deg(M/B)}\}$ then $$\tau_m=-(m+m_2+\ldots +m_{\deg(M/B)})+\deg(M/B)m.$$ In light of Proposition \ref{prop:dense2}, potential density follows if we can produce a nondegenerate multisection $M$ of genus $0$. We now restrict our attention to the subgroup of the Tate-Shafarevich group $\Sh_0(J)\subset \Sh(J)$, corresponding to fibrations with a {\em differentiable} section. It is known that $\Sh_0(J)=\Sh(J)$ whenever $J$ has degenerate fibers, and in particular, whenever $J$ is non-isotrivial \cite{Shaf1}, \cite{Shaf2} VII \S 8. (Elliptic K3 surfaces always have degenerate fibers--see \S\ref{subsect:shioda}.) The subgroup $\Sh_0(J)$ admits an elegant interpretation in terms of the cohomology of the compact K\"ahler manifold $\oJ$: $$\Sh_0(J)\simeq [H^2(\oJ,\cO_{\oJ})/\text{ image }H^2(\oJ,\bZ)].$$ The image of $H^2(\oJ,\bZ)$ in $H^2(\oJ,\cO_{\oJ})$ corresponds to the transcendental classes $$H^2(\oJ,\bZ)_{\tran}= H^2(\oJ,\bZ)/(H^2(\oJ,\bZ)\cap H^1(\oJ,\Omega^1_{\oJ})= H^2(\oJ,\bZ)/\mathrm{NS}(\oJ),$$ the integral classes modulo the N\'eron-Severi group. The torsion elements can be expressed $$\Sh_0(J)_{\tors}=H^2(\oJ,\bZ)_{\tran}\otimes \bQ/\bZ.$$ {\em proof} The cocycle governing $J^m(X_F)$ is $m$ times the cocycle governing $X_F=J^1(X_F)$, so the formula follows. Moreover, $[\overline{J^m(X)}]=0$ exactly when $\overline{J^m(X)}$ has a section, which is equivalent to $J^m(X)(F)\ne \emptyset$. Any such class is represented by a degree-$m$ divisor $M_F\subset X_F$, which is effective by Riemann-Roch. The closure of $M_F$ in $X$ is the curve $M$. Conversely, any curve $M$ of relative degree $m$ yields a point in $J^m(X)(F)$. $\square$ Given any such principle homogeneous space $X_F$, we may consider the relative minimal model, $\eta:X\ra \bP^1$. Since $X(\hat{F}_b)\ne \emptyset$ for each $b\in \bP^1$, $\eta$ has no multiple fibers, and $X$ is a K3 surface by Proposition \ref{prop:JacK3}. Conversely, any elliptic K3 surface $\eta:X \ra \bP^1$, with Jacobian $J(X_F)\simeq J_F$, yields an element of $\Sh(J_F)$ by Proposition \ref{prop:nomult}. Furthermore, Proposition \ref{prop:gpscheme} implies that the smooth locus of $\eta$ is a principle homogeneous space for the smooth locus for $J\ra \bP^1$. Let $F=\bC(\bP^1)$ be the function field of the base and $X_F$ the generic fiber, an elliptic curve over $F$. Then the Jacobian $J(X_F):=J^0(X_F)$ is defined and we have the identity $0\in J(X_F)(F)$. Let $\iota: \oJ \ra \bP^1$ be the {\em Jacobian fibration} (see \cite{BPV} V.9 and \cite{Kod}), defined by the following properties: \begin{enumerate} \item{$\oJ_F\simeq J(X_F)$;} \item{$\oJ$ is smooth and projective;} \item{$\iota:\oJ\ra \bP^1$ is relative minimal.} \end{enumerate} Let $J\ra \bP^1$ be the open subset where $\iota$ is smooth, a group-scheme over $\bP^1$. Similarly, let $J^m(X_F)$ denote the degree-$m$ component of the Picard scheme of $X_F$ over $F$ and $\overline{J^m(X)}\ra \bP^1$ its relatively minimal model over $\bP^1$. Multiplication by $N$, $\mu_N:J^m(X_F) \ra J^{mN}(X_F)$, extends to a rational map $\mu_N:\overline{J^{m}(X)} \dashrightarrow \overline{J^{mN}(X)}.$ Assume that the homomorphism $$\alpha:\Gamma \ra \Aut(\bZ/m\bZ^{\oplus 2})$$ governing $J[m]$ is surjective. Let $Y\ra \bP^1$ be a twist of $X\ra \bP^1$ coming from an element of $H^1(\bP^1,J[m])$. Let $M\subset Y$ be a corresponding multisection which is a $J[m]$-homogeneous space. This is governed by a homomorphism $$\phi:\Gamma \ra \Aff(\bZ/m\bZ^{\oplus 2})$$ so that $q\circ \phi=\alpha$, where $q$ is defined by \ref{eq:ext}. For each $\gamma\in \Gamma$, the order of $\phi(\gamma)$ is at least as large as the order of $\alpha(\gamma)$. This allows us to compare the monodromy actions on the normalizations $M^{\nu}$ and $J[m]^{\nu}$ near a point $b\in \bP^1 \setminus U$. The local monodromy near $b$ is cyclic and generated by a single element $\gamma_b \in \Gamma$. The ramification order of $J[m]^{\nu}\ra \bP^1$ over $b$ is equal to the order of $\alpha(\gamma_b)$. The ramification order of $M^{\nu} \ra \bP^1$ over $b$ is equal to the order of $\phi(\gamma_b)$. The group theoretic analysis above implies that the total ramification degree of $J[m]^{\nu}$ is no larger than that of $M^{\nu}$. On the other hand, the (set-theoretic) number of points in $M_b$ equals the number of points of $J[m]_b$. We may therefore apply the following: \begin{prop} Let $M\ra \bP^1$ and $J[m]\ra \bP^1$ be finite flat maps from irreducible curves to $\bP^1$, so that for each $b\in \bP^1$ there is a bijection $$M_b \simeq J[m]_b$$ respecting multiplicities (e.g., vanishing orders for a uniformizer of $\cO_{\bP^1,b}$). Assume that the total ramification degree $M^{\nu}$ exceeds the total ramification degree of $J[m]$. \end{prop} {\bf How do we force $M$ irreducible? Use nontriviality of the twist and the following:} The notation of the following proposition refers to exact sequence \ref{eq:ext}: \begin{prop} Any subgroup $H\subset \Aff(\bZ/p\bZ^{\oplus 2})$, with $H\neq \sigma(\Aut(\bZ/p\bZ^{\oplus 2})$ and $q(H)=\Aut(\bZ/p\bZ^{\oplus 2})$, acts transitively on $\bZ/p\bZ^{\oplus 2}$. \end{prop} {\em proof:} Given an element $$v \ra A_1v+C_1, A_1\in \Aut(\bZ/p\bZ^{\oplus 2}), C_1\neq 0\in \bZ/p\bZ^{\oplus 2},$$ there exists an $(A_2,C_2)$ with $A_2(C_1)$ independent from $C_1$. {\bf FINISH THIS} \begin{prop}\label{prop:genus} Let $B$ be a smooth, projective curve over $K$ and let $G$ be a finite group. Let $\cG\ra B$ be a finite flat group scheme, isomorphic to $G$ over the geometric generic point of $B$. Let $\cP\ra B$ be a projective curve so that $\cP(K(B))$ is a principal homogeneous space for $\cG(K(B))$. If $\cP$ is irreducible then the genus of the normalization $\cP^{\nu}$ is greater than the genus of each component of $\cG^{\nu}$. \end{prop} {\em proof}: In this proof, we may assume $K$ to be algebraically closed. We analyze the ramifications of the covers $$f_1:\cG^{\nu} \ra B \quad f_2:\cP^{\nu} \ra B,$$ which are both finite and flat of degree $|G|$. Let $U\subset B$ be the open subset where both are smooth. Pick $b\in B\setminus U$ and write $f_1^{-1}(b)=\{g_1,\ldots,g_m\}$ and $f_2^{-1}(b)=\{p_1,\ldots,p_n\}$. Take $\Gamma=\pi_1(U)$ or the appropriate unramified quotient of the Galois group of ${\overline K(B)}/K(B)$. Associated to $\cP^{\nu}$ and $\cG^{\nu}$ are representations $$\alpha:\Gamma \ra \Aut(G), \quad \phi:\Gamma \ra \Aff(G)$$ satisfying the compatibility $\alpha=q\circ \phi$ (see section \ref{subsect:GalCoh}). Let $\lambda$ be the generator for the local fundamental group of $U$ near $b$, determined up to conjugation. This is the class of a small loop around $b$ or the generator of the Galois group $$\Gal({\overline K_b}/K_b)\simeq \hat{\bZ}, \quad K_b=K({\hat \cO_{B,b}})\simeq K((t)).$$ The element $\phi(\lambda)\in \Aff(G)$ acts on $G$ with $n$ orbits corresponding to the $p_i$. The orbit attached to $p_i$ has $e_i$ elements, where $e_i$ is the ramification order at $p_i$. On the other hand, $\alpha(\lambda)\in \Aut(G)$ acts on $G$ with $m$ orbits corresponding to the $g_1,\ldots,g_m$. The orbit attached to $g_j$ has $\epsilon_j$ elements, where $\epsilon_j$ is the ramification order at $g_j$. The orbit decomposition for the action of $\phi(\lambda)$ on $G$ must be compatible with the action of $\phi(\Gamma)$ on $G$???? \bibitem[Mo]{Mo} D. Morrison, D.R. Morrison, On $K3$ surfaces with large Picard number, {\em Invent. Math.} {\bf 75} (1984), no. 1, 105--121. \subsubsection{Leray spectral sequence} Let $\eta:X\ra \bP^1$ be an elliptic K3 surface {\em with section). The Leray spectral sequence for $\eta$ degenerates at $E_2$ and yields a filtration $$0\subset F^2\subset F^1 \subset F^0:=H^2(X,\bZ).$$ We analyze the graded pieces, in particular, how they sit in with respect to the Hodge filtration of $X$. To this end, we use the exact sequence $$0 \ra \bZ \ra \cO_X \ra \cO_X^* \ra 0,$$ taking the Leray spectral sequences of each of the terms. First, we have $$F^2=H^2(\bP^1,\eta_*\bZ)=H^2(\bP^1,\bZ)=\bZ$$ and $$H^2(\bP^1,\eta_*\cO_X)=H^2(\bP^1,\cO_X)=0,$$ so $F^2$ comes from the Picard group $H^1(X,\cO_X^*)$. It is generated by the class $[\eta^{-1}(p)]$ of the generic fiber. Second, we have $$F^1/F^2=H^1(\bP^1,\bR^1\eta_*\bZ),$$ the orthogonal complement to the classes of curves in fibers of $\eta$. Finally, $$F^0/F^1=H^0(\bP^1,\bR^2\eta_*\bZ)$$ is the quotient of the cohomology obtained by taking intersections with all fibral curves. Using the nondegenerate interesection pairing on $H^2(X,\bZ)$, the final graded piece is isomorphic (over $\bQ$) to the subspace of $H^2(X,\bQ)$ generated by fibral curves. Retain the notation from section \ref{subsect:GalCoh}. The $m$-torsion of an abelian group $G$ is denoted $G[m]$. Thus $J[m]$ is a flat group scheme over $\bP^1$ and a twisted form of $(\bZ/m\bZ)^{\oplus 2}$ over the open subset $U\subset \bP^1$ over which $\oJ\ra \bP^1$ is smooth. The cocycle governing this twist corresponds to a homomorphism $$\alpha:\Gamma \ra \SL((\bZ/m\bZ)^{\oplus 2}),$$ where $\Gamma$ is the fundamental group/Galois group $\pi_1(U)$. \begin{prop} Fix a Jacobian elliptic K3 surface $\oJ \ra \bP^1$ and retain the notation above. An element $[X]\in \Sh(J)$ with $m[X]=0$ determines a representation $$\phi:\Gamma \ra \ASL_2(\bZ/m\bZ)$$ so that $\phi\circ q=\alpha$. \end{prop} This representation is not canonically determined: it depends on the choice of the lift to $H^1_{\et}(J[m])$. \noindent {\em proof:} We first analyze the situation over the generic point of the base. Let $F=\bC(\bP^1)$ and $\Xi=\Gal({\overline F}/F)$, so that we have the following exact sequence of $\Xi$-modules $$0\ra J[m]({\overline F}) \ra J(\overline F) \stackrel{\times m}{\ra} J(\overline F) \ra 0$$ which induces maps in cohomology $$0\ra J[m](F) \ra J(F) \stackrel{\times m}{\ra} J(F) \ra H^1_{\Xi}(J[m](\overline F)) \ra H^1_{\Xi}(J({\overline F}))[m] \ra 0.$$ This shows that each $[X]\in \Sh(J)$ with $m[X]=0$ arises from a $J[m]$-principal homogeneous space, i.e., the structure group admits a restriction to the subgroup of $m$-torsion. Of course, the restriction is only unique modulo elements of $J(F)/mJ(F)$. We realize this a bit more geometrically. Let $\Sigma \in \overline{J^m(X)}$ be a section over $\bP^1$ and the coresponding point of $J^m(X)(F)$. Consider the multiplication map $$\mu_m:X_F \stackrel{\times m}{\ra} J^m(X_f),$$ the preimage $\mu_m^{-1}(\Sigma)$, and its closure $M$ in $X$. Any two points in $M_{b'}=\eta^{-1}(b')$, $b'\in B$ generic, differ by $m$-torsion, hence $M$ is a torsion multisection. We remark that $\Sigma$ is determined uniquely up to a section of $J$, and $M$ is determined up to the quotient $J/mJ$. We claim $M_U:=M\cap \eta^{-1}(U)$ is \'etale over $U$. Clearly, $M_U\times_U M_U$ is smooth and unramified over $M_U$, since it is contained in the $m$-torsion points of a smooth elliptic fibration. However, being \'etale is a local condition in the faithfully flat topology \cite{EGAIV} 17.7.4, and $M_U \ra U$ is faithfully flat. Thus we can conclude that our $J[m]$-principal homogeneous space is \'etale over $U$, and thus is classified by elements of the \'etale cohomology $H^1_{\et}(U,J[m])$. Regarding $J[m]$ as a $\Gamma=\pi_1(U)$ module, this can be expressed as an element of $H^1_{\Gamma}(J[m])$, or as an element $\phi \in \Hom(\Gamma,\ASL_2(\bZ/m\bZ))$ so that $q\circ \phi=\alpha$ (see Proposition \ref{prop:cocycle}). $\square$ \noindent {\bf Case $I^*_0$:} In this situation, $$ \alpha(\gamma_b)=-\left(\begin{matrix} 1 & 0 \\ 0 & 1\end{matrix} \right), \quad \phi(\gamma_b)= \left( \begin{array}{c|c} \begin{matrix} -1 & 0 \\ 0 & -1 \end{matrix} & \begin{array}{c} v_1 \\ v_2 \end{array} \\ \hline \begin{array}{cc} 0 & 0 \end{array} & 1 \end{array} \right) $$ Note that conjugation yields $$ \tau_{(v_1/2,v_2/2)}^{-1}\phi(\gamma_b)\tau_{(v_1/2,v_2/2)} =\left( \begin{array}{c|c} \begin{matrix} -1 & 0 \\ 0 & -1 \end{matrix} & \begin{array}{c} 0 \\ 0 \end{array} \\ \hline \begin{array}{cc} 0 & 0 \end{array} & 1 \end{array} \right), $$ so we may restrict to the case $v=0$. Then $\phi(\gamma_b)$ has $(p^2+1)/2$ orbits and we have $$ \sum_{f(m)=b}e_m=(p^2-1)/2. $$ \noindent {\bf Case $I_a, a\neq 0$:} We may assume $p\gg 0$ so that $(p,a)=1$. After conjugating by a suitable diagonal matrix in $\SL_2(\Zp)$, we may take $$\alpha(\gamma_b)=\left(\begin{matrix} 1 & 1 \\ 0 & 1\end{matrix} \right), \quad Z(\alpha(\gamma_b))\supset \{ \left(\begin{matrix} 1 & x \\ 0 & 1\end{matrix} \right): x\in \Zp \}.$$ It suffices to consider the cases $v=(v_1,0),$ and $v=(0,v_2),v_2\neq 0$. If $v=(v_1,0)$ then $$\phi(\gamma_b)= \left( \begin{array}{c|c} \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix} & \begin{array}{c} v_1 \\ 0 \end{array} \\ \hline \begin{array}{cc} 0 & 0 \end{array} & 1 \end{array} \right) $$ which has $p$ fixed points (the points $\{(w,-v_1)\},w\in \Zp$) and $p-1$ orbits with $p$ elements. In this case, $$ \sum_{f(m)=b}e_m=p^2-(2p-1)=(p-1)^2. $$ If $v=(0,v_2),v_2\neq 0$ then $$\phi(\gamma_b)= \left( \begin{array}{c|c} \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix} & \begin{array}{c} 0 \\ v_2 \end{array} \\ \hline \begin{array}{cc} 0 & 0 \end{array} & 1 \end{array} \right) $$ has $p$ orbits each with $p$ elements, and $$ \sum_{f(m)=b}e_m=p^2-p. $$ \noindent {\bf Case $I^*_a$:} We assume $p\gg a$ so that $(p,a)=1$. After conjugating by a suitable diagonal matrix in $\SL_2(\Zp)$, we may take $$ \alpha(\gamma_b)=-\left(\begin{matrix} -1 & 1 \\ 0 & 1\end{matrix} \right), \quad \phi(\gamma_b)= \left( \begin{array}{c|c} \begin{matrix} -1 & 1 \\ 0 & -1 \end{matrix} & \begin{array}{c} v_1 \\ v_2 \end{array} \\ \hline \begin{array}{cc} 0 & 0 \end{array} & 1 \end{array} \right) $$ Conjugating $\phi(\gamma_b)$ yields $$ \tau_{((2v_1+v_2)/4,v_2/2)}^{-1}\phi(\gamma_b)\tau_{((2v_1+v_2)/4,v_2/2)} =\left( \begin{array}{c|c} \begin{matrix} -1 & 1 \\ 0 & -1 \end{matrix} & \begin{array}{c} 0 \\ 0 \end{array} \\ \hline \begin{array}{cc} 0 & 0 \end{array} & 1 \end{array} \right), $$ so we may restrict to the case $v=0$. Computing $$ \left( \begin{array}{c|c} \begin{matrix} -1 & 1 \\ 0 & -1 \end{matrix} & \begin{array}{c} 0 \\ 0 \end{array} \\ \hline \begin{array}{cc} 0 & 0 \end{array} & 1 \end{array} \right)^n\left( \begin{matrix} w_1 \\ w_2 \\ 1\end{matrix} \right)= \left( \begin{matrix} (w_1-nw_2)(-1)^n \\ w_2(-1)^n \\ 1\end{matrix} \right) $$ we find one fixed point (the origin), $(p-1)/2$ orbits with two elements ($\pm(w_1,0), w_1\neq 0$), and $(p-1)/2$ orbits with $2p$ elements. Thus we have $$ \sum_{f(m)=b}e_m=p^2-p. $$ \noindent {\bf Case $II$} In this situation $$\alpha(\gamma_b)=\left(\begin{matrix} 1 & 1 \\ -1 & 0 \end{matrix} \right), \quad \phi(\gamma_b)= \left( \begin{array}{c|c} \begin{matrix} 1 & 1 \\ -1 & 0 \end{matrix} & \begin{array}{c} v_1 \\ v_2 \end{array} \\ \hline \begin{array}{cc} 0 & 0 \end{array} & 1 \end{array} \right). $$ Note that conjugation yields $$ \tau_{(v_1+v_2,-v_1)}^{-1}\phi(\gamma_b)\tau_{(v_1+v_2,-v_1)} =\left( \begin{array}{c|c} \begin{matrix} 1 & 1 \\ -1 & 0 \end{matrix} & \begin{array}{c} 0 \\ 0 \end{array} \\ \hline \begin{array}{cc} 0 & 0 \end{array} & 1 \end{array} \right), $$ so we may restrict to the case $v=0$. Since the matrix $\alpha(\gamma_b)$ is semisimple with eigenvalues equal to primitive sixth roots of unity, there are $(p^2-1)/6$ orbits with six elements and the origin, which is fixed. We obtain $$ \sum_{f(m)=b}e_m=5/6(p^2-1). $$ \noindent {\bf Case $II^*$} In this situation $$\alpha(\gamma_b)=\left(\begin{matrix} 0 & -1 \\ 1 & 1 \end{matrix} \right), \quad \phi(\gamma_b)= \left( \begin{array}{c|c} \begin{matrix} 0 & -1 \\ 1 & 1 \end{matrix} & \begin{array}{c} v_1 \\ v_2 \end{array} \\ \hline \begin{array}{cc} 0 & 0 \end{array} & 1 \end{array} \right). $$ Note that conjugation yields $$ \tau_{(-v_2,-v_1-v_2)}^{-1}\phi(\gamma_b)\tau_{-v_2,-(v_1+v_2)} =\left( \begin{array}{c|c} \begin{matrix} 0 & -1 \\ 1 & 1 \end{matrix} & \begin{array}{c} 0 \\ 0 \end{array} \\ \hline \begin{array}{cc} 0 & 0 \end{array} & 1 \end{array} \right), $$ so we may restrict to the case $v=0$. Repeating the argument of Case $II$, we find $$ \sum_{f(m)=b}e_m=5/6(p^2-1). $$ \noindent {\bf Case $III$} Here $$\alpha(\gamma_b)=\left(\begin{matrix} 0 & 1 \\ -1 & 0 \end{matrix} \right), \quad \phi(\gamma_b)= \left( \begin{array}{c|c} \begin{matrix} 0 & 1 \\ -1 & 0 \end{matrix} & \begin{array}{c} v_1 \\ v_2 \end{array} \\ \hline \begin{array}{cc} 0 & 0 \end{array} & 1 \end{array} \right) $$ and after conjugating $$ \tau_{(v_1+v_2)/2,(v_2-v_1)/2}^{-1}\phi(\gamma_b)\tau_{(v_1+v_2)/2,(v_2-v_1)/2} =\left( \begin{array}{c|c} \begin{matrix} 0 & 1 \\ -1 & 0 \end{matrix} & \begin{array}{c} 0 \\ 0 \end{array} \\ \hline \begin{array}{cc} 0 & 0 \end{array} & 1 \end{array} \right), $$ so we may take $v=0$. Since the matrix $\alpha(\gamma_b)$ is semisimple with eigenvalues equal to primitive fourth roots of unity, there are $(p^2-1)/4$ orbits with four elements and the origin, which is fixed. We obtain $$ \sum_{f(m)=b}e_m=3/4(p^2-1). $$ \noindent {\bf Case $III^*$} Here $$\alpha(\gamma_b)=\left(\begin{matrix} 0 & -1 \\ 1 & 0 \end{matrix} \right), \quad \phi(\gamma_b)= \left( \begin{array}{c|c} \begin{matrix} 0 & -1 \\ 1 & 0 \end{matrix} & \begin{array}{c} v_1 \\ v_2 \end{array} \\ \hline \begin{array}{cc} 0 & 0 \end{array} & 1 \end{array} \right) $$ and after conjugating $$ \tau_{(v_1-v_2)/2,(v_1+v_2)/2}^{-1}\phi(\gamma_b)\tau_{(v_1-v_2)/2,(v_1+v_2)/2} =\left( \begin{array}{c|c} \begin{matrix} 0 & -1 \\ 1 & 0 \end{matrix} & \begin{array}{c} 0 \\ 0 \end{array} \\ \hline \begin{array}{cc} 0 & 0 \end{array} & 1 \end{array} \right), $$ so we may take $v=0$. Repeating the argument of Case $III$, we obtain $$ \sum_{f(m)=b}e_m=3/4(p^2-1). $$ \noindent {\bf Case $IV$} Here $$\alpha(\gamma_b)=\left(\begin{matrix} 0 & 1 \\ -1 & -1 \end{matrix} \right), \quad \phi(\gamma_b)= \left( \begin{array}{c|c} \begin{matrix} 0 & 1 \\ -1 & -1 \end{matrix} & \begin{array}{c} v_1 \\ v_2 \end{array} \\ \hline \begin{array}{cc} 0 & 0 \end{array} & 1 \end{array} \right) $$ and after conjugating $$ \tau_{(2v_1+v_2)/3,(-v_1+v_2)/3}^{-1}\phi(\gamma_b)\tau_{(2v_1+v_2)/3,(-v_1+v_2)/3} =\left( \begin{array}{c|c} \begin{matrix} 0 & 1 \\ -1 & -1 \end{matrix} & \begin{array}{c} 0 \\ 0 \end{array} \\ \hline \begin{array}{cc} 0 & 0 \end{array} & 1 \end{array} \right), $$ so we may take $v=0$. Since the matrix $\alpha(\gamma_b)$ is semisimple with eigenvalues equal to primitive third roots of unity, there are $(p^2-1)/3$ orbits with four elements and the origin, which is fixed. We obtain $$ \sum_{f(m)=b}e_m=2/3(p^2-1). $$ \noindent {\bf Case $IV^*$} Here $$\alpha(\gamma_b)=\left(\begin{matrix} -1 & -1 \\ 1 & 0 \end{matrix} \right), \quad \phi(\gamma_b)= \left( \begin{array}{c|c} \begin{matrix} -1 & -1 \\ 1 & 0 \end{matrix} & \begin{array}{c} v_1 \\ v_2 \end{array} \\ \hline \begin{array}{cc} 0 & 0 \end{array} & 1 \end{array} \right) $$ and after conjugating $$ \tau_{(v_1-v_2)/3,(v_1+2v_2)/3}^{-1}\phi(\gamma_b)\tau_{(v_1-v_2)/3,(v_1+2v_2)/3} =\left( \begin{array}{c|c} \begin{matrix} -1 & -1 \\ 1 & 0 \end{matrix} & \begin{array}{c} 0 \\ 0 \end{array} \\ \hline \begin{array}{cc} 0 & 0 \end{array} & 1 \end{array} \right), $$ so we may take $v=0$. Repeating the argument of Case $IV$, we obtain $$ \sum_{f(m)=b}e_m=2/3(p^2-1). $$ \documentclass[12pt]{article} \usepackage{amsmath} \usepackage{epsfig} \usepackage{theorem} \usepackage{amssymb} \newcommand{\ra}{\rightarrow} \newcommand{\bA}{{\mathbb A}} \newcommand{\bC}{{\mathbb C}} \newcommand{\bG}{{\mathbb G}} \newcommand{\bN}{{\mathbb N}} \newcommand{\bP}{{\mathbb P}} \newcommand{\bQ}{{\mathbb Q}} \newcommand{\bR}{{\mathbb R}} \newcommand{\bZ}{{\mathbb Z}} \newcommand{\cC}{{\mathcal C}} \newcommand{\cD}{{\mathcal D}} \newcommand{\cE}{{\mathcal E}} \newcommand{\cF}{{\mathcal F}} \newcommand{\cG}{{\mathcal G}} \newcommand{\cH}{{\mathcal H}} \newcommand{\cK}{{\mathcal K}} \newcommand{\cL}{{\mathcal L}} \newcommand{\cM}{{\mathcal M}} \newcommand{\cO}{{\mathcal O}} \newcommand{\cP}{{\mathcal P}} \newcommand{\cS}{{\mathcal S}} \newcommand{\cX}{{\mathcal X}} \newcommand{\cY}{{\mathcal Y}} \newcommand{\oJ}{{\overline J}} \newcommand{\oA}{{\overline A}} \newcommand{\Sym}{\mathrm{Sym}} \newcommand{\Aut}{\mathrm{Aut}} \newcommand{\Aff}{\mathrm{Aff}} \newcommand{\Gal}{\mathrm{Gal}} \newcommand{\Hom}{\mathrm{Hom}} \newcommand{\rank}{\mathrm{rank}} \newcommand{\et}{\text{\'et}} \newcommand{\Gr}{\mathrm{Gr}} \newcommand{\Tr}{\mathrm{Tr}} \newcommand{\Chow}{\mathrm{Chow}} \newcommand{\Spec}{\mathrm{Spec}} \newcommand{\Sh}{\mathrm{Sh}} \newcommand{\tran}{\mathrm{tran}} \newcommand{\tors}{\mathrm{tors}} \newcommand{\NS}{\mathrm{NS}} \newcommand{\SL}{\mathrm{SL}} \newcommand{\ASL}{\mathrm{AffSL}} \newcommand{\SLtwop}{\mathrm{SL}_2(\mathbb{Z}/p\mathbb{Z})} \newcommand{\SLtwoZ}{\mathrm{SL}_2(\mathbb{Z})} \newcommand{\Ztwop}{(\mathbb{Z}/p\mathbb{Z})^{\oplus 2}} \newcommand{\Zp}{\mathbb{Z}/p\mathbb{Z}} \newtheorem{thm}{Theorem}[section] \newtheorem{prop}[thm]{Proposition} {\theorembodyfont{\rm} \newtheorem{defn}[thm]{Definition} \newtheorem{conj}[thm]{Conjecture} \newtheorem{rem}[thm]{Remark} \newtheorem{lem}[thm]{Lemma} \newtheorem{cor}[thm]{Corollary} \newtheorem{question}[thm]{Question} \newtheorem{example}[thm]{Example}} \begin{document} \begin{prop}\label{prop:reducible} Let $T\in \SLtwoZ$ be an element of finite order $n$, generating a subgroup $H$. Then $n=2,3,4,$ or $6$. If $p$ is a prime $\neq 2,3$, then the reduction $T(\text{mod }p)$ also has order $n$. Let $H'\subset \ASL_2(\Zp)$ so that $q(H')=H$. Then exact sequence \ref{eq:ext} a split exact sequence \begin{equation} \label{eq:ext2} 1 \ra V \ra H' \ra H \ra 1, \quad V:=H' \cap \Ztwop. \end{equation} For each splitting $\sigma':H \hookrightarrow H'$, $\sigma'(H)$ fixes a point of $V$ and is conjugate to a subgroup of $\sigma(\SL_2(\Zp))$, where $\sigma$ is the canonical splitting of exact sequence \ref{eq:ext}. The orbit decomposition of $\Ztwop$ under the action of $H'$ is one of the following: \begin{enumerate} \item{If $\dim_{\Zp}(V)=0$, $H'$ has one fixed point and $(p^2-1)/n$ orbits with $n$ elements.} \item{If $\dim_{\Zp}(V)=1$, $H'$ has one orbit with $p$ elements (the subspace $V$), and $(p-1)/n$ orbits with $pn$ elements.} \item{If $\dim_{\Zp}(V)=2$, $H'$ has one orbit with $p^2$ elements.} \end{enumerate} \end{prop} {\em Proof:} If $T\in \SLtwoZ$ has finite order $n$, it is semisimple and its eigenvalues are primitive $n$th roots of unity. The characteristic polynomial of $T$ is quadratic, so $n=2,3,4,$ or $6$. As $p\ne 2,3$, the reduction of $T(\text{mod }p)$ still has eigenvalues which are $n$th roots of unity, and $T(\text{mod }p)$ has order $n$. The exact sequence \ref{eq:ext2} is clearly induced from exact sequence \ref{eq:ext}; it is split because $|V|$ is prime to $n=|H|$. Now $\sigma'(H)$ is conjugate to a subgroup of $\sigma(\SL_2(\Zp))$, provided $\sigma'(T)$ fixes some point $v\in \Ztwop$. The stabilizer in $\ASL_2(\Zp)$ of a point of $\Ztwop$ is conjugate to $\sigma(\SL_2(\Zp))$. Consider the action of $\sigma'(T)$ on polynomials over $\Ztwop$ of degree $\le 1$ $$c_0+c_1x_1+c_2x_2, \quad c_0,c_1,c_2 \in \Zp.$$ We know $\sigma'(T)$ fixes the constants $c_0$ and has order prime to $p$, so its action decomposes as a direct sum of irreducibles $$\left<1\right> \oplus \left,$$ and the induced action on the second factor is semisimple. The fixed point is $v=(v_1,v_2)$; the orbit analysis in the next paragraph will show that $v\in V$. It remains to analyze the orbit decomposition. In each case, we first conjugate so that $\sigma'(H)\subset \SL_2(\Zp)$. If $V=0$, $H'\subset \SL_2(\Zp)$, generated by a semisimple matrix $\sigma'(T)$ of order $n>1$. The fixed point is the origin and every other orbit has $n$ elements. If $V=\Ztwop$ then $H'$ contains the full translation group, so the action is transitive. Now assume $V$ is one dimensional. Of course, $V$ is an eigenspace for $\sigma'(T)$. The group $H'$ is generated by translations by elements of $V$ and the action of $\sigma'(T)$. Again, the only fixed point under the action of $\sigma'(T)$ is the origin, so any orbit not containing the origin has order divisible by $n$. No element of $\Ztwop$ is fixed under translation by $V$, so each orbit has order divisible by $p$. The description of the orbits follows. $\square$ \begin{prop} Let $\oJ \ra \bP^1$ be a relatively minimal Jacobian elliptic fibration with generic fiber $J_F$. Let $P_F$ be a nontrivial $J_F[p]$-principal homogeneous space, and let $P\ra \bP^1$ be the normalization of $\bP^1$ in $P_F$. Let $p$ be a sufficiently large prime. Suppose that $\oJ$ is isotrivial with $j$-invariant $0$. Then the degenerate fibers of $\oJ$ are $n_0$ fibers of type $I_0^*$, $n_2$ fibers of types $II$ or $II^*$, and $n_4$ fibers of types $IV$ or $IV^*$. If $$1/2n_0+5/6n_2+2/3n_4>2$$ then each irreducible component of $P$ has positive geometric genus. Suppose that $\oJ$ is isotrivial with $j$-invariant $1728$. Then the degenerate fibers of $\oJ$ are $n_0$ fibers of type $I_0^*$ and $n_3$ fibers of types $III$ or $III^*$. If $$1/2n_0+3/4n_3>2$$ then each irreducible component of $P$ has positive geometric genus. \end{prop} Let $G$ be an abelian group and $\Aut(G)$ its automorphism group. Let $\Aff(G)$ be the semidirect product of $G$ by $\Aut(G)$, so that we have an exact sequence \begin{equation} 1 \ra G \ra \Aff(G) \stackrel{q}{\ra} \Aut(G) \ra 1 \label{eq:ext} \end{equation} admitting a splitting $\sigma:\Aut(G)\ra \Aff(G)$. We can interpret $\Aff(G)$ is the permutations of $G$ generated by left translations $$\tau_g:x \ra gx \quad g\in G$$ and automorphisms. Given $g_1,g_2\in G$ and $a_1,a_2\in \Aut(G)$ we have $$ \tau_{g_1}a_1\tau_{g_2}a_2= \tau_{g_1}\tau_{a_1(g_2)} a_1 a_2.$$ Let $E/F$ be a Galois extension with Galois group $\Gamma$. Let $G$ be an abelian group with trivial $\Gamma$-action. Let $P$ be a $G$-homogeneous space over $F$, trivialized on passage to $E$. Such spaces are classified by cocycles $H^1_{\Gamma}(G)$, which are equivalent to the set of group homomorphisms $\Hom(\Gamma,G)$. Let $G'$ be a twist of $G$ over $F$, with nontrivial Galois action trivialized on passage to $E$. How do we interpret $G'$-homogeneous spaces that are trivialized on passage to $E$ in terms of homomorphisms? Such homogeneous spaces coincide with cocycles in $H^1_{\Gamma}(G')$, collections $(g(\gamma))_{\gamma\in \Gamma}$ of elements in $G$, satisfying the cocycle relation $$g(\gamma \gamma')=\gamma(g(\gamma'))+g(\gamma).$$ Each cocyle representative corresponds to a homomorphism $\Gamma \stackrel{\phi}{\ra} \Aff(G)$ so that the composition $q\circ \phi=\alpha$. Indeed, given $(g_{\gamma})_{\gamma\in \Gamma}$ we define $\phi(\gamma)=\tau_{g(\gamma)}\alpha(\gamma)$ so that \begin{eqnarray*} \phi(\gamma\gamma')&=& \tau_{g(\gamma\gamma')}\alpha(\gamma\gamma')\\ &=&\tau_{g(\gamma)} \tau_{\gamma(g(\gamma'))} \alpha(\gamma)\alpha(\gamma')\\ &=&\tau_{g(\gamma)}\alpha(\gamma) \tau_{g(\gamma')}\alpha(\gamma')\\ &=&\phi(\gamma)\phi(\gamma'). \end{eqnarray*} The coboundaries, with $g_{\gamma}=\gamma(g_0)-g_0$ for some $g_0$, satisfy \begin{eqnarray*} \phi(\gamma)&=&\tau_{g(\gamma)}\alpha(\gamma)= \tau_{-g_0+\gamma(g_0)}\alpha(\gamma)\\ &=&\tau_{-g_0}\tau_{\gamma(g_0)}\alpha(\gamma)= \tau_{g_0}^{-1}\alpha(\gamma)\tau_{g_0}\alpha(\gamma)^{-1}\alpha(\gamma)\\ &=&\tau_{g_0}^{-1}\alpha(\gamma)\tau_{g_0}. \end{eqnarray*} Then $\phi$ factors through the stabilizer of $-g_0$ in $\Aff(G)$ and is conjugate to the homomorphism induced by the splitting $\sigma$ (which stabilizes $0$). \begin{prop} \label{prop:cocycle} Let $G'$ be a $\Gamma$-twisted form of an abelian group $G$, with classifying cocycle $\alpha\in H^1_{\Gamma}(\Aut(G))=\Hom(\Gamma,\Aut(G)).$ Then $H^1_{\Gamma}(G')$ corresponds to $G$-conjugacy classes of homomorphisms $$\phi:\Gamma \ra \Aff(G), \text{ with }\quad q\circ \phi=\alpha.$$ The identity corresponds to $\sigma\circ \alpha$. \end{prop} First, we compute the genus of an irreducible component $f:M^{\nu}\ra \bP^1$ of the normalization of $\bP^1$ in $J_F[p]-\{0\}$. Each such component corresponds to an nonzero $H$-orbit $\cM \subset \Ztwop$, which has $n$ elements. Hence the Riemann-Hurwitz formula is $$2g(M^{\nu})-2=-2n+\sum_{m\in M^{\nu}} e_m$$ where $e_m$ is the ramification at $m\in M^{\nu}$. For each $b\in \bP^1$, we compute the contribution of $\sum_{f(m)=b}e_m$ to the total ramification. Only points corresponding to singular fibers of $\oJ$ contribute; the $p$-torsion is unramified at smooth fibers. Let $\gamma_b \in \Gamma$ be the class of a loop around $b$, or a local generator for the absolute Galois group of the completion $\hat{F}_b$. Each point $m\in f^{-1}(b)$ corresponds to an orbit of $\alpha(\gamma_b)$ on $\cM$, and the number of elements of the orbit is equal to $e_m+1$. As $\alpha(\gamma_b)$ is the $(\text{mod }p)$ reduction of $\rho(\gamma_b)$, we obtain $$ \sum_{f(m)=b} e_m=n(1-1/\text{order}(\rho(\gamma_b)). $$ \begin{rem}\label{rem:except} The classification theory of linear series on K3 surfaces (\cite{SD} \S 2.6,2.7) implies that for any polarization $f$ with basepoints, there exists a smooth elliptic curve $E\subset X$ with $f.E=1$. We may write $f=nE+\Sigma$ where $\Sigma$ is a section: $$ \begin{array}{c|cc} & E & \Sigma \\ \hline E & 0 & 1 \\ \Sigma & 1 & -2 \end{array}. $$ The line bundles $f=nE+\Sigma$ are ample but have fixed component $\Sigma$. \end{rem} The assumption that $|f|$ is basepoint-free implies that the $f.C>1$. Suppose that $y_1,\ldots,y_N\in X$ are generic, defined over a number field. Then there exists a {\em nonsingular} $D\in |f|$ of genus $N$, defined over a number field and containing $y_1,\ldots,y_N$. Since $D^{(N)}$ is birational to $J^N(D)$, $D^{(N)}$ has potentially dense rational points. Consider the symmetric product $X^{(N)}$ and the induced abelian fibration \begin{eqnarray*} X^{(N)} &\longrightarrow& (\bP^1)^{(N)}\simeq \bP^N\\ x_1+\ldots+x_N & \longrightarrow & \eta(x_1)+\ldots+\eta(x_N). \end{eqnarray*} The symmetric product $D^{(N)}\subset X^{(N)}$ gives a multisection, which is nondegenerate by the argument of Proposition \ref{prop:bigcase}, once we choose $y_1,\ldots,y_N$ generically. We may apply Proposition \ref{prop:dense} to conclude the proof. We give one last consequence of the proof of Theorem \ref{thm:ratmult}. \begin{prop} \label{prop:JacMult} Let $\oJ \ra \bP^1$ be a nonisotrivial Jacobian elliptic K3 surface. Suppose that $M$ is one of the multisections given by Theorem \ref{thm:ratmult}. Then $M$ is nontorsion. \end{prop} {\em Proof:} We retain the notation of the proof of Theorem \ref{thm:ratmult}, with $X=\oJ$, so that $[X']$ has order $p$ in $\Sh(J_F)$. If $M$ is torsion then $M'\subset X'$ is torsion as well. If $N$ is the order of $M'$ then $\mu_N(M')$ is a section of $J^N(X')$, so $p|N$ by Proposition \ref{prop:multiply}. Write $N=np$ and consider the image of $M'$ under multiplication $\mu_n$. As $\mu_n(M')$ is a $p$-torsion multisection dominated by $M'$, it also has relative degree $p$. By Proposition \ref{prop:multPHS}, the generic fiber $\mu_n(M')_F$ is contained in a principal homogeneous space for the $p$-torsion. However, the argument for Theorem \ref{thm:ratmult} shows that no component of such a principal homogeneous space has relative degree $p$ over the base. $\square$ \begin{cor} Let $\oJ \ra \bP^1$ be a nonisotrivial, Jacobian elliptic K3 surface, defined over a number field. Then rational points are potentially dense on $\oJ$. \end{cor} This follows from Propositions \ref{prop:JacMult} and \ref{prop:dense}. In light of the exact seqences above, the reduction map $$H^1(\Gamma,J_F[m]) \ra H^1(\hat{\Gamma}_b,J_{\hat{F}_b}[m])$$ can often shed light on Tate-Shafarevich group. The following proposition can sometimes be used to describe its image, e.g., in the case of additive reduction: \begin{prop}[Corollary of \cite{Shaf1}, \S 1.2.] \label{prop:simplify} Let $\hat{F}=\bC((t))$ with absolute Galois group $\hat{\Gamma}$, and $J_{\hat{F}}$ an elliptic curve over $\hat{F}$. Then for each $m$ $$|H^1(\hat{\Gamma},J_{\hat{F}}[m])|=|J_{\hat{F}}(\hat{F})[m]|.$$ \end{prop} \begin{rem} \label{rem:thinair} One might be tempted to combine this with Proposition \ref{prop:transellip} to prove potential density for all elliptic K3 surfaces $\eta:X\ra \bP^1$. An impediment is that the elliptic curves produced by Proposition \ref{prop:getcurve} may all be fibers of $\eta$. For instance, consider general elliptic K3 surfaces with one section $\Sigma$. The Picard lattice is $$ \begin{array}{c|cc} & E & \Sigma \\ \hline E & 0 & 1 \\ \Sigma & 1 & -2 \end{array}. $$ and the nef cone is generated by $E$ and $2E+\Sigma$. Each effective divisor takes the form $f=h+a\Sigma$, where $h$ is nef and $a\ge 0$ \cite{SD}, and $h\neq 0$ whenever $h^0(X,f)>1$. But then $f-E$ is always effective, so the curve produced by Proposition \ref{prop:getcurve} might always be $E$. \end{rem} \end{document}