Let \(C\) be a complete, smooth, geometrically irreducible curve over \(\mathbb{F}_ q\), \(q\) odd, with function field \(L\) and jacobian \(J\). Let \(g\) be the genus of \(L\) and let \(h\) denote the number of \(\mathbb{F}_ q\)-points of \(J\). Also, let \((E,{\mathcal O})\) be an elliptic curve over \(L\), and assume its 2-torsion points are rational over \(L\). Denote by \({\mathcal E}\) the complete Néron model of \(E\), by \(p:{\mathcal E}\to C\) the \(\mathbb{F}_ q\)- rational projection, and by \(\sigma_ 0:C\to{\mathcal E}\) the zero section. There are three more \(\mathbb{F}_ q\)-rational sections \(\sigma_ i\), \(i=1,2,3\), corresponding to the 2-torsion points on the generic fiber. The four \(\sigma_ i\)’s intersect the fibers \(p^{-1}(v)\) in four distinct points. Write \(q_ v=q^{\deg(v)}\) for each place \(v\) of \(L\). A key role is played by the residue fields \(K^{(v)}=K({\mathcal E}_ v,\sigma_ 0)\) of the local rings of the components of \(p^{-1}(v)\) containing \(\sigma_ 0(v)\). \(K^{(v)}\) is isomorphic to the function field of such a component of the corresponding fiber. For each place \(v\) of \(L\) such that \(p^{-1}(v)\) is irreducible one can define a quaternion algebra \(A_ v\) over \(K^{(v)}\) which is ramified at \(\sigma_ i(v)\cap p^{-1}(v)\), \(i=1,2\), and which is unramified at all other places of \(K^{(v)}\). Write \(S_{(v)}\) for the subring of \(K^{(v)}\) consisting of functions with only poles at \(\sigma_ 0(v)\cap p^{- 1}(v)\), and let \(M^ 1_ v\) denote the set of elements of a maximal \(S_{(v)}\)-order \(M_ v\) of \(A_ v\) with reduced norm equal to one. For the \(M^ 1_ v\) one can define and calculate an Euler-Poincaré characteristic \(\chi_ v(M^ 1_ v)\), essentially by using the Euler- Poincaré [cf. J. P. Serre in: Prospects Math., Ann. Math. Stud. 70, 77-169 (1971; Zbl 0235.22020)] and Tamagawa measures on \(SL_ 2\). With obvious notation for the zeta-functions, define \(K_ v(s)=\zeta({\mathcal E}_ v,s)\zeta_{K^{(v)}}(s){-1}\). Then, for smooth \({\mathcal E}_ v\), \(K_ v(s)=1\), so \(K_ v(s)\) can only be non-trivial for the singular fibers of \(p\). On the other hand, using results of J. Milne, one has for the ‘motivic’ Euler-Poincaré characteristic \(\chi({\mathcal E},\mathbb{Z})\) of \({\mathcal E}\) with \(\mathbb{Z}\)-coefficients (in the étale cohomology), \[ \chi({\mathcal E},\mathbb{Z})={\bigl[H^ 0({\mathcal E},\mathbb{Z})_{\text{tor}}\bigr]\bigl[H^ 2({\mathcal E},\mathbb{Z})_{\text{cotor}}\bigr]\bigl[H^ 4({\mathcal E},\mathbb{Z})\bigr]\over\bigl[H^ 1({\mathcal E},\mathbb{Z})\bigr]\bigl[H^ 3({\mathcal E},\mathbb{Z})\bigr]\bigl[H^ 5({\mathcal E},\mathbb{Z})\bigr]}, \] where \([\cdots]\) denotes the order. We can now state the main result of the paper \((c_ 2({\mathcal E})\) is the second Chern class of the tangent bundle of \({\mathcal E})\):
Theorem. One has the following relation between the weighted product of the \(\chi_ v(M^ 1_ v)\) and \(\chi({\mathcal E},\mathbb{Z})\): \[ \chi({\mathcal E},\mathbb{Z})=q^{1-g-c_ 2({\mathcal E})}h(q-1)^{-1}\prod_ vq_ v^{- 3}(1-q_ v^{-1})^{-2}|\chi_ v(M^ 1_ v)|\cdot \prod_ wq_ w^{-3}K_ w(2)\cdot\prod_ x\zeta({\mathcal E}_ x,2)(1-q_ x^{-1}), \] where the first product extends over all irreducible fibers of \(p\), the second over all singular irreducible fibers, and the third over all reducible fibers.
The paper closes with an appendix on the association of an algebraic family of quaternion algebras to the smooth fibers of an elliptic surface with level two structure.