On the \(\ell\)-adic cohomology of Jacobian elliptic surfaces over finite fields (Q2374204)

Let \(k\) be a finite field of characteristic \(p > 0\) and \(C/k\) be a smooth projective geometrically integral curve with function field \(K = k(C)\). Let \(E_0/K\) be an elliptic curve with associated Kodaira-Neron surface \(\pi: S_0 \to C\), the minimal regular completion of the Neron model \(\mathcal{E}_0 \to C\). For \(\ell \neq p\) prime, let \(\mathcal{E}_0(\ell)^c \to C\) be the regular completion of the Neron model \(\mathcal{E}_0 \to C\) of the etale group scheme \(E_0(\ell)/K\) given by the \(\ell\)-primary component of the torsion subgroup of \(E_0\). The inclusion map \(E_0(\ell) \hookrightarrow E_0\) extends to a morphism \(h: \mathcal{E}_0(\ell)^c \to S_0\). The author studies the restriction map \(h^*: H^2(S_0,\mathbb{Z}_\ell(1)) \to H^2(\mathcal{E}_0(\ell)^c,\mathbb{Z}_\ell(1))\) in \(\ell\)-adic etale cohomology. Let \(F^1H^2(S_0,\mathbb{Z}_\ell(1))\) be the first term of the filtration of \(H^2(S_0,\mathbb{Z}_\ell(1))\) coming from the Leray spectral sequence of \(\pi: S_0 \to C\) and the \(\ell\)-adic sheaf \(\mathbb{Z}_\ell(1)\) on \(S_0\) and \(F^1H^2(S_0,\mathbb{Z}_\ell(1))^0\) the kernel of the restriction map \(0^*: F^1H^2(S_0,\mathbb{Z}_\ell(1)) \to H^2(C,\mathbb{Z}_\ell(1))\) by the zero section of \(S_0\). Suggested by the Tate conjecture \(c_1: \mathrm{Pic}(S_0) \otimes_\mathbb{Z} \mathbb{Z}_\ell = H^2(S_0, \mathbb{Z}_\ell(1))\) or \(\mathrm{Pic}^0(S_0/C) \otimes_\mathbb{Z} \mathbb{Z}_\ell = F^1H^2(S_0,\mathbb{Z}_\ell(1))^0\), the author proves that the maps \[ h^*: F^1H^2(S_0,\mathbb{Z}_\ell(1))^0 \to H^2(\mathcal{E}_0(\ell)^c,\mathbb{Z}_\ell(1)) \] and \[ F^1H^2(S_0,\mathbb{Z}_\ell(1))^0 \to H^1(C,T_\ell\mathcal{E}_0^0) \to H^1(C,T_\ell\mathcal{E}_0) \to \mathrm{Pic}(\mathcal{E}_0(\ell)^c)^{\mathrm{inv}} \hookrightarrow H^2(\mathcal{E}_0(\ell)^c,\mathbb{Z}_\ell(1)) \] coincide with \(\mathrm{Pic}(\mathcal{E}_0(\ell)^c)^{\mathrm{inv}} := \{\xi \in \mathrm{Pic}(\mathcal{E}_0(\ell)^c) \mid s^*(\xi) = p_1^*(\xi) + p_2^*(\xi)\}\) where \(p_i: \mathcal{E}_0(\ell)^c \times_C \mathcal{E}_0(\ell)^c \to \mathcal{E}_0(\ell)^c\) are the projection maps and \(s\) the addition map. As a corollary, one gets that for almost all primes \(\ell \neq p\), the map \(h^*\) is injective on \(F^1H^2(S_0,\mathbb{Z}_\ell(1))^0\) yielding an isomorphism \(F^1H^2(S_0,\mathbb{Z}_\ell(1))^0 = \mathrm{Pic}(\mathcal{E}_0(\ell)^c)^{\mathrm{inv}}\).