The \(\ell\)-parity conjecture for abelian varieties over function fields of characteristic \(p>0\) (Q2877489)

Denote by \(A\) an abelian variety over a function field \(K\) of characteristic \(p>0\) and let \(\ell\) be a prime number (\(\ell=p\) allowed). Fix \(\overline{K}\) a separable closure of \(K\) and let \(C/\mathbb{F}_q\) be the proper smooth connected curve with function field \(K\), and let \(f:\mathcal{A}\rightarrow C\) the Neron model of \(A/K\). Denote by \(U\) the open subset of \(C\) where we have removed the places of bad reduction. We can associate finite dimensional \(\mathbb{Q}_{\ell}\)-vector spaces in \(\ell\)-adic étale cohomology for the \(\ell\)-adic sheaf \(V_{\ell}(\mathcal{A})\) from \(\ell^n\)-torsion of \(\mathcal{A}\) for \(\ell\neq p\) noted by \(H^i_{\mathrm{et},c}(\overline{U},V_{\ell}(\mathcal{A}))\) and a isocrystal associated to \(\mathcal{A}\) in rigid cohomology denoted by \(H^i_{\mathrm{rig},c}(U,D^{\dag}(\mathcal{A}))\) when \(\ell=p\). We denote these cohomology groups by \(H^i_{\ell}\) in the following. \(H^i_{\ell}\) is endowed with a Frobenius operator \(\varphi_{\ell}\) (when \(\ell\neq p\) it is induced by the geometric Frobenius). Denote by \(I_{2,\ell}:=\mathrm{ker}(\mathrm{id}-\varphi_{\ell},H^1_{\ell})\) and by \(I_{3,\ell}\) the part of \(H^{1}_{\ell}\) where \(\varphi_{\ell}\) acts unipotently, and if the semisimplicity conjecture holds for \(H^1_{\ell},\varphi\) then \(I_{2,\ell}=I_{3,\ell}\).NEWLINENEWLINE\textit{K. Kato} and \textit{F. Trihan} [Invent. Math. 153, No. 3, 537--592 (2003; Zbl 1046.11047)] observed in particular that \(\dim_{\mathbb{Q}_{\ell}}(I_{2,\ell})\) is equal to the corank of the \(\ell\) discrete Selmer group \(\mathrm{Sel}_{\ell^{\infty}}(A/K)\) of \(A/K\) and that the \(\dim_{\mathbb{Q}_{\ell}}(I_{3,\ell})\) is equal to the order at \(s=1\) of the Hasse-Weil \(L\)-function of \(A/K\).NEWLINENEWLINETherefore, the \(\ell\)-parity conjecture for abelian varieties claims that NEWLINE\[NEWLINE\dim_{\mathbb{Q}_{\ell}}(I_{2,\ell})\equiv \dim_{\mathbb{Q}_{\ell}}(I_{3,\ell})\pmod{2}.NEWLINE\]NEWLINE The idea to prove the \(\ell\)-parity conjecture is to construct a perfect symmetric pairing and compatible with the Frobenius action NEWLINE\[NEWLINE(\cdot,\cdot)_{\ell}:I_{3,\ell}\times I_{3,\ell}\rightarrow \mathbb{Q}_{\ell}NEWLINE\]NEWLINE such that from a result of \textit{J. Nekovář} [Doc. Math., J. DMV 12, 243--274 (2007; Zbl 1201.11067)] one obtains \(\mathrm{det}(-\varphi_{\ell})=(-1)^{\dim_{\mathbb{Q}_{\ell}}(\mathrm{ker}(1-\varphi_{\ell}))}\), implying the parity conjecture.NEWLINENEWLINEThe goal of Trihan and Yasuda in the paper under review is to construct the above pairing. They construct the perfect pairing compatible with Frobenius from the pairing in étale cohomology between a cohomology module with coefficient in a sheaf from the Tate module of \(A\) and another étale cohomology module with coefficients in a sheaf from the Tate module of the dual abelian variety \(A^t\) constructed by \textit{R. Kiehl} and \textit{R. Weissauer} [Weil conjectures, perverse sheaves and \(l\)-adic Fourier transform. Berlin: Springer (2001; Zbl 0988.14009)] and for the rigid geometry they use a result of \textit{R. Crew} [Ann. Sci. Éc. Norm. Supér. (4) 31, No. 6, 717--763 (1998; Zbl 0943.14008)] also involving a pairing between one rigid cohomology module constructed from \(A\) and the other module constructed from \(A^t\). For example, when \(\ell\neq p\) they translate the \(H^1_{\ell}\) to \(\mathcal{H}^1_{\ell}:=H^1_{et}(\overline{C}, j_*V_{\ell}(\mathcal{A}))\) where \(j:U\hookrightarrow C\) the canonical inclusion, and define \(\mathcal{I}_{2,\ell}:=\mathrm{ker}(\mathrm{id}-\varphi_{\ell};\mathcal{H}_{\ell}^1)\) and \(\mathcal{I}_{3,\ell}\) the part of \(\mathcal{H}^1_{\ell}\) where \(\varphi_{\ell}\) acts unipotently. In particular, the authors prove that for any \(k\in\{2,3\}\) \(\mathcal{I}_{k,\ell}\cong I_{k,\ell}\).NEWLINENEWLINEFixing a polarization of \(A\) one induces an isogeny between \(A\) and \(A^t\) and an isomorphism \(\lambda:\mathcal{H}^1_{\ell}\cong \mathcal{H}^1_{\ell,t}\) where \(t\) means that the cohomology module is the one related with the dual abelian variety in the above perfect pairing. Thus the authors prove, under the fix \(\lambda\), a nondegenerate symmetric pairing compatible with the Frobenius action NEWLINE\[NEWLINE\mathcal{H}^1_{\ell}\times\mathcal{H}^1_{\ell}\rightarrow\mathbb{Q}_{\ell}.NEWLINE\]NEWLINE The above pairing restricts to \(\mathcal{I}_{3,\ell}\) and by the above result of Nekovář the authors can conclude the proof of the \(\ell\)-parity conjecture in this setting.NEWLINENEWLINEMoreover, the authors extend the above method to any smooth \(\ell\)-adic sheaf \(F_{\ell}\) (\(\ell\neq p\)) and any overconvergent \(F\)-isocrystal \(F_p\), both of pure weight -1 and endowed with a non-degenerate pairing , and to compatible families of such objects relating the analytic rank of the \(L\)-function associated to \(F_{\ell}\) to a rank defined from the cohomology of the Selmer complex associated to \(F_{\ell}\).NEWLINENEWLINEFinally, for \(\ell\neq p\) the authors generalize the techniques to a proper smooth variety \(X\) of dimension \(d\) over a finite field, \(U\) an open dense subscheme of \(X\) and \(F\) a smooth \(\overline{\mathbb{Q}}_{\ell}\) sheaf on \(U\) of pure weight \(w\). We can associated a complex \(L\)-function and denote by \(r_{\mathrm{an}}(F,n)\) the order of the zero of \(L\) at \(n\). Assume \(d\) and \(w\) are odd (for an even version see the paper under review). Also assume that \(F\) is endowed with a skew-symmetric non-degenerate pairing \(F\times F\rightarrow\mathbb{Q}_{\ell}(w)\), then the authors prove in the paper the following sort of \(\ell\)-adic pairing conjecture: NEWLINE\[NEWLINEr_{\mathrm{an}}(F,(d+w)/2)\equiv \dim_{\mathbb{Q}_{\ell}}(H^{2(\frac{d+w}{2})-2}_{\mathrm{et}}(X,G(n))\pmod{2},NEWLINE\]NEWLINE where \(G\) denote the intermediate extension of the sheaf \(F\) over \(U\) to \(X\).NEWLINENEWLINEWith this result they specialize to the primitive part, twisted conveniently, of \(H^{2i}_{\mathrm{et}}(\overline{V},\mathbb{Q}_{\ell}(i))\) which appears as a direct sum decomposition, each of them has a non-degenerate symmetric pairing, ans \(V\) is a projective and smooth variety of pure dimension \(d\) over a finite field of characteristic \(p\), proving for any integer \(i\) with \(0\leq i\leq d\): NEWLINE\[NEWLINEr(i)\equiv \dim_{\mathbb{Q}_{\ell}}H^{2i}_{\mathrm{et}}({V},\mathbb{Q}_{\ell}(i))\pmod{2},NEWLINE\]NEWLINE where \(r(i)\) is the order of the pole of the Zeta function of \(V\) at \(s=n\).NEWLINENEWLINERecall that the Artin-Tate conjecture [\textit{J. Tate}, Proc. Symp. Pure Math. 55, Pt. 1, 71--83 (1994; Zbl 0814.14009)] affirms that \( r(i)= \dim_{\mathbb{Q}_{\ell}}H^{2i}_{\mathrm{et}}({V},\mathbb{Q}_{\ell}(i))\) always, and moreover \(r(i)\) is equal to the rank of the numerical equivalence classes of cycles of codimension \(i\); this conjecture is called \(T(i)\). The authors observed that the conjecture \(T(1)\) for projective smooth surfaces over finite fields is equivalent to the BSD conjecture for abelian varieties over function fields in one variable over finite fields of characteristic \(p>0\), whose proof is independent of the above results of the paper and is a consequence of the results in [Kato and Trihan (loc. cit.); \textit{Q. Liu} et al., Invent. Math. 159, No. 3, 673--676 (2005; Zbl 1077.14023); \textit{D. Ulmer}, ``Curves and Jacobians over function fields'', Preprint, \url{http://people.math.gatech.edu/~ulmer/research/preprints/C.pdf}].