Congruences between derivatives of geometric \(L\)-functions. With an appendix by David Burns, King Fai Lai and Ki-Seng Tan
The main theorem of this article establishes the truth of a deep conjecture concerning the leading terms of \(L\)-functions of Abelian extensions of global function fields. The conjecture, which we shall discuss in more detail below, is a refinement of a conjecture of Lichtenbaum relating the Weil-étale cohomology of curves over finite fields to the associated zeta function. It can also be seen as a function field analogue of a conjecture concerning \(L\)-functions of Galois extensions of number fields, namely the equivariant Tamagawa number conjecture of Burns-Flach in the setting of Artin \(L\)-functions at \(s= 0\). The theorem also implies the truth of a number of related conjectures, as is discussed in the article. Explicitly, all the following now hold in the function field setting: Chinburg's \(\Omega(3)\) conjecture for Abelian extensions, Chinburg's \(\Omega(1)\) conjecture for tamely ramified Abelian extensions, the congruence conjecture of [\textit{D. Burns}, Invent. Math. 169, No. 3, 451--499 (2007; Zbl 1133.11063)] (whence the title of the article), Rubin's integral refinement of Stark's conjecture, the refined class number formulas of Gross, Tate, and Aoki-Lee-Tan, and finally the refined \(p\)-adic abelian Stark conjecture of Gross. Let us now give a brief formulation of the theorem, omitting many details. We fix a finite Abelian extension \(F/k\) of global function fields with Galois group \(G\). The constructions take place in the Picard category \({\mathcal P}(R)\) of graded line bundles on \(\text{Spec}(R)\) with \(R= \mathbb Z[G]\) and \(R= \mathbb Q[G]\). The category \({\mathcal P}(R)\) has unit object \(\mathbf{1}_R\). Further, there is the determinant functor \({\mathbf d}_R\) of Grothendieck-Knudsen-Mumford which assigns to a perfect complex of \(R\)-modules an object of \({\mathcal P}(R)\). If \(C^*\) is a perfect \(\mathbb Q[G]\)-complex acyclic outside two consecutive degrees, then any isomorphism \(\lambda\) between the cohomology groups in those consecutive degrees induces a morphism \(\vartheta_\lambda:{\mathbf d}_{\mathbb Q[G]}(C^*)\to\mathbf{1}_{\mathbb Q[G]}\). In the arithmetic setting, there is a natural choice for the complex \(C^*\) and for the isomorphism \(\lambda\). Namely, \(C^*= \mathbb Q[G]\otimes^{\mathbb L}_{\mathbb Z[G]} D^*\), where \(D^*\) is the complex \[ R\,\Hom_{\mathbb Z}(R\Gamma_{\text{Wét}}(C_F,j_!\mathbb Z), \mathbb Z[-2]). \] Here, \(C_F\) is the irreducible smooth projective curve associated to \(F\), Wét denotes the Weil-étale topology of Lichtenbaum, \(j: \text{Spec}({\mathcal O}_{F,\Sigma})\to C_F\) is the natural open immersion, and \(\Sigma\) is a finite set of places of \(k\) containing the ramified ones. The natural choice for \(\lambda\) is the valuation map from \(\Sigma\)-units to the augmentation kernel in the free Abelian group on places of \(F\) above \(\Sigma\). The main theorem then says that, interpreted correctly, \[ \vartheta_\lambda({\mathbf d}_{\mathbb{Z}[G]}(D^*))= Z^*_{F/k,\Sigma}(1)\cdot\mathbf{1}_{\mathbb{Z}[G]}) \] in \({\mathcal P}(\mathbb{Z}[G])\), where \(Z^*_{F/k,\Sigma}(1)\) is the equivariant leading term element defined in terms of the Zeta functions \(Z_\Sigma(\chi, t)\) associated to the characters \(\chi\) of \(G\). The proof of the theorem makes use of a result of \textit{R. Crew} [Invent. Math. 88, 395--403 (1987; Zbl 0615.14013)] on \(L\)-functions of \(p\)-adic characters associated to a smooth curve over a finite field. Indeed, the equality in the theorem has an analogue in an Iwasawa theoretic setting, and the author shows that the analogous equality over the Iwasawa algebra can be reduced to Crew's result. In order to then apply the argument to finite extensions, the author employs descent arguments lie developed earlier with Venjakob. The paper concludes with an appendix that the author wrote with King Fai Lai and Ki-Seng Tan. The goal of the appendix is to show that the result of Crew mentioned-above, which originally was proven with recourse to crystalline cohomology, can in fact be established from work of Weil relating zeta functions in the function field situation to \(\ell\)-adic homology, with \(\ell\) different from the residue characteristic. This has the consequence that the results of the main article itself rely nowhere on crystalline cohomology.
scientific article; zbMATH DE number 5899561
scientific article