Degeneration of \(l\)-adic Eisenstein classes and of the elliptic polylog.

*(English)*Zbl 0955.11027Let \(F= \mathbb{Q}(\mu_N)\), \(N\geq 3\), be a cyclotomic field. For any \(\omega\in\mu_N\), \(\omega\neq 1\), for any integer \(k\geq 1\), A. Beilinson has constructed a “cyclotomic element in \(K\)-theory”, say \(B_k(\omega)\), which lives in \(K_{2k+1}(F)\) and which is subject to “comparison theorems”, according to the philosophy of motives. This means that there are various regular maps which go from \(K\)-theory to relevant cohomologies, and these maps send \(B_k(\omega)\) to remarkable elements in the cohomology groups: the image of \(B_k(\omega)\) by the Beilinson regulator from \(K\)-theory to Deligne cohomology is given by special values of classical polylog functions, and for all primes \(\ell\), its images by the higher Chern classes \(K_{2k+1} (F)\otimes \mathbb{Z}_\ell\to H_{\text{cont}}^1 (F,\mathbb{Z}_\ell (k+1))\) agree with “Soulé’s cyclotomic elements”. The interest in the comparison theorems comes from their arithmetical applications, namely the Tamagawa number conjecture of Bloch and Kato for the motives \(\mathbb{Q}(n)\) over a number field (which is actually a conjecture on special values of the Dedekind zeta function).

The first comparison theorem above was proved by A. A. Beilinson [J. Sov. Math. 30, 2036-2070 (1985); translation from Itogi Nauki Tekhn., Ser. Sovrem. Probl. Mat. 24, 181-238 (1984; Zbl 0588.14013)]. A sketch of proof of the second one was also given by him in his 1990 MIT preprint, “Polylog and cyclotomic elements”. The complete proof was achieved by A. Huber and G. Wildeshaus [“Classical motivic polylog according to Beilinson and Deligne”, Doc. Math., J. DMV 3, 27-133 (1998; Zbl 0906.19004)].

In this paper, the authors present an alternative proof, which is shorter and technically “less demanding”. The first key idea, building on Anderson and Harder, is to construct mixed motives for cyclotomic fields via modular curves: Let \(M\) be the modular curve over \(F^*\) which parametrizes elliptic curves with full level-\(N\)-structure, let \({\mathcal H}\) be the Tate module of the universal elliptic curve over \(M\). The “Eisenstein symbol” gives elements in \[ H_{\text{ét}}^1 (M, \text{Sym}^k{\mathcal H}(1))= H_{\text{ét}}' (M_{\overline{F}}, \text{Sym}^k{\mathcal H}(1))^{G(\overline{F}/F)}. \] Over \(\mathbb{C}\), they are related to Eisenstein series. In the \(\ell\)-adic word, the functor \(i^* Rj_*\) (where \(j:M\to \overline{M}\) is the compactification of \(M\) and \(i: \operatorname {Spec}F\to \overline{M}\) is the inclusion of the cusp \(\infty\)) induces a little diagram which we don’t need to write down here, but which means that an Eisenstein symbol whose residue in \(H_{\text{ét}}^0 (\operatorname {Spec} F,i^* R^1j_* \operatorname {Sym}^k{\mathcal H}(1))= \mathbb{Q}_\ell\) vanishes, induces an element of \[ H_{\text{ét}}^1 (\operatorname {Spec} F,i^* j_*\operatorname {Sym}^k{\mathcal H}(1))= H_{\text{Cont}}^1 (F,\mathbb{Q}_\ell (k_1)). \] The construction works in \(K\)-theory as well: Beilinson has constructed the Eisenstein symbol as an element of \(K\)-groups of powers of the universal elliptic curve, and the cup-product construction in \(\ell\)-adic cohomology is a different way of writing down the diagram which was not written down earlier.

The second step is to compute the degeneration of Eisenstein symbols with residue zero. In the \(\ell\)-adic case, the authors use the machinery of polylog. All Eisenstein symbols appear as components of the “cohomological polylog” \({\mathcal P}ol^{\text{coh}}\) on the universal elliptic curve, more precisely as linear combinations of fibers \(\psi^* {\mathcal P}ol^{\text{coh}}\), where \(\psi\) is a formal linear combination of torsion sections. If \(\psi^* {\mathcal P}ol^{\text{coh}}\) has residue 0 at infinity, the degeneration \(i^* j_* \psi^* {\mathcal P}ol^{\text{coh}}\) is equal to the degeneration of the usual “elliptic polylog” \(i^* j_* \psi^* {\mathcal P}ol(-1)\). This degeneration is known and is equal to the classical polylog. Its value at torsion sections is again known and given by a linear combination of cyclotomic elements in \(\ell\)-adic cohomology. This yields the comparison theorem.

The paper also contains three appendices reviewing the construction of the elliptic and the classical polylog in unified form (A), studying the degeneration of the elliptic into the classical polylog (B), and treating the relation of the cohomological polylog with the Eisenstein symbol (C).

The first comparison theorem above was proved by A. A. Beilinson [J. Sov. Math. 30, 2036-2070 (1985); translation from Itogi Nauki Tekhn., Ser. Sovrem. Probl. Mat. 24, 181-238 (1984; Zbl 0588.14013)]. A sketch of proof of the second one was also given by him in his 1990 MIT preprint, “Polylog and cyclotomic elements”. The complete proof was achieved by A. Huber and G. Wildeshaus [“Classical motivic polylog according to Beilinson and Deligne”, Doc. Math., J. DMV 3, 27-133 (1998; Zbl 0906.19004)].

In this paper, the authors present an alternative proof, which is shorter and technically “less demanding”. The first key idea, building on Anderson and Harder, is to construct mixed motives for cyclotomic fields via modular curves: Let \(M\) be the modular curve over \(F^*\) which parametrizes elliptic curves with full level-\(N\)-structure, let \({\mathcal H}\) be the Tate module of the universal elliptic curve over \(M\). The “Eisenstein symbol” gives elements in \[ H_{\text{ét}}^1 (M, \text{Sym}^k{\mathcal H}(1))= H_{\text{ét}}' (M_{\overline{F}}, \text{Sym}^k{\mathcal H}(1))^{G(\overline{F}/F)}. \] Over \(\mathbb{C}\), they are related to Eisenstein series. In the \(\ell\)-adic word, the functor \(i^* Rj_*\) (where \(j:M\to \overline{M}\) is the compactification of \(M\) and \(i: \operatorname {Spec}F\to \overline{M}\) is the inclusion of the cusp \(\infty\)) induces a little diagram which we don’t need to write down here, but which means that an Eisenstein symbol whose residue in \(H_{\text{ét}}^0 (\operatorname {Spec} F,i^* R^1j_* \operatorname {Sym}^k{\mathcal H}(1))= \mathbb{Q}_\ell\) vanishes, induces an element of \[ H_{\text{ét}}^1 (\operatorname {Spec} F,i^* j_*\operatorname {Sym}^k{\mathcal H}(1))= H_{\text{Cont}}^1 (F,\mathbb{Q}_\ell (k_1)). \] The construction works in \(K\)-theory as well: Beilinson has constructed the Eisenstein symbol as an element of \(K\)-groups of powers of the universal elliptic curve, and the cup-product construction in \(\ell\)-adic cohomology is a different way of writing down the diagram which was not written down earlier.

The second step is to compute the degeneration of Eisenstein symbols with residue zero. In the \(\ell\)-adic case, the authors use the machinery of polylog. All Eisenstein symbols appear as components of the “cohomological polylog” \({\mathcal P}ol^{\text{coh}}\) on the universal elliptic curve, more precisely as linear combinations of fibers \(\psi^* {\mathcal P}ol^{\text{coh}}\), where \(\psi\) is a formal linear combination of torsion sections. If \(\psi^* {\mathcal P}ol^{\text{coh}}\) has residue 0 at infinity, the degeneration \(i^* j_* \psi^* {\mathcal P}ol^{\text{coh}}\) is equal to the degeneration of the usual “elliptic polylog” \(i^* j_* \psi^* {\mathcal P}ol(-1)\). This degeneration is known and is equal to the classical polylog. Its value at torsion sections is again known and given by a linear combination of cyclotomic elements in \(\ell\)-adic cohomology. This yields the comparison theorem.

The paper also contains three appendices reviewing the construction of the elliptic and the classical polylog in unified form (A), studying the degeneration of the elliptic into the classical polylog (B), and treating the relation of the cohomological polylog with the Eisenstein symbol (C).

Reviewer: T.Nguyen Quang Do (Besançon)

##### MSC:

11R70 | \(K\)-theory of global fields |

19F27 | Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects) |

11R42 | Zeta functions and \(L\)-functions of number fields |

11G55 | Polylogarithms and relations with \(K\)-theory |