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The boundary of the Eisenstein symbol. (English) Zbl 0729.11027
For an elliptic curve \(E\) over a field \(F\) (supposed to have a nontrivial discrete valuation \(v\), valuation ring \(\mathcal O\) and perfect residue field \(k\)) with a finite subgroup scheme \(P\subset E\) defined over \(F\), and for any integer \(n\geq 1\), one has the Eisenstein symbol map \[ \mathcal E^ n_ P: \mathbb Q[P]^ 0\to H_{\mathcal M}^{n+1}(E^ n,\mathbb Q(n+1))_{\text{sgn}}, \] where \(H^ i_{\mathcal M}(-,\mathbb Q(j))\) is motivic cohomology, \(\mathbb Q[P]^ 0\) is the \(\mathbb Q\)-vector space of \(\text{Gal}(\bar F/F)\)-invariant functions \(\beta: P(\bar F)\to \mathbb Q\) satisfying \(\sum_{x\in P(\bar F)}\beta (x)=0\), \(E^ n\) is identified with the kernel of the sum map \(E^{n+1}\to E\) (thus giving an action of the symmetric group \(\mathcal S_{n+1}\) on \(E^ n)\), and where the subscript ‘sgn’ denotes the image under the projector \[ \prod_{\text{sgn}}=\frac{1}{(n+1)!}\sum_{\sigma \in\mathcal S_{n+1}}\text{sgn}(\sigma)\cdot \sigma. \] Write \(E/k\) for the special fibre of the minimal regular model \(E/{\mathcal O}\) of \(E\) and suppose that \(E/k\) is a Néron \(N\)-gon for some \(N\geq 1\). Furthermore suppose that \(P\) extends to a finite flat subgroup scheme \(P/\mathcal O\) of the Néron model of \(E\) over \(\mathcal O\).
Also, let \(\overset \circ E\) denote the connected component of the Néron model of \(E\) over \(\mathcal O\). An isomorphism \(\overset \circ E/k\overset \sim \rightarrow\mathbb G_ m\) induces a bijection between \(\mathbb Z/N\mathbb Z\) and the set of components \(C_{\nu}\) of \(E/k\). Thus \(E/k=\cup_{\nu \in\mathbb Z/N\mathbb Z}C_{\nu}\). For \(\beta\in\mathbb Q[P]^ 0\) let \(d_{\beta}(\nu)\) be the degree of the restriction of the flat extension of \(\beta\) to \(C_{\nu}\). The localization sequence for the pair (\(\overset \circ E^ n/{\mathcal O},\overset \circ E^ n/k)\) gives a boundary map \[ \partial^ n: H_{{\mathcal M}}^{n+1}(E^ n,\mathbb Q(n+1))_{\text{sgn}}\to H^ n_{\mathcal M}(\overset \circ E^ n/k,\mathbb Q(n))_{\text{sgn}}. \] The target space is a 1-dimensional \(\mathbb Q\)-vector space generated by an element of the form \(\Phi^ n_ n=\prod_{\text{sgn}}(y_ 0\cup...\cup y_ n)\), where \(y_ 0y_ 1...y_ n=1\), and \(y_ i\), \(1\leq i\leq n\), is a coordinate on the ith copy of \(\mathbb G_{m/k}\). The main result of the paper is following theorem:
\[ \partial^ n\circ{\mathcal E}^ n_ P(\beta)=C^ n_{P,N}\left(\sum_{\nu\in\mathbb Z/N\mathbb Z}d_{\beta}(\nu)B_{n+2}\left(\langle\frac{\nu}{N}\rangle\right)\right)\cdot \Phi^ n_ n, \] where \(C^ n_{P,N}\) is an explicit nonzero constant, \(B_ k(X)\) is the \(k\)th Bernoulli polynomial, and \(0\leq \langle x\rangle<1\) is a representative of \(x\in\mathbb Q/\mathbb Z\).
For the proof one may restrict to the situation where \(E/k\) is an untwisted Néron \(N\)-gon with \(N\geq 3\), \(P=\mu_ n\times\mathbb Z/N\mathbb Z\subset E(F)\) is a level \(N\) structure on \(E\), and \(P/k\) gives the standard level \(N\) structure on \((E/k)^{\text{smooth}}=\mathbb G_ m\times\mathbb Z/N\mathbb Z\). Then \(C^ n_{P,N}\) turns out to be \(\pm N^ n(n+1)/(n+2)!\).
The theorem is shown to follow from an explicit formula for the boundary map \[ \partial^ n_ v: H_{{\mathcal M}}^{n+1}(U^{n'}/F,\mathbb Q(n+1))^{P^ n}_{\text{sgn}}\to H^ n_{{\mathcal M}}(U^{n'}/k,\mathbb Q(n))^{P^ n}_{\text{sgn}}, \] where \(H^{\bullet}_{{\mathcal M}}(U^{n'},{\mathbb{Q}}(*))^{P^ n}_{\text{sgn}}\) is a suitable \(P(\bar F)^ n\)-invariant sgn-part of the motivic cohomology of \(U^{n'}=\{(x_ 1,...,x_ n)\in E^ n| x_ i\not\in P\), \(\forall i,0\leq i\leq n\}\subset E^ n\), with \(x_ 0=-x_ 1-...-x_ n\). One defines a map \[ \Theta^ n_ P: \mathbb Q[P]^{0\otimes (n+1)}\to H_{{\mathcal M}}^{n+1}(U^{n'},\mathbb Q(n+1))^{P^ n}_{\text{sgn}} \] and then the formula for \(\partial_{\nu}\circ \Theta^ n_ P(\otimes \beta_ i)\) involves, among other things, a sum of expressions containing \(\zeta\in \mu N\), \(\zeta\neq 1\), and this leads, on account of their distributional property, to the Bernoulli polynomials. The explicit calculation uses the fact that the boundary maps in Milnor and Quillen \(K\)-theory agree. Then the theorem is verified for the case \(n=1\) and \(F\) a number field.
The general case consists in the “weight decomposition” of \(H^{\bullet}_{{\mathcal M}}(U^{n'}/F\), \(\mathbb Q(*))^{P^ n}_{\text{sgn}}\) under the “\(L^{-1}\)”-multiplication. Actually, this “\(L^{-1}\)”-multiplication (\(L\geq 1\) an integer) induces a Galois covering \([\times L]: \tilde U^{n'}\to U^{n'}\) and a homomorphism on (motivic) cohomology that plays a role throughout. The main step is a result, due to Beilinson and Deninger, which identifies \(H^{\bullet}_{{\mathcal M}}(E^ n,\mathbb Q(*))_{\text{sgn}}\) with the \(L^{-n}\)-eigenspace (for a certain endomorphism) of \(H^{\bullet}_{{\mathcal M}}(U^{n'},\mathbb Q(*))^{P^ n}_{\text{sgn}}\). The Eisenstein symbol \({\mathcal E}^ n_ P(\beta)\) is then defined as the projection of \(\Theta^ n_ P(\beta \otimes \alpha^{\otimes n})\), \(\alpha= \sum_{x\in P(\bar F)}(0)-(x)\), into the \(L^{-n}\)-eigenspace, viewed as an element of \(H_{{\mathcal M}}^{n+1}(E^ n,\mathbb Q(n+1))\). If \(F\) is a number field and \(v\) is a place of bad reduction of \(E\) one obtains a description of the ‘integral’ cohomology \[ H_{{\mathcal M}}^{n+1}(E^ n/F,\mathbb Q(n+1))_{\mathbb Z}\subset H_{{\mathcal M}}^{n+1}(E^ n/F,\mathbb Q(n+1)). \] Also, in the modular case, one obtains a new proof of a result of Beilinson which says that the boundary map
\[ \partial: H_{{\mathcal M}}^{n+1}(E^ n,\mathbb Q(n+1))_{\text{sgn}} \rightarrow \{f: \text{GL}_ 2(\mathbb Z/N\mathbb Z)\to \mathbb Q\mid f(g\begin{pmatrix} *&*\\0&1 \end{pmatrix})= f(g)=(-1)^ nf(-g)\}, \] where \(E\) is the universal elliptic curve with level \(N\) structure, defined over the function field of the modular curve of level \(N\), \(N\geq 3\), is an isomorphism on the image of the Eisenstein symbol.

MSC:
11G05 Elliptic curves over global fields
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
19F15 Symbols and arithmetic (\(K\)-theoretic aspects)
19D45 Higher symbols, Milnor \(K\)-theory
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