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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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