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An introduction to Kato’s Euler systems. (English) Zbl 0952.11015
Scholl, A. J. (ed.) et al., Galois representations in arithmetic algebraic geometry. Proceedings of the symposium, Durham, UK, July 9-18, 1996. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 254, 379-460 (1998).
The paper gives a fairly self-contained introduction to Kato’s Euler systems. These Euler systems may be applied to prove finiteness results for elliptic curves, in particular over $$\mathbb Q$$. The paper consists of five sections (the second with an appendix on higher $$K$$-theory of modular varieties) and a list of references.
To fix the notations and introduce some basic notions, let $$f:E\rightarrow S$$ be an elliptic curve over a (suitable) base scheme $$S$$. Write $$e:S\rightarrow E$$ for the zero section. Then there is a standard invertible sheaf $$\omega= \omega_{E/S}=f_*\Omega^1_{E/S}=e^*\Omega^1_{E/S}$$. This $$\omega$$ can be considered as a sheaf on the moduli stack $$\mathcal M$$ of elliptic curves. A (meromorphic) modular form of weight $$k$$ is then a section of $$\omega^ {\otimes k}_{E/S}$$, i.e. an element of $$\Gamma(\mathcal M,\omega^{\otimes k})$$. Of special importance is the discriminant $$\Delta(E/S)$$ which gives a trivialisation of $$\omega^{\otimes 12}_{E/S}$$. This fact may be held responsable for the existence of so called Kato-Siegel functions:
For an integer $$D$$ with $$(D,6)=1$$ there is exactly one rule $$\vartheta_D$$ which associates to each elliptic curve $$E\rightarrow S$$ a section $$\vartheta_D^{(E/S)}\in\mathcal O^*(E-\text{ker}[\times D])$$ such that $$\vartheta_D^{(E/S)}$$ has divisor $$D^2(e)-\text{ker}[\times D]$$, where $$\times D$$ means multiplication by $$D$$. The $$\vartheta_D^{(E/S)}$$ are compatible with base change and natural under isogenies. Furthermore, $$\vartheta_{-D}= \vartheta_D$$, $$\vartheta_1=1$$ and $$[\times D]_*\vartheta_D=1$$. Over $$\mathbb C$$ with $$E/{\mathbb C}$$ given by the lattice $${\mathbb Z}+\tau{\mathbb Z}$$, $$\text{Im}(\tau)>0$$, one has $\vartheta_D^{(E/{\mathbb C})}=(-1)^{{D-1}\over 2}\Theta(u,\tau)^{D^2} \Theta(Du,\tau)^{-1},$ where $${\Theta(u,\tau)=q^{1\over 12} (t^{1\over 2}-t^{-1\over 2})\prod_{n>0}(1-q^nt)(1-q^nt^{-1})}$$ with $$q=e^ {2\pi i\tau}$$, $$t=e^{2\pi i u}$$.
Let $$x\in E/S$$ be a torsion section of order $$N$$ with $$(N,D)=1$$. The vertical logarithmic derivative $$\text{d}\log_v\vartheta_D\in\Gamma(E-\text{ker}[\times D],\Omega^1_{E/S})$$ gives rise to a weight one modular form (Eisenstein series) ${}_D\text{Eis}(x) ={}_D\text{Eis}(E/S,x):=x^*\text{d}\log_v\vartheta_D\in\Gamma(S, x^*\Omega^1_{E/S})=\Gamma(S,\omega_{E/S}).$ Then, for $$D\equiv 1\bmod N$$, one gets $\text{Eis}(E/S,x):{1\over D^2-D}\cdot{}_D\text{Eis} (E/S,x)\in\Gamma \Biggl(S\otimes{\mathbb Z} \biggl[{1\over D(D-!)} \biggr],\omega\Biggr)$ independent of $$D$$. Over $$\mathbb C$$ one recovers the classical Eisenstein series $\text{Eis}(E/{\mathbb C},x)=\sum_{m_i\in{a_i\over N}+{\mathbb Z}}\left. {1\over(m_1\omega_1+m_2\omega_2)|(m_1\omega_1+m_2\omega_2)|^s}\right|_{s=0} du,$ where $$u$$ is the variable in the complex plane and where $$x\in E({\mathbb C})- \{e\}$$ is the torsion point $$(a_1\omega_1+a_2\omega_2)/N\in N^{-1}\Lambda/ \Lambda$$, with $$\Lambda$$ the lattice defining $$E/{\mathbb C}$$.
The second section deals with norm relations in algebraic $$K$$-theory. The Euler system in $$K_2$$ of modular curves is constructed. For an elliptic curve $$E/S$$ one denotes by $$S(N)(T)$$ the level $$N$$ structures on $$E\times_ST$$, i.e. isomorphisms $$\alpha:({\mathbb Z}/N)^2_T{\buildrel\sim\over\longrightarrow}\text{ker}[\times N]_T$$ (in a functorial way). For integers $$D,D''$$ and a prime $$\ell$$ such that $$(6\ell,DD'')=1$$ one writes $$\vartheta= \vartheta^{(E/S)}_D$$ and $$\vartheta''=\vartheta^{(E/S)}_{D''}$$. Also one writes $$N_{?/?}$$ for the push-forward maps $$\pi_{?/?*}$$ on $$K_2$$, i.e. for a proper map of schemes $$\pi_{?/?}:X''\rightarrow X$$ one has $$\pi_{?/?*}:K_2X''\rightarrow K_2X$$. If $$S=Y(N)/H$$, $$H\subset GL_2 ({\mathbb Z}/N)$$, is a modular curve of level prime to $$\ell$$, $$E=\mathcal E^ {\text{univ}}\rightarrow S$$ the universal elliptic curve, and $$z$$, $$z^ {\prime}$$ are torsion sections of $$E/S(\ell)$$ whose projections onto $$\text{ker}[\times\ell]$$ are linearly independent, then $N_{S(\ell)/S}\{\vartheta(z),\vartheta''(z'')\}=(1-T_{\ell} \circ\langle\ell\rangle_*+\ell\langle\ell\rangle_*)\{\vartheta(\ell z), \vartheta''(\ell z''\},$ where $$\{.,.\}$$ denotes a symbol in $$K_2$$, and where $$T_{\ell}$$ is a Hecke operator. One may generalize to norm relations for $$\Gamma(\ell^n)$$. Also, one may deduce norm relations for products of the Eisenstein series $${}_D\text{Eis}(E/S,x)$$ and $${}_D\text{Eis}(E/S,x'')$$. One should pass these facts to Galois cohomology. The problem becomes to show that the corresponding cohomology classes are non-trivial (if the corresponding value of the $$L$$-function do not vanish). To this end Kato formulated a general reciprocity theory. This is the subject of the third section.
In the third section the dual exponential map is introduced. Let $$K$$ be a finite extension of $${\mathbb Q}_p$$ with ring of integers $$\mathfrak o$$ and fixed algebraic closure $$\overline{K}$$. Denote by $$G_K$$ the Galois group $$\text{Gal} (\overline{K}/K)$$. and let $$\zeta_{p^n}$$ be a primitive $$p^n$$-th root of unity in $$\overline{K}$$. Write $$K_n=K(\zeta_{p^n})$$ with valuation ring $$\mathfrak o_n$$. For a continuous finite-dimensional representation $$V$$ of $$G_K$$ which is supposed to be de Rham in Fontaine’s terminology, let $$DR(V)=(B_{\text{dR}}\otimes_{{\mathbb Q}_p} V)^{G_K}$$ be the associated filtered $$K$$-vector space with decreasing filtration $$DR^i(V)$$ coming from the filtration on $$B_{\text{dR}}$$). Then Kato’s dual exponential map is defined as the composite $H^1(K,V)\rightarrow H^1(K,B^0_{\text{dR}}\otimes_{{\mathbb Q}_p}V) =H^1(K,\text{Fil}^0(B_{\text{dR}}\otimes KDR(V)))\simeq DR^0(V).$ Apply this to $$V=H^1(Y_{\overline{K}},{\mathbb Q}(p))(1)$$ for a smooth $$\mathfrak o$$-scheme $$Y$$, which is the complement in a smooth proper $$\mathfrak o$$-scheme $$X$$ of a divisor $$Z$$ with relatively normal crossings. Write $$H^i_{\text{dR}}(Y/\mathfrak o)=H^i(X,\Omega^{\bullet}_{X/\mathfrak o}(Z))$$. One calculates $$DR^0(V)=\text{Fil}^1H^1_{\text{dR}}(Y/\mathfrak o) \otimes_{\mathfrak o}K$$. The explicit reciprocity theorem now becomes the commutativity of a diagram starting with $$\displaystyle{\lim_{\buildrel\longleftarrow\over n}(K_2(Y\otimes\mathfrak o_n)\otimes\mu_{p^n} ^{\otimes -1})}$$ and going (via the dual exponential) to $$K_m\otimes_{\mathfrak o} \text{Fil}^1H^1_{\text{dR}}(Y/\mathfrak o)$$. The proof of this result (in fact, of a theorem of which the reciprocity theorem is a corollary) consumes a lot of pages and uses the Faltings-Fontaine-Hyoda approach to $$p$$-adic Hodge theory.
In the fourth section a sketch is given of Kato’s Rankin-Selberg integral method to compute the projection of the image of the dual exponential into a Hecke eigenspace. The presentation is adelic. One transplants the Eisenstein series into the adelic setting to obtain $$E_{k,s}$$ and $$E_k:=E_{k,0}$$, where $$E_{k,s}$$ is absolutely convergent for $$k+2$$ $$\text{Re}(s)>2$$. The $$E_{k,s}$$ depend on several arguments: a locally constant function $$\phi:{\mathbb A}^2_f\rightarrow {\mathbb C}$$ of compact support and with an action of the group $$G_f=GL_{2, {\mathbb A}_f}$$, and the ’lattice’ parameter $$\tau$$. One can then write $$E_{k,s}(\phi)$$ as a function $$E_{k,s,f}$$ where $$f:G_f\rightarrow{\mathbb C}$$ is a suitable locally constant function such that $$E_{k,s,f}$$ can be expressed in terms of $$f$$. Then, for adelic modular forms $$F$$, $$G$$ of weights $$k+\ell$$, $$k$$ respectively, at least one is a cusp form, define $\Omega=E_{\ell,s,f}G \overline{F}y^{k+\ell+s-2}|\det(g)|^{-k-\ell-s} d\tau\wedge d\overline{\tau}.$ This is a left $$G^+_{{\mathbb Q}}$$-invariant form on $$\mathfrak H\times G_f$$. One is interested in computing $${\int_{G^+_{{\mathbb Q}}\mathfrak H\times G_f}\Omega dg}$$, which is by definition the inner product $$\langle E_{\ell,s,f}G,F \rangle$$. In particular, one takes for $$F$$ a cusp form, belonging to an irreducible representation $$\pi=\otimes''\pi_p$$ of $$G_f$$, satisfying some more suitable conditions, and let $$G=E_k(\phi'')$$ a suitable Eisenstein series. Then the inner product $$\langle E_{\ell,s}(\phi)E_k( \phi''),F\rangle$$ can be decomposed explicitly into local integrals. The upshot is that one obtains a formula for the inner product in terms of values of $$L$$-functions related to $$\pi$$. A supplementary character may also be included.
Let $$D,D''$$ be integers prime to $$6Np$$ and define $$\mathcal R''_p$$ as the set of squarefree integers prime to $$NpDD''$$. Also, let $$\mathcal R_p=\{r=r_0p^m\mid r_0\in\mathcal R''_p$$, $$m\geq 1\}$$. Suppose that for any $$r\in\mathcal R_p$$ one has points $$z_r,z_r''\in\mathcal E^{\text{univ}} (Y(Nr)) \simeq({\mathbb Z}/Nr)^2$$ having suitable properties, in particular, for $$r=p^m$$, the orders of $$z_r,z_r''$$ are divisible by a prime other than $$p$$. One defines for any $$r\in\mathcal R_p$$: $\overline{\sigma}_r=\{\vartheta_D(z_r),\vartheta _{D''}(z_r'')\}\in K_2(Y(Nr)),$ and also $\sigma_r=N_{Y(Nr)/Y(N) \otimes{\mathbb Q}(\mu_r)}\overline{\sigma}_r\in K_2(Y(N)\otimes{\mathbb Q}(\mu_r)).$ The $$\sigma$$’s define an almost Euler system for $$K_2$$ in the sense that:
(i) $$N_ {{\mathbb Q}(\mu_{rp})/{\mathbb Q}(\mu_r)}\sigma_{rp}=\sigma_r$$;
(ii) If $$\ell$$ is prime and $$(\ell,NDD''r)=1$$ then $N_{{\mathbb Q}(\mu_{\ell r})/{\mathbb Q} (\mu_r)}\sigma_{\ell r}=(1-T_{\ell}\langle\ell\rangle_*\otimes\text{Frob}_ {\ell}+\ell\langle\ell\rangle_*\otimes\text{Frob}^2_{\ell})\sigma_r.$ Define $${\mathbb T}_{p,N}=H^1(Y(N)\otimes_{\mathbb Q}\overline{\mathbb Q},{\mathbb Z}_p(1))$$. Then, by Abel-Jacobi and corestriction, the family $$\sigma_{r_0p^n}\otimes [\zeta_{p^n}]^{-1}$$ maps to the Galois cohomology $$H^1({\mathbb Q}(\mu_r), {\mathbb T}_{p,N})$$. Write $$\xi_r=\xi_r(Y(N))\in H^1({\mathbb Q}(\mu_r), {\mathbb T}_{p,N})$$ for its image. Then it follows that the $$\xi$$’s also satisfy Euler system-like identities. Looking at an elliptic curve $$E/{\mathbb Q}$$ with conductor $$N_E$$ and Weil parametrisation $$\varphi_E:X_0(N_E)\rightarrow E$$, considering the composite morphism $$\varphi_{E,N}:X(N)\rightarrow X_0(N_E) {\buildrel\varphi_E\over\longrightarrow}E$$, one gets Galois-equivariant maps $\varphi_{E,N*}:H^1(X(N)\otimes_{\mathbb Q}\overline{\mathbb Q},{\mathbb Z}_p(1))\rightarrow H^1(E\otimes_{\mathbb Q}\overline{\mathbb Q},{\mathbb Z}_p(1))=T_p(E),$ the Tate module, and also a restriction to $${\mathbb T}_{p.N}$$. The (proof of the) Manin-Drinfeld theorem now implies that one can eventually define elements $$\xi_r(E)\in H^1 ({\mathbb Q}(\mu_r),T_p(E))$$ from the $$\xi_r$$ via $$\varphi_{E,N*}$$.
The main result of the fifth section is now that the family $$\{\xi_r(E)\}$$ is an Euler system for $$T_p(E)$$. In the reasoning behind these facts the dual exponential plays an important role. Finally, one may relate the Euler system to the $$L$$-function by using the Rankin-Selberg integral described above. For a character $$\lambda:\text{Gal}({\mathbb Q}(\mu_r)/{\mathbb Q})\rightarrow {\mathbb C}^{\times}$$ such that $$\lambda(-1)=\pm 1$$, one gets an interesting formula relating the sum (over the Galois group) of the dual exponentials of the conjugates of the Euler system elements times the character values in terms of $$L(E,\lambda,1)$$, the periods of $$E$$ and $$\omega_E$$.
For the entire collection see [Zbl 0905.00052].

##### MSC:
 11G05 Elliptic curves over global fields 11F11 Holomorphic modular forms of integral weight 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 14G05 Rational points 14H52 Elliptic curves