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Formal groups and \(L\)-series. (English) Zbl 0741.14026
Two theorems on formal groups arising from matrix-valued Dirichlet series and their relation to the formal completions of the Néron models along the zero section of certain commutative group varieties are proved. A theorem of T. Honda [Osaka J. Math. 5, 199–213 (1968; Zbl 0169.37601)] that the \(L\)-series \(L(\chi,s)\) associated to the Dirichlet character \(\chi\) of a quadratic extension \(K/\mathbb{Q}\) with discriminant \(d_ K\) gives rise to a formal group over \(\mathbb{Z}\) which becomes strongly isomorphic to \(x+y+\sqrt{d_ K}xy\) over the ring \({\mathfrak O}_ K\) of integers of \(K\) is generalized to abelian tori of higher dimension. A theorem of Honda and Cartier [the same paper of Honda as above, and P. Cartier in Actes Congr. Int. Math. 1970, No. 2, 291–299 (1971; Zbl 0298.14013)] that the formal group associated to the \(L\)-series of an elliptic curve \(E\) over \(\mathbb{Q}\) is isomorphic to the formal completion of the group law of the Néron model \({\mathcal E}\) of \(E\) along its zero section is generalized to abelian varieties over \(\mathbb{Q}\) with real multiplication. In more precise terms, two theorems are formulated as follows:
Theorem 1. Let \(T\) be an algebraic torus of dimension \(d\) over \(\mathbb{Q}\) with character
\[ X(T)=\operatorname{Hom}_{\overline{\mathbb Q}} (T_{\overline{\mathbb Q}},G_{m,\overline{\mathbb Q}})\]
and with the Galois representation \(\rho: G_{\mathbb{Q}}\to\operatorname{Aut}(X(T))\simeq \mathrm{GL}_d(\mathbb{Z})\). Then \(T\) has a Néron model \({\mathcal T}\) over \(\mathbb{Z}\). Assume that \(T\) is abelian, that is, \(\rho\) is commutative. Let \(G=\operatorname{Im}(\rho)\) denote the Galois group of the finite abelian splitting field of \(T\) corresponding to \(\text{Ker}(\rho)\). Let \(\chi_ 1,\ldots,\chi_ d\) be a basis of \(X(T)\) as a \(\mathbb{Z}\)-module. Then for each prime \(p\) where \(T\) has good reduction, there is a matrix \(A_ p\in \mathrm{GL}_ d(\mathbb{Z})\) associated to the Galois Frobenius \(\sigma_ p\in G\). There one obtains a \(d\)-dimensional formal group law \(\hat L\) over \(\mathbb{Z}\) associated to the matrix-valued Dirichlet series \(\sum^ \infty_{n=1}A_nn^{-s}= \prod_{p: \text{good}}(I_ d-A_ pp^{-s})^{-1}\), where \(I_ d\) denotes the identity matrix of order \(d\). On the order hand, one has the formal completion \(\hat{\mathcal T}\) of \(\mathcal T\) along the zero section. Then \(\hat{\mathcal T}\) and \(\hat L\) are isomorphic over \(\mathbb{Z}_ S\) where \(S\) is an appropriate finite set of “bad” primes \(p\) depending only on \(G\) and \(d\).
Theorem 2. Let \(A\) be an abelian variety over \(\mathbb{Q}\) of dimension \(g\geq 1\). Let \(\text{End}(A)\) be the ring of endomorphisms of \(A\) over \(\mathbb{Q}\). Assume that there is a homomorphism \(\theta:F\to\text{End} A\otimes\mathbb{Q}=\text{End}^0(A)\) where \(F\) is a totally real number field of degree \(g=[F:\mathbb{Q}]\). Let \({\mathfrak O}_ A=\theta^{-1}(\text{End}(A))\subset {\mathfrak O}_F\) be an order and let \(R: \mathfrak O_A\to M_g(\mathbb{Z})\) be any faithful representation of \({\mathfrak O}_A\). The local \(L\)-series of \(F\) at a prime \(p\) is defined as follows: \[ L_p(A,F,s)=\det_{F_ l}(1-(\sigma_ p^{-1})^*p^{- s}\mid H^ 1_{\mathfrak l}(A)^{I_p})^{-1}=(1-c_ pp^{- s}+pc_{p^ 2}p^{-2s})^{-1} \] with \(c_p\in{\mathfrak O}_ F\) and \(c_{p^2}=0\) or 1. Here \(\ell\) is a prime different from \(p\) and \(F_ \ell=F\otimes\mathbb{Q}_\ell=\prod_{{\mathfrak l}\mid\ell}F_{\mathfrak l}\). One obtains a formal group law \(\hat L\) over \(\mathbb{Z}\) associated to the matrix-valued Dirichlet series of \(A\):
\[ \sum^\infty_{n=1}A_nn^{-s}=\prod_ p(I_ g-C_ pp^{-s}+pC_{p^2}p^{-2s})^{- 1}\]
where \(C_p=R(c_p)\) and \(C_{p^2}=R(c_{p^2})\). On the other hand, one has the formal completion \(\hat {\mathcal A}\) of the Néron model \({\mathcal A}\) of \(A\) along its zero section.
Assume that \({\mathfrak O}_ A={\mathfrak O}_F\). Then \(\hat A\) and \(\hat L\) are isomorphic over \(\mathbb{Z}_S\) where \(S\) is an appropriately chosen finite set of “bad” primes.
An application of theorem 2 to modular forms is also discussed.
Corollary. Let \(N\geq 1\) be a square-free integer. Let \(A=J_0(N)^{\text{new}}\) be the new part of the modular variety \(J_0(N)\) over \(\mathbb{Q}\) associated to \(\Gamma_0(N)\). Then the formal completion of the Néron model \(\hat{\mathcal A}\) of \(A\) along its zero section is isomorphic over \(\mathbb{Z}\) to the formal group obtained from the matrix-valued \(L\)-series: \[ L=\prod_{p\mid N}(1-R(U_p)p^{-s})^{-1}\prod_{p\nmid N}(1-R(T_p)p^{-s}+Ip^{1-2s})^{-1} \] where \(T_p\) for \(p\nmid N\) denotes the Hecke operator, \(U_p\) for \(p\mid N\) the Atkin-Lehner operator, and \(R\) denotes the natural representation of \(\text{End}(A)\) on a \(\mathbb{Z}\)-basis of \(\omega_{{\mathcal A}^ 0/\mathbb{Z}}\).
The above result are proved using the method developed by T. Honda in [J. Math. Soc. Japan 22, 213–246 (1970; Zbl 0202.03101)].

14L05 Formal groups, \(p\)-divisible groups
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
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