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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)].

##### MSC:
 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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