Iwasawa L-functions for multiplicative abelian varieties. (English) Zbl 0716.14008

Jones, John W.

Duke Math. J. 59, No. 2, 399-420 (1989).

Iwasawa L-functions for abelian varieties with multiplicative reductions are studied, extending some results proved by B. Mazur in Invent. Math. 18, 183-266 (1972; Zbl 0245.14015) for abelian varieties with good ordinary reductions.
Let \(p\neq 2\) be a prime, \(\Gamma ={\mathbb{Z}}_ p\) (as an additive topological group) with a generator \(\gamma\), and \(\Lambda:=\lim_{{\leftarrow \nu}}{\mathbb{Z}}_ p[\Gamma /p^{\nu}\Gamma] \). Then the map which sends T to \(\gamma\)-1 induces an isomorphism from \({\mathbb{Z}}_ p[[T]]\) to \(\Lambda\). The Iwasawa L-function for an elliptic curve was defined as the characteristic polynomial of the p-Selmer group of the curve. To generalize this definition to abelian varieties, one needs “good” \(\Lambda\)-modules which are finitely generated modules M over \(\Lambda\). Such a module is quasi-isomorphic to the direct sum \(\Lambda^{\rho}\oplus {\mathbb{Z}}/p^{\mu_ i}{\mathbb{Z}}[[T]]\oplus (\oplus_{j}{\mathbb{Z}}_ p[[T]](F_ j)^{n_ j})\) where \(\rho\) is the free rank of M, \(F_ j\) is an irreducible distinguished polynomial for each j. The invariants \((\rho,\mu_ i,\{F_ j^{n_ j}\})\) determine M completely up to quasi-isomorphism (i.e., up to finite kernel and cokernel). The \(\mu\)-invariant of M is \(\mu:=\sum_{i}\mu_ i \), the characteristic polynomial of M is \(F_ M(T):=p^{\mu}\prod_{j}(F_ j(T))^{n_ j} \) and \(f_ M(t)\) is the polynomial satisfying \(f_ M(T+1)=F_ M(T).\)
Let K be a number field with ring of integers \({\mathfrak O}_ K\). Let \(A_{/K}\) be an abelian variety defined over K, A its Néron model over \({\mathfrak O}_ K\), \(\tilde A\) the dual abelian variety of A, \(A^ 0\) the connected component of A and \(A_{p^{\infty}}:=\cup_{\nu}A_{p^{\nu}} \). Let \(\Phi\) be defined by the short exact sequence \(0\to A^ 0\to A\to \Phi \to 0\). Let L/K be a \(\Gamma\)-extension of K, T the set of all primes in K ramifying in L, \(\log_ p\) a p-adic logarithm of L/K, \(\kappa: Gal(L/K)\to 1+p^ e{\mathbb{Z}}_ p\subset {\mathbb{Z}}^*_ p\) a fixed continuous character compatible with \(\log_ p\). If \(\nu\in T\), \(e_{\nu}\) denotes the dimension of a maximal split subtorus in the reduction of A at \(\nu\) and \(e:=\sum_{\nu}e_{\nu}.\)
Assume that A satisfies the following hypothesis:
1. \({Sh}_{p^{\infty}}(K)\) is finite.
2. Every prime of K for which A has bad reduction splits finitely in L.
3. The reduction of A is semistable at every place of K dividing p and is an extension of an ordinary abelian variety by a torus for every \(t\in T.\)
4. For every place \(t\in T\), the universal norm of \(A(L_ t)\) is of finite index in \(A(K_ t).\)
There is a p-adic height pairing \(<, >_ p\) on A such that \(<, >_ p:=<, >_{\gamma}\log_ p\kappa (\gamma)\), where \(<, >_{\gamma}\) is a p-adic height pairing defined by the author [“p-adic heights for semistable abelian varieties”, Compos. Math. (to appear)] and is equivalent to Schneider’s analytic height [P. Schneider, Invent. Math. 69, 401-409 (1982; Zbl 0509.14048)]. A necessary and sufficient condition for \(<, >_ p\) to be nondegenerate is obtained. Further, define the groups \({\mathcal I}:=Image[H^ 1({\mathfrak O}_ K,A^ 0_{p^{\infty}})\to H^ 1({\mathfrak O}_ K-T,A^ 0_{p^{\infty}})]\) and \({\mathcal I}_{\infty}:=Image[H^ 1({\mathfrak O}_ L,A^ 0_{p^{\infty}})\to H^ 1({\mathfrak O}_ L-T,A^ 0_{p^{\infty}})]\). (They are quasi-isomorphic to the classical p- Selmer group of A over K and L, respectively.) Write \(A_{p^{\infty}}(L)=A_{p^{\infty}}^{\inf}(L)\oplus A_{p^{\infty}}^{fin}(L)\) where \(A_{p^{\infty}}^{\inf}\) is the divisible subgroup of \(A_{p^{\infty}}(L)\). Then one can define \(A_{p^{\infty}}^{fin}(K)\) to be the K-rational points of \(A_{p^{\infty}}^{fin}(L)\). Define the \({\mathcal L}_{\nu}\)-invariant of A with respect to L/K at a place \(\nu\in T\) by \({\mathcal L}_{\nu}(A):=(A(K_{\nu})/NA(K_{\nu}))/(\Phi ({\mathfrak O}_{K_{\nu}})| \log_ p\kappa (\gamma)|_ p^{e_{\nu}})\) and define the global \({\mathcal L}\)-invariant of A with respect to L/K by \({\mathcal L}(A):=\prod_{\nu \in T}{\mathcal L}_{\nu}(A).\)
The main result of the paper is to define a “good” \(\Lambda\)-module, H, which is subject to a quasi-exact sequence \[ 0\to {\mathcal I}_{\infty}\to H\to ({\mathbb{Q}}_ p/{\mathbb{Z}}_ p)^ e\to 0\quad or\quad 0\to ({\mathbb{Q}}_ p/{\mathbb{Z}}_ p)^ e\to H\to {\mathcal I}_{\infty}\to 0 \] where \(\Gamma\) acts trivially on the \(({\mathbb{Q}}_ p/{\mathbb{Z}}_ p)^ e\) term. Let \(f_ H(t)=(t-1)^ ef_{{\mathcal I}}(t)\), and define a p-adic L-function \(L_ H(s):=f_ H(\kappa (\gamma)^{1-s})\). (This is a candidate for the p- adic L-function of an ordinary abelian variety A which is semistable at p.) Let \(\rho =ord_{s=1}L_ H(s)\) and \(r=rank_{{\mathbb{Z}}}A(K)\). Then the main result of this paper is formulated in the following theorem:
One has \(\rho \geq r+e\). If \(<\,\;>_ p\) is nondegenerate, then \(\rho =r+e\) and the \(\rho\)-th derivative of \(L_ H(s)\) has the following value at \(s=1:\) \[ L_ H^{(\rho)}(1)\approx {\mathcal L}(A)\frac{\det<\,\;>_ p{Sh}_ K}{A_{p^{\infty}}^{fin}(K)\tilde A_{p^{\infty}}^{fin}(K)}\cdot \prod_{\ell \nmid \infty}m_{\ell}, \] where \(m_{\ell}\) denotes the number of connected components in the fibre of A over \(\ell\) and \(a\approx b\) means that a and b have the same p-norm.
A functional equation for \(L_ H(s)\) is also proved. That is, \(f_ H(t)=(-1)^{\rho}t^{\lambda}f_ H(1/t)\) where \(\lambda\) is the \(\lambda\)-invariant of H and \(\rho\) is the multiplicity of the root of 1 in \(f_ H(t)\), and similarly, \(L_ H(s)=(-1)^{\rho}\kappa (\gamma)^{\lambda (1-s)}L_ H(2-s)\). Several candidates for such a \(\Lambda\)-module are tested, e.g., \(H^ 1({\mathfrak O}_ L,A^ 0_{p^{\infty}})\), \(H^ 1({\mathfrak O}_ L,A_{p^{\infty}})\), and Greenberg’s module.

Reviewer: N.Yui

MSC:

14G10	Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14K05	Algebraic theory of abelian varieties
14G40	Arithmetic varieties and schemes; Arakelov theory; heights
11G40	\(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture

Keywords:

functional equation for L-function; derivative of L-function; Birch and Swinnerton-Dyer conjecture; Iwasawa L-functions for abelian varieties with multiplicative reductions; p-adic height pairing

Citations:

Zbl 0245.14015; Zbl 0509.14048

References:

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