Iwasawa theory of elliptic curves with complex multiplication. \(p\)-adic \(L\) functions.

*(English)*Zbl 0674.12004
Perspectives in Mathematics, Vol. 3. Boston etc.: Academic Press, Inc. (Harcourt Brace Jovanovich, Publishers) ix, 154 p. $ 19.50 (1987).

From the introduction: \(p\)-adic \(L\) functions are analytical functions of \(p\)-adic characters that, one way or another, interpolate special values of classical (complex) \(L\) functions. The first such examples were the \(p\)-adic \(L\) functions of T. Kubota and H. W. Leopoldt [J. Reine Angew. Math. 214-215, 328–339 (1964; Zbl 0186.09103)], interpolating Dirichlet \(L\) series. M. M. Vishik and Yu. I. Manin [Math. USSR, Sb. 24(1974), 345–371 (1976); translation from Mat. Sb., Nov. Ser. 95(137), 357–383 (1974; Zbl 0329.12016)] and N. M. Katz [Ann. Math. (2) 104, 459–571 (1976; Zbl 0354.14007)] constructed \(p\)-adic \(L\) functions which interpolate special values of Hecke \(L\) series associated with a quadratic imaginary field \(K\), in which \(p\) splits. (To fix notation write \(p={\mathfrak pp}\bar{\;}.)\) The \(p\)-adic \(L\) function of Manin-Vishik and Katz is the first object studied in this work.

Our point of view is nevertheless different, and goes back to the two fundamental papers by J. Coates and A. Wiles [Invent. Math. 39, 223–251 (1977; Zbl 0359.14009) and J. Aust. Math. Soc., Ser. A 26, 1–25 (1978; Zbl 0442.12007)]. The program may be summarized in two main steps: Fix an abelian extension \(F_ 1\) of \(K\), and let \(K_{\infty}\) be the unique \(\mathbb Z_p\) extension of \(K\) unramified outside \(\mathfrak p\) (one of the two factors of \(p\) in \(K\)). We assume that \(F_ 1\) is the ray class field modulo \(\mathfrak{fp}\), where \(\mathfrak f\) is an integral ideal relatively prime to \(\mathfrak p\). The \(p\)-adic \(L\) function, then, is essentially a \(p\)-adic integral measure on \(\mathcal G=\text{Gal}(F_ 1K_{\infty}/K)\).

Now in the first step we are given a norm-coherent sequence \(\beta\) of semilocal units in the completion of the tower \(F_{\infty}=F_ 1K_{\infty}\) at \(\mathfrak p\). Out of each such sequence we construct a certain measure \(\mu_{\beta}\) on \(\mathcal G\). We describe this construction in chapter I. In the second step, carried out in chapter II, we introduce special global units, the elliptic units. They come in norm coherent sequences, so we can view them inside the local units. When the procedure from chapter I is applied to them we obtain the \(p\)-adic \(L\) function.

Other results obtained in chapters I and II include a new proof of Wiles’ explicit reciprocity law, a \(p\)-adic analogue of Kronecker’s limit formula, and a functional equation for the \(p\)-adic \(L\) function.

The immense interest in Katz’ \(p\)-adic \(L\) functions arises from their significance to class field theory (abelian extensions of \(K\)) and the arithmetic of elliptic curves with complex multiplication. In the last two chapters we give a sample of results in these two directions.

Chapter III is mainly concerned with the “Main Conjecture” in the style of cyclotomic Iwasawa theory. The fundamental idea is that the zeros of the \(p\)-adic \(L\) function ought to be those \(p\)-adic characters of \(\mathcal G\) whose reciprocals appear in the representation of \(\mathcal G\) on a certain free \(\mathbb Z_p\)-module of finite rank. More precisely, this module \(\mathcal X\) is the Galois group of the maximal abelian \(p\)-extension of \(F_{\infty}\) which is unramified outside \(\mathfrak p\). We prove that the Iwasawa invariants of \(\mathcal X\) and the Iwasawa invariants of the \(p\)-adic \(L\) function are equal, but we do not go into the recent evidence for this conjecture discovered by K. Rubin, nor do we give Gillard’s proof of the vanishing of the \(\mu\)-invariant.

While elliptic curves are deliberately kept behind the scene in chapter III, their arithmetic, and in particular the conjecture of Birch and Swinnerton-Dyer, is the main topic of chapter IV. First we show how Kummer theory and descent are used to relate the Galois group previously denoted by \(\mathcal X\) to the Selmer group over \(F_{\infty}\). Then we give a complete proof of two beautiful theorems of Coates-Wiles and of R. Greenberg. These theorems are generalized here to treat elliptic curves with CM by an arbitrary quadratic imaginary field, not necessarily of class number 1.

Our point of view is nevertheless different, and goes back to the two fundamental papers by J. Coates and A. Wiles [Invent. Math. 39, 223–251 (1977; Zbl 0359.14009) and J. Aust. Math. Soc., Ser. A 26, 1–25 (1978; Zbl 0442.12007)]. The program may be summarized in two main steps: Fix an abelian extension \(F_ 1\) of \(K\), and let \(K_{\infty}\) be the unique \(\mathbb Z_p\) extension of \(K\) unramified outside \(\mathfrak p\) (one of the two factors of \(p\) in \(K\)). We assume that \(F_ 1\) is the ray class field modulo \(\mathfrak{fp}\), where \(\mathfrak f\) is an integral ideal relatively prime to \(\mathfrak p\). The \(p\)-adic \(L\) function, then, is essentially a \(p\)-adic integral measure on \(\mathcal G=\text{Gal}(F_ 1K_{\infty}/K)\).

Now in the first step we are given a norm-coherent sequence \(\beta\) of semilocal units in the completion of the tower \(F_{\infty}=F_ 1K_{\infty}\) at \(\mathfrak p\). Out of each such sequence we construct a certain measure \(\mu_{\beta}\) on \(\mathcal G\). We describe this construction in chapter I. In the second step, carried out in chapter II, we introduce special global units, the elliptic units. They come in norm coherent sequences, so we can view them inside the local units. When the procedure from chapter I is applied to them we obtain the \(p\)-adic \(L\) function.

Other results obtained in chapters I and II include a new proof of Wiles’ explicit reciprocity law, a \(p\)-adic analogue of Kronecker’s limit formula, and a functional equation for the \(p\)-adic \(L\) function.

The immense interest in Katz’ \(p\)-adic \(L\) functions arises from their significance to class field theory (abelian extensions of \(K\)) and the arithmetic of elliptic curves with complex multiplication. In the last two chapters we give a sample of results in these two directions.

Chapter III is mainly concerned with the “Main Conjecture” in the style of cyclotomic Iwasawa theory. The fundamental idea is that the zeros of the \(p\)-adic \(L\) function ought to be those \(p\)-adic characters of \(\mathcal G\) whose reciprocals appear in the representation of \(\mathcal G\) on a certain free \(\mathbb Z_p\)-module of finite rank. More precisely, this module \(\mathcal X\) is the Galois group of the maximal abelian \(p\)-extension of \(F_{\infty}\) which is unramified outside \(\mathfrak p\). We prove that the Iwasawa invariants of \(\mathcal X\) and the Iwasawa invariants of the \(p\)-adic \(L\) function are equal, but we do not go into the recent evidence for this conjecture discovered by K. Rubin, nor do we give Gillard’s proof of the vanishing of the \(\mu\)-invariant.

While elliptic curves are deliberately kept behind the scene in chapter III, their arithmetic, and in particular the conjecture of Birch and Swinnerton-Dyer, is the main topic of chapter IV. First we show how Kummer theory and descent are used to relate the Galois group previously denoted by \(\mathcal X\) to the Selmer group over \(F_{\infty}\). Then we give a complete proof of two beautiful theorems of Coates-Wiles and of R. Greenberg. These theorems are generalized here to treat elliptic curves with CM by an arbitrary quadratic imaginary field, not necessarily of class number 1.

##### MSC:

11S40 | Zeta functions and \(L\)-functions |

11R23 | Iwasawa theory |

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

11G40 | \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture |

11G15 | Complex multiplication and moduli of abelian varieties |

11R37 | Class field theory |