##
**Modular units.**
*(English)*
Zbl 0492.12002

Grundlehren der mathematischen Wissenschaften. 244. New York - Heidelberg - Berlin: Springer-Verlag. xiii, 358 p. DM 98.00; $ 45.70 (1981).

Let \(N\) be an integer greater than 1 and let \(\Gamma(N)\) denote the subgroup of the modular group \(\Gamma(1) = \mathrm{SL}_2 (\mathbb Z)\) consisting of the matrices \(\gamma\equiv 1\pmod N\). \(\Gamma(1)\) acts on the upper half plane \(\mathfrak H\) and the modular function \(j\) defines a complex analytic isomorphism \(\Gamma(1)\backslash\mathfrak H \to \mathbb P^1(\mathbb C) - \infty\) whose one-point compactification is the Riemann surface \(\mathbb P^1(\mathbb C)\). The subgroup \(\Gamma(N)\) acts on \(\mathfrak H\) and there is a complex analytic isomorphism \(\Gamma(N)\backslash\mathfrak H \to Y(N)\) lifting that defined by \(j\), where \(Y(N)\) is an affine curve, whose compactification \(X(N)\) is obtained by adding the inverse image of \(\infty\) on the \(j\)-line: thus \(X(N) =Y(N)\cup X^\infty(N)\), where the points in \(X^\infty(N)\) are called cusps, after the shape of the fundamental domain for \(\Gamma(N)\). If \(\mathfrak H^* = \mathfrak H \cup \mathbb Q\cup \{\infty\}\), then \(X^\infty(N)\) is the set of equivalence classes of \(\mathbb Q\cup \{\infty\}\) with respect to the action of \(\Gamma(N)\).

The affine ring of regular functions on \(Y(N)\) over \(\mathbb C\) is the integral closure of \(\mathbb C[j]\) in the function field of \(X(N)\) over \(\mathbb C\) and one can also consider, for example, curves defined over \(\mathbb Q(\mu_N)\) (the cyclotomic field of \(N\)-th roots of unity), in which case one obtains the integral closure of \(\mathbb Q[j]\). In what follows we are concerned primarily with functions defined over \(\mathbb Q(\mu_N)\).

The cuspidal divisor class group \(\mathcal C(N)\) consists of the group of divisors of degree 0 whose support is the set \(X^\infty(N)\) of cusps, modulo the group of divisors of functions on \(X(N)\) having neither poles nor zeros outside the cusps; that is, the subgroup \(\mathrm{Pic}^\infty X(N)\) of \(\mathrm{Pic } X(N)\). The units consist of those functions having no zeros nor poles in the upper half plane.

The primary object of the theory presented in this book is the study of \(\mathcal C(N)\) as a module over a certain Cartan group \(C(N)\); namely the reduction \(\bmod N\) of a subgroup of \(\mathrm{GL}_2 (\mathbb Z_N)\), where \(\mathbb Z_N = \prod_{p\mid N} \mathbb Z_p\). The theory is analogous to the study of the ideal class group of \(\mathbb Q(\mu_N)\) as a module over the group ring \(\mathbb Z[G]\), \(G\approx (\mathbb Z/N\mathbb Z)^*\), in the cyclotomic case. Unlike our present knowledge of the cyclotomic case, \(\mathcal C(N)\) admits a complete description. In chapter 5, the authors show that the divisor class group generated by the cusps can be represented as a quotient of the group ring of the Cartan group \(C(N)\) by an analogue of the Stickelberger ideal. Full use is made of the characterization of the units given in chapters 2 and 3 and there are beautiful connections with algebraic geometry. The order of the cuspidal divisor class group is computed in a manner analogous to that used by Iwasawa in the cyclotomic case, the role of the Bernoulli numbers \(B_{1,\chi}\) in the latter case being played by the second Bernoulli numbers \(B_{2,\chi}\) in the case of \(\mathcal C(N)\). Chapter 5 closes with an analysis of the eigenspace decompositions on \(X(p)\), \(p\) a prime, and it turns out that they involve ordinary Bernoulli numbers and Gauß sums.

Chapter 6 deals with the cuspidal divisor class group for the modular curve \(X_1(N)\) obtained as before from the subgroup \(\Gamma_1(N)\) of \(\Gamma(1)\) of matrices \(\gamma\equiv \begin{pmatrix} 1 & b \\ 0 & 1\end{pmatrix}\pmod N\), where \(b\) is arbitrary.

Chapters 7 to 13 are more specialised. In Chapter 7 the authors study the modular units on Tate curves; that is on the elliptic curves \(Y^2 -XY = X^3 - h_2X - h_3\) where \(h_2 =5\,\sum_{n=1}^\infty q^n/(1-q^n)\), \(h_3 = \sum_{n=1}^\infty (5n^3+7n^5)\cdot q^n/12 \cdot (1-q^n)\).

Chapter 8 is concerned with applications to Diophantine equations and in particular to \[ \frac{X_3-X_1}{X_2-X_1} + \frac{X_2-X_3}{X_2-X_1} = 1 \]

which is satisfied by the \(\lambda\)-function.

The remaining chapters contain an exposition of the theory of Robert’s elliptic units in arbitrary class-fields with a unit index computation due to Kersey. The modular units are an example of a universal even distribution. That is, of a mapping \(\varphi\colon \mathbb Q/\mathbb Z\to A\) to an Abelian group such that for every \(N\) and some positive integer \(k\), \(N^k \sum_{j=1}^{N-1} \varphi(x+\tfrac{j}{N}) = \varphi(Nx)\). Distributions of that kind occur in a number of contexts in number theory and the book begins with an account of the basic theory. Not only is the geometric theory of the cuspidal divisor class group analogous to the theory of cyclotomic fields, but also the most exciting developments in recent years have their origins in work of A. Wiles [Invent. Math. 58, 1–35 (1980; Zbl 0436.12004)], who first showed how the connection between the two theories can be made.

The present book is to be welcomed both as an exposition of a fascinating subject, much of which appeared first in a series of papers in the Math. Ann. and also as an introduction to the dramatic developments due to Mazur and Wiles [see S. Lang, Bull. Am. Math. Soc., New Ser. 6, 253–316 (1982; Zbl 0482.12002)].

The affine ring of regular functions on \(Y(N)\) over \(\mathbb C\) is the integral closure of \(\mathbb C[j]\) in the function field of \(X(N)\) over \(\mathbb C\) and one can also consider, for example, curves defined over \(\mathbb Q(\mu_N)\) (the cyclotomic field of \(N\)-th roots of unity), in which case one obtains the integral closure of \(\mathbb Q[j]\). In what follows we are concerned primarily with functions defined over \(\mathbb Q(\mu_N)\).

The cuspidal divisor class group \(\mathcal C(N)\) consists of the group of divisors of degree 0 whose support is the set \(X^\infty(N)\) of cusps, modulo the group of divisors of functions on \(X(N)\) having neither poles nor zeros outside the cusps; that is, the subgroup \(\mathrm{Pic}^\infty X(N)\) of \(\mathrm{Pic } X(N)\). The units consist of those functions having no zeros nor poles in the upper half plane.

The primary object of the theory presented in this book is the study of \(\mathcal C(N)\) as a module over a certain Cartan group \(C(N)\); namely the reduction \(\bmod N\) of a subgroup of \(\mathrm{GL}_2 (\mathbb Z_N)\), where \(\mathbb Z_N = \prod_{p\mid N} \mathbb Z_p\). The theory is analogous to the study of the ideal class group of \(\mathbb Q(\mu_N)\) as a module over the group ring \(\mathbb Z[G]\), \(G\approx (\mathbb Z/N\mathbb Z)^*\), in the cyclotomic case. Unlike our present knowledge of the cyclotomic case, \(\mathcal C(N)\) admits a complete description. In chapter 5, the authors show that the divisor class group generated by the cusps can be represented as a quotient of the group ring of the Cartan group \(C(N)\) by an analogue of the Stickelberger ideal. Full use is made of the characterization of the units given in chapters 2 and 3 and there are beautiful connections with algebraic geometry. The order of the cuspidal divisor class group is computed in a manner analogous to that used by Iwasawa in the cyclotomic case, the role of the Bernoulli numbers \(B_{1,\chi}\) in the latter case being played by the second Bernoulli numbers \(B_{2,\chi}\) in the case of \(\mathcal C(N)\). Chapter 5 closes with an analysis of the eigenspace decompositions on \(X(p)\), \(p\) a prime, and it turns out that they involve ordinary Bernoulli numbers and Gauß sums.

Chapter 6 deals with the cuspidal divisor class group for the modular curve \(X_1(N)\) obtained as before from the subgroup \(\Gamma_1(N)\) of \(\Gamma(1)\) of matrices \(\gamma\equiv \begin{pmatrix} 1 & b \\ 0 & 1\end{pmatrix}\pmod N\), where \(b\) is arbitrary.

Chapters 7 to 13 are more specialised. In Chapter 7 the authors study the modular units on Tate curves; that is on the elliptic curves \(Y^2 -XY = X^3 - h_2X - h_3\) where \(h_2 =5\,\sum_{n=1}^\infty q^n/(1-q^n)\), \(h_3 = \sum_{n=1}^\infty (5n^3+7n^5)\cdot q^n/12 \cdot (1-q^n)\).

Chapter 8 is concerned with applications to Diophantine equations and in particular to \[ \frac{X_3-X_1}{X_2-X_1} + \frac{X_2-X_3}{X_2-X_1} = 1 \]

which is satisfied by the \(\lambda\)-function.

The remaining chapters contain an exposition of the theory of Robert’s elliptic units in arbitrary class-fields with a unit index computation due to Kersey. The modular units are an example of a universal even distribution. That is, of a mapping \(\varphi\colon \mathbb Q/\mathbb Z\to A\) to an Abelian group such that for every \(N\) and some positive integer \(k\), \(N^k \sum_{j=1}^{N-1} \varphi(x+\tfrac{j}{N}) = \varphi(Nx)\). Distributions of that kind occur in a number of contexts in number theory and the book begins with an account of the basic theory. Not only is the geometric theory of the cuspidal divisor class group analogous to the theory of cyclotomic fields, but also the most exciting developments in recent years have their origins in work of A. Wiles [Invent. Math. 58, 1–35 (1980; Zbl 0436.12004)], who first showed how the connection between the two theories can be made.

The present book is to be welcomed both as an exposition of a fascinating subject, much of which appeared first in a series of papers in the Math. Ann. and also as an introduction to the dramatic developments due to Mazur and Wiles [see S. Lang, Bull. Am. Math. Soc., New Ser. 6, 253–316 (1982; Zbl 0482.12002)].

Reviewer: J. Vernon Armitage (Durham)

### MSC:

11G16 | Elliptic and modular units |

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

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

11R18 | Cyclotomic extensions |

14G25 | Global ground fields in algebraic geometry |

11F11 | Holomorphic modular forms of integral weight |