*(English)*Zbl 0804.14013

This book is an introduction to the arithmetic study of elliptic curves and to modular forms. The remarkable fact is the elementary presentation of rather deep and far reaching results and conjectures which yield a quick access to the heart of this very classical and nevertheless still very active field of mathematical research, also for non-experts. The book ends with a chapter on the Taniyama-Weil- (or Shimura-Taniyama- )conjecture. The last section explains the connection with Fermat’s last theorem, therefore it yields a relatively easy access in the style of a very carefully written textbook of about 400 pages, which comes close to the latest developments in the proof of Fermat’s last theorem.

After elementary introduction to plane algebraic curves and classification of cubic curves, $j$-invariant etc. two classical theorems are proved: Mordell’s theorem, i.e. for an elliptic curve $E$ over $\mathbb{Q}$, the group $E\left(\mathbb{Q}\right)$ is finitely generated, and the Lutz-Nagell theorem, which gives necessary conditions to be satisfied by the coordinates of rational torsion points. – It follows a chapter on the analytic point of view for the theory of elliptic curves. Besides the usual topics like Weierstrass $\wp $-function and elliptic integrals, it explains for example the use of the arithmetic-geometric Gauss mean to compute the periods of elliptic integrals.

After this the author turns to $L$-functions and modular forms. – $L$- functions are first introduced, following the historical order, to prove Dirichlet’s theorem on arithmetic progressions. Analytic properties of these functions are proved in clearly emphasizing that these “motivic $L$-functions” are “automorphic”, i.e. attached to automorphic forms, and that automorphic $L$-functions have more manageable analytic properties. This is done before a general introduction to modular forms and Hecke-operators which is the objective of the following two chapters. Then follows a chapter where the $L$-function of an elliptic curve is introduced and Hasse’s theorem (which is a special case of the Weil conjectures for the congruence zeta function) is proved.

It follows a rather long chapter on Eichler-Shimura theory which is of different nature, compared with the previous style of the book. After an overview it summarizes and assembles background material, mostly omitting proofs, on topics like:

(a) compact Riemann surfaces together with a Riemann-Roch theorem for compact Riemann surfaces, Abel’s theorem, Jacobi’s inversion theorem;

(b) varieties and curves;

(c) abelian varieties and Jacobian varieties.

This is used to construct the modular curves ${X}_{0}\left(N\right)$, the action of the Hecke algebra on ${H}_{1}({X}_{0}\left(N\right),\u2102)$ gnd on $J\left({X}_{0}\left(N\right)\right)$ and to establish the assignment of a modular elliptic curve $E$ to a normalized newform whose Fourier coefficients are integers, given by the Eichler-Shimura theory, with the property the $L$-series of $E$ is essentially the $L$-function of $f$ (up to finitely many Euler factors).

In the last chapter XII various well known conjectures concerning $L$- functions of elliptic curves are stated and commented, culminating in the conjecture: Every elliptic curve has a modular parametrization (Taniyama). The idea to relate this to prove Fermat’s last theorem is explained in the last section and Ribet’s theorem is quoted. Thus, $L$- functions are perhaps the most interesting and exiting topics of the book, one other remarkable fact is that there are many interesting and illuminating examples discussed in some detail.

##### MSC:

14H52 | Elliptic curves |

14-02 | Research monographs (algebraic geometry) |

11-02 | Research monographs (number theory) |

14G10 | Zeta-functions and related questions |

11F66 | Langlands L-functions; one variable Dirichlet series, etc. |

11G40 | $L$-functions of varieties over global fields |

11D41 | Higher degree diophantine equations |

11G05 | Elliptic curves over global fields |

14G05 | Rational points |