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Introduction to elliptic curves and modular forms. (English) Zbl 0553.10019
Graduate Texts in Mathematics, 97. New York etc.: Springer-Verlag. viii, 248 pp. DM 112.00 (1984).
This book provides an introduction to the theory of elliptic curves and modular forms. The author motivates his choice of subject matter with a recent result of J. B. Tunnell [Invent. Math. 72, 323–334 (1983; Zbl 0515.10013)] on the congruent number problem. A positive integer $$n$$ is called congruent if there exists a right triangle with all three sides rational and area $$n$$. It has been an ancient problem to find a simple criterion for determining whether or not a given n is congruent or not. This problem is related in an elementary fashion to the existence of points of infinite order on certain elliptic curves with complex multiplication. Via the Birch–Swinnerton-Dyer conjecture and the results of Coates-Wiles, one is led to examine the value of the Hasse-Weil $$L$$-function at $$s=1$$. Then theorems of Shimura, Waldspurger, and Tunnell on modular forms of half-integral weight lead to Tunnell’s theorem:
Let $$n\geq 1$$ be an odd square-free integer. Consider the following two conditions: (A) $$n$$ is congruent, (B) the number of triples of integers $$(x,y,z)$$ satisfying $$2x^ 2+y^ 2+8z^ 2=n$$ is equal to twice the number of triples of integers $$(x,y,z)$$ satisfying $$2x^ 2+y^ 2+32z^ 2=n$$. Then (A) implies (B). If the weak Birch-Swinnerton-Dyer conjecture is true for the elliptic curve $$Y^ 2=X^ 3-n^ 2 X$$, then (B) implies (A). A similar result holds for $$n$$ even.
The central concepts in the proof of Tunnell’s theorem form the major topics in this book. Chapter I provides an introduction to various aspects of the theory of elliptic curves, such as the addition law, points of finite and infinite order, and the connection between congruent numbers and points of infinite order. Chapter II develops those aspects of the Hasse-Weil $$L$$-function necessary to understand the Birch–Swinnerton-Dyer conjecture and the theorem of Coates-Wiles for elliptic curves with complex multiplication. In Chapter III the author gives the basics of the theory of modular forms. Finally, Chapter IV outlines recent results on modular forms of half-integral weight.
With the exception of proofs for the more difficult theorems, the treatment is self-contained. This book is well-written and will certainly appeal to many who are interested in the interplay among number theory, elliptic curves, and modular forms.
Reviewer: L. D. Olson

##### MSC:
 11G05 Elliptic curves over global fields 11-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory 11F37 Forms of half-integer weight; nonholomorphic modular forms 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 11D09 Quadratic and bilinear Diophantine equations 11G15 Complex multiplication and moduli of abelian varieties