Algebraic geometry and arithmetic curves /

Every serious student of modern number theory must eventually confront the profound synthesis of algebraic geometry and arithmetic embodied in Grothendieck's theory of schemes. For decades English-speaking students have studied R. Hartshorne's Algebraic Geometry (1977) to learn the language of schemes. But Hartshorne concentrates on pure geometry as he mostly suppresses strictly arithmetical issues by working over algebraically closed fields. For students of number theory, that makes the long march through Hartshorne a mere prelude to relearning the subject in greater generality, perhaps by reading the voluminous writings of Grothendieck and his followers. But Liu (Universite Bordeaux) allows the student interested in Diophantine matters to learn algebraic geometry with the appropriate emphasis right from the start. Liu takes beginners all the way from first principles to the modern theory of stable reduction. Although other books do offer a fast passage to modern number theory (Joseph H. Silverman's books on elliptic curves, for example), only Liu provides a systematic development of algebraic geometry aimed at arithmetic. Upper-division undergraduates through professionals. D. V. Feldman University of New Hampshire