Background on Curves and Jacobians

This chapter introduces the main characters of this book — curves and their Jacobians. To this aim we give a brief introduction to algebraic and arithmetic geometry. We first deal with arbitrary varieties and abelian varieties to give the general definitions in a concise way. Then we concentrate on Jacobians of curves and their arithmetic properties, where we highlight elliptic and hyperelliptic curves as main examples. The reader not interested in the mathematical background may skip the complete chapter as the chapters on implementation summarize the necessary mathematical properties. For full details and proofs we refer the interested reader to the books [CAFL 1996, FUL 1969, LOR 1996, SIL 1986, STI 1993, ZASA 1976]. Throughout this chapter let K denote a perfect field (cf. Chapter 2) and K its algebraic closure. Let L be an extension field of K. Its absolute Galois group AutL(L) is denoted by GL.