The arithmetic (= number theory) of curves of genus 0 is well understood. For genus 1, there is a rich body of theory and conjecture, and in recent years notable success has been achieved in transforming the latter into the former. For curves of higher genus there is a rich body of theory with some spectacular successes (e.g. Faltings' proof of ‘Mordell's Conjecture’), but our command is still rudimentary. For genus 1, a concrete question about an individual curve can usually be answered by one means or another: for higher genus, this is far from the case.

For curves of genus 0 and 1 the road between general theory and particular cases is no one-way street. Numerous individual cases were investigated by amateurs as well as as by distinguished mathematicians such as Diophantos, Fermat, Euler, Sylvester, Mordell, Selmer, Birch and Swinnerton-Dyer. Regularities which emerged, sometimes quite unexpectedly, suggested theorems, which could sometimes be proved. The new theorems suggested new questions. For higher genus, existing theory is notoriously unadapted to the study of individual curves, and few have been elucidated. What is needed is a corpus of explicit concrete cases and a middlebrow arithmetic theory which would provide both a practicable means to obtain them and a framework to understand any unexpected regularities.

Such a theory will, of course, draw freely from the existing body of knowledge, but the emphasis on feasibility gives new perspectives.

Introduction. It can happen that an abelian variety of dimension 2 is isogenous to a product of two (not necessarily distinct) abelian varieties of dimension 1, i.e. elliptic curves. The first example of this seems to have given by Legendre at the age of 80 in the troisième supplément to his Théorie des fonctions elliptiques: in a review Jacobi (1832) gave one form of Theorem 1.1 below. There are infinitely many cases indexed by a natural number as parameter, but Jacobi's is particularly straightforward. We have already met it in Chapter 9, where it had to be excluded from the discussion. We shall treat it in this chapter. It occurs quite frequently for naturally arising curves. The other cases of reducibility appear to be much more difficult to handle, and we do not make the attempt. There is a treatment in Frey & Kani (1991) and a rather inconclusive discussion of the construction of explicit examples in Frey (1995). For a different approach and further references, see Kuhn (1988). See also Ruppert (1990) [use of the complex structure], Grant (1994c) [use of L-functions], Stoll (1995), pp. 1343–1344 [use of L-functions], and Flynn, Poonen & Schaefer (1995) [simple proof of irreducibility in a special case with probable wide applicability].

The straightforward case. Since what we are doing in this section is geometry, we work in the algebraic closure. We say that two curves Y2 = F(X) are equivalent if they are taken into one another by a fractional linear transformation of X and the related transformation of Y [see (1.1.3)].