Chapter 14 - Reducible jacobians

Summary

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)].