Descent by 3-isogeny and 3-rank of quadratic fields Jaap Top September 1991 Abstract In this paper families of elliptic curves admitting a rational isogeny of degree 3 are studied. It is known that the 3-torsion in the class group of the field defined by the points in the kernel of such an isogeny is related to the rank of the elliptic curve. Families in which almost all the curves have rank at least 3 are constructed. In some cases this provides lower bounds for the number of quadratic fields which have a class number divisible by 3. 1 Introduction By the 3-rank r 3 (K ) of a number field K we will mean the dimension (as a vector space over the field with 3 elements F 3 ) of the 3-torsion in the class group of K . As usual one writes r 3 (d) := r 3 (Q( √ d)). A remarkable classical result of Scholz says that the numbers r 3 (d) and r 3 (−3d) differ by at most 1. Most other literature on the function r 3 (d) concerns its possible values. So far, the record seems to be r 3 (d) = 6. In his thesis written in 1987, Quer exhibited 3 negative integers d for which this value is taken [Qu1],[Qu2]. The method exploited is that one can show that under certain conditions 2r 3 (d) + 1 is an upper bound for the rank of the group of Q-rational points on the elliptic curve given by y 2 = x 3 + d. Given d such that this curve has rank 12, one obtains r 3 (d) ≥ 6. By class field theory the 3-torsion in the class group of Q( √ d) corresponds to degree 3 Galois extensions of Q( √ d) which are unramified at all finite primes. Unramified extensions with –more generally– Galois group the alternating group A n have been constructed by Fr¨ohlich and by Uchida [Uch]. An example in case n = 3 (with in fact even more special properties) is given by the fields Q( √ −27m 2 − 4m), for any integer m which cannot be written in the form n 3 − n 2 . This last example appears in a recent paper of Brinkhuis [Br]. From the example given above it follows easily that infinitely many fields Q( √ d) exist for which r 3 (d) ≥ 1. The same holds for r 3 (d) ≥ 2 (Shanks, [Sha]) and even 1