The conjecture of Birch and Swinnerton-Dyer had a tremendous influence on the development of arithmetic geometry. Which other results of Swinnerton-Dyer have had a lasting influence?

[edit, in answer to Yemon Choi]

The influence of BSD has been multifold. There was the initial work to get an exact formula for the leading term and extend it to abelian varieties, which lead to progress on duality in Galois cohomology of number fields and integral models of abelian varieties. It served as a prototype for general conjectures about special values of L-functions (Tate, Beilinson, Bloch-Kato). The attempts to prove it have opened new fields of research (like the Gross-Zagier theorem that paved the way to Kudla's program, some kind of arithmetic mirror symmetry, or Coates-Wiles result that gave a boost to Iwasawa theory), etc.