# Influential results by Swinnerton-Dyer

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.

• That you can't bid 8 diamonds as a sacrifice in bridge. – literature-searcher Dec 28 '18 at 13:18
• Sir Peter Swinnerton-Dyer passed away on December 26th: en.wikipedia.org/wiki/Peter_Swinnerton-Dyer In the area of rational points, Swinnerton-Dyer had a huge influence, e.g., his papers on rational points on cubic hypersurfaces. – Jason Starr Dec 28 '18 at 13:19
• His name appeared in several lines in Modern Chess Openings, 10th edition, mostly in offbeat variations such as the Ponziani. I don't know whether his lines have survived to the 15th edition. shropshirechess.org/History/1950s.htm – Gerry Myerson Dec 28 '18 at 18:57
• @YemonChoi I think that being one of the \$1M Millennium Problems consitutes pretty good evidence for a conjecture having had tremendous influence. – Joe Silverman Dec 28 '18 at 21:30
• @YemonChoi Fair enough, we can agree to differ. My feeling is that the Millennium prize problems were selected because they've both generated a huge amount of research and because their resolution is likely to have a great influence. So I'm not a computer scientist or a topologist or a PDE researcher, but I'm happy to accept that P=NP, the Poincare conjecture, and solving the Navier-Stokes equation have had a tremendous influence on their areas. The description of the Millennium problems online (easy to google) will describe why it's important. ... – Joe Silverman Dec 29 '18 at 0:47

In their paper, Atkin and Swinnerton-Dyer proved the startling fact that for the three values $$m = 5, 7, 11$$ and every value of $$r = 0, 1, ... ,m -1$$ the generating function $$\sum_{n\geq 0}p(mn+r)q^n,$$ with $$p(n)$$ the number of partitions of $$n$$, is congruent modulo $$m$$ to a simple infinite product.