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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.

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    $\begingroup$ That you can't bid 8 diamonds as a sacrifice in bridge. $\endgroup$ – literature-searcher Dec 28 '18 at 13:18
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    $\begingroup$ 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. $\endgroup$ – Jason Starr Dec 28 '18 at 13:19
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    $\begingroup$ 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 $\endgroup$ – Gerry Myerson Dec 28 '18 at 18:57
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    $\begingroup$ @YemonChoi I think that being one of the $1M Millennium Problems consitutes pretty good evidence for a conjecture having had tremendous influence. $\endgroup$ – Joe Silverman Dec 28 '18 at 21:30
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    $\begingroup$ @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. ... $\endgroup$ – Joe Silverman Dec 29 '18 at 0:47
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Just to show the limited value of citation counts, the most cited paper of Sir Peter Swinnerton-Dyer on MathSciNet is not his 1965 paper with Birch, but a 1954 paper with Atkin on Some properties of partitions:

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.

as discussed in: Winquist and the Atkin-Swinnerton-Dyer partition congruences for modulus 11

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    $\begingroup$ An issue for citation counts on papers that appeared before, say 1970, is that MathSciNet doesn't have citations from papers that appeared before journals started to send them electronic lists of references. So for example, the BSwD 1965 paper that you mention has only two citations from the 1980s, and none earlier, listed in MSN. GoogleScholar lists a couple of dozen citations between 1965 and 1980. (Not that GS is so great.) Anyway, there are sources such as Science Citation Index, which did an exhaustive citation list before the 1980s, but MathSciNet is not accurate. $\endgroup$ – Joe Silverman Dec 28 '18 at 21:28
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"The Hasse problem for rational surfaces" by Birch and Swinnerton-Dyer is certainly influential in the study of obstructions to the Hasse principal, as far as I understand there are examples of families surfaces for which the Hasse principal fails which are constructed here including an example for a del Pezzo surface to add to an example of Iskovskih. Given the interest in showing that for many families of surfaces the Brauer-Manin Obstruction fully accounts for obstructions to the Hasse-principal, having these counterexamples to the Hasse principal at hand is certainly important.

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