現代数学の系譜工学物理雑談古典ガロア理論も読む49

[過去ﾛｸﾞ] 現代数学の系譜工学物理雑談古典ガロア理論も読む49 (658ﾚｽ)
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574(3): 現代数学の系譜雑談古典ガロア理論も読む ◆e.a0E5TtKE 2018/01/18(木)23:11 ID:gGT+ehE7(6/7) AAS
>>570 補足

Swinnerton-Dyerさんが出てくるね（＾＾

外部ﾘﾝｸ:en.wikipedia.org
Littlewood conjecture
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Connection to further conjectures[edit]
It is known that this would follow from a result in the geometry of numbers, about the minimum on a non-zero lattice point of a product of three linear forms in three real variables: the implication was shown in 1955 by J. W. S. Cassels and Swinnerton-Dyer.[1]
This can be formulated another way, in group-theoretic terms. There is now another conjecture, expected to hold for n ? 3: it is stated in terms of G = SLn(R), Γ = SLn(Z), and the subgroup D of diagonal matrices in G.

Conjecture: for any g in G/Γ such that Dg is relatively compact (in G/Γ), then Dg is closed.

This in turn is a special case of a general conjecture of Margulis on Lie groups.
(引用終り)

外部ﾘﾝｸ:en.wikipedia.org
Peter Swinnerton-Dyer
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Sir Henry Peter Francis Swinnerton-Dyer, 16th Baronet KBE FRS (born 2 August 1927), commonly known as Peter Swinnerton-Dyer, is an English mathematician specialising in number theory at University of Cambridge.
As a mathematician he is best known for his part in the Birch and Swinnerton-Dyer conjecture relating algebraic properties of elliptic curves to special values of L-functions, which was developed with Bryan Birch during the first half of the 1960s with the help of machine computation, and for his work on the Titan operating system.
(引用終り)

外部ﾘﾝｸ:en.wikipedia.org
Birch and Swinnerton-Dyer conjecture

575(1): 現代数学の系譜雑談古典ガロア理論も読む ◆e.a0E5TtKE 2018/01/18(木)23:43 ID:gGT+ehE7(7/7) AAS
>>574 補足

この文が、だれがいつ書いたのか不明だが・・・
”that contributed to Lindenstrauss' Fields Medal in 2010.”とあってね
へー、「Lindenstrauss' Fields Medal in 2010」なのか〜、と思った次第
私も、不勉強だね〜。全然ピントこなかったな〜（＾＾

外部ﾘﾝｸ:www.york.ac.uk
University of York
外部ﾘﾝｸ[pdf]:www.york.ac.uk
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Littlewood's Conjecture (1930)
Littlewood's Conjecture is at the heart of multiplicative Diophantine approximation and has motivated
many recent breakthrough developments such as the work of Einsiedler, Katok and Lindenstrauss [5]
that contributed to Lindenstrauss' Fields Medal in 2010. The conjecture is well known for its strong
links with dynamical systems and ergodic theory (indeed, the measure rigidity conjecture of Margulis [7]
regarding the dynamics on SL3(R)=SL3(Z) implies Littlewood's Conjecture) and is currently a part of
a major research trend world-wide. It has been in the spotlight at many recent major workshops and
conferences including the 2010 ICM in Hyderabad.
(引用終り)

600: 現代数学の系譜雑談古典ガロア理論も読む ◆e.a0E5TtKE 2018/01/20(土)10:16 ID:gQefYikW(3/21) AAS
>>574

Current statusのところに図があって、これなかなか綺麗な図だなと（＾＾
外部ﾘﾝｸ:en.wikipedia.org
Birch and Swinnerton-Dyer conjecture
(抜粋)
Current status

A plot of Π _{p<= X}{{N_{p}}/{p}} for the curve y2 = x3 ? 5x as X varies over the first 100000 primes.
The X-axis is log(log(X)) and Y-axis is in a logarithmic scale so the conjecture predicts that the data should form a line of slope equal to the rank of the curve, which is 1 in this case.
For comparison, a line of slope 1 is drawn in red on the graph.
(引用終り)

625(1): 現代数学の系譜雑談古典ガロア理論も読む ◆e.a0E5TtKE 2018/01/20(土)22:36 ID:gQefYikW(14/21) AAS
>>574 補足

外部ﾘﾝｸ:en.wikipedia.org
Littlewood conjecture
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References
3 M. Einsiedler; A. Katok; E. Lindenstrauss (2006-09-01). "Invariant measures and the set of exceptions to Littlewood's conjecture". Annals of Mathematics. 164 (2): 513?560. arXiv:math.DS/0612721?Freely accessible. doi:10.4007/annals.2006.164.513. MR 2247967. Zbl 1109.22004.
(引用終り)

これ、arXiv:mathのリンクから下記に入ると、”Ann. of Math. (2) 164 (2006)”版が公開されているね〜（＾＾
外部ﾘﾝｸ:arxiv.org
外部ﾘﾝｸ[pdf]:arxiv.org
Invariant measures and the set of exceptions to Littlewood's conjecture
Manfred Einsiedler, Anatole Katok, Elon Lindenstrauss
(Submitted on 22 Dec 2006)
We classify the measures on SL (k,R)/SL (k,Z) which are invariant and ergodic under the action of the group A of positive diagonal matrices with positive entropy. We apply this to prove that the set of exceptions to Littlewood's conjecture has Hausdorff dimension zero.
Subjects: Dynamical Systems (math.DS); Number Theory (math.NT)
Journal reference: Ann. of Math. (2) 164 (2006), no. 2, 513--560
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Part 2. Positive entropy and the set of exceptions to Littlewood’s Conjecture
7. Definitions

11. The set of exceptions to Littlewood’s Conjecture

The following well-known proposition
gives the reduction of Littlewood’s conjecture to the dynamical question which
we studied in Section 10; see also [24, §2] and [46, §30.3]. We include the proof
for completeness.
Proposition 11.1. The tuple (u, v) satisfies
(11.1) liminf n→∞ n ||nu|| ||nv|| = 0,
if and only if the orbit A+τu,v is unbounded where A+ is the semigroup
（略）
(引用終り)

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