bio | website | jmilne.org/math |
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location | Ann Arbor, MI, USA, and New Zealand. | |
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visits | member for | 4 years, 11 months |
seen | Oct 18 '10 at 13:35 | |
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Arithmetic geometry (especially Shimura varieties and abelian varieties).
Nov 10 |
awarded | Pundit |
Oct 30 |
awarded | Nice Answer |
Oct 28 |
awarded | Good Answer |
Oct 22 |
awarded | Yearling |
Oct 18 |
comment |
Are there any good computer programs for drawing (algebraic) curves?
If you have access to Mathematica and it produces the diagram you want, fine, but for more control use gnuplot+tikz. |
Oct 17 |
comment |
Awfully sophisticated proof for simple facts
Brauer groups and cohomology are certainly overkill for Wedderburn's theorem: if $D$ is a finite division algebra and $L$ is a maximal subfield, then the Noether-Skolem theorem shows that the multiplicative group of $D$ is a union of conjugates of that of $L$; hence $D$=$L$. |
Oct 14 |
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Chapters 1--4 of the Artin-Tate notes on Class Field Theory
As far as I know, the notes for the first part of the seminar were never written up. Lang missed this part of the seminar because he started as a philosophy student. |
Oct 7 |
comment |
who fixed the topology on ideles?
Yes, I think the answer is Weil. In his short 1936 paper "Remarques sur des resultats recent de C. Chevalley" he complains about Chevalley's topology, and writes down other constructions and topologies (and mentions Grossencharacters) but doesn't write down anything immediately recognizable (to me) as the ideles. In his 1951 paper (J. Math. Soc. Japan) he defines without comment the ideles with the natural (modern) topology. That is also where he wrote "La recherche d'une interpretation pour C_k ... me semble constituer l'un des problemes fondamentaux..." See also his Commentaries in his CW. |
Oct 3 |
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Is a torsion free abelian group finitely generated, if all of its localizations at primes p are finitely generated over Zp?
Have you seen the erratum jmilne.org/math/CourseNotes/errata.html#AV ? |
Sep 24 |
answered | Math keyboard: does it exist ? |
Sep 21 |
comment |
Are there motives which do not, or should not, show up in the cohomology of any Shimura variety?
Blasius has pointed out that the naive generalization of the modularity conjecture fails --- there exist elliptic curves over number fields that are not quotients of the albanese of any Shimura variety --- but I don't know of any reason why the more general version (4) can't be true. (Blasius 2004 MR2058605). |
Sep 11 |
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What heuristic evidence is there concerning the unboundedness or boundedness of Mordell-Weil ranks of elliptic curves over $\Bbb Q$?
Cassels said it best (JLMS 1966, p.257):... it has been widely conjectured [on the basis of calculations] that there is an upper bound for the rank depending only on the groundfield. This seems to me implausible because the theory makes it clear that an abelian variety can only have high rank if it is defined by equations with very large coefficients. . (For there must be a lot of alternative factorizations to be possible in the arguments of §24.) |
Sep 10 |
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Is the category of commutative group schemes abelian?
Thanks - fixed. |
Sep 10 |
revised |
Is the category of commutative group schemes abelian?
added 1041 characters in body |
Sep 10 |
awarded | Nice Answer |
Sep 9 |
answered | Is the category of commutative group schemes abelian? |
Sep 1 |
comment |
Is there a classification of embeddings of SL_2 into SP_6 as algebraic groups over Q and R respectively?
Assume the base field has characteristic zero. The category of representations of a semisimple Lie algebra is Tannakian, and the algebraic group attached to this category is the simply connected algebraic group with the given Lie algebra. A homomorphism of Lie algebras defines a tensor functor of the Tannakian categories, and hence a homomorphism of the corresponding simply connected algebraic groups. |
Aug 31 |
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Why aren't there more classifying spaces in number theory?
Continued: It was then found that $H^{1}$ coincided with the group of crossed homomorphisms modulo principal crossed homomorphisms, and $H^{2}$ with the group of equivalence classes of "factor sets", which had been introduced much earlier (e.g., I. Schur, \"{U}ber die Darstellung der endlichen$\ldots$ , 1904; O. Schreier, \"{U}ber die Erweiterungen von Gruppen, 1926; R. Brauer, \"{U}ber Zusammenh\"{a}nge$\ldots$ , 1926). For more on the history, see MacLane 1978 (Origins of the cohomology of groups. Enseign. Math. (2) 24 (1978), no. 1-2, 1--29. MR0497280). |
Aug 31 |
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Why aren't there more classifying spaces in number theory?
From my Class Field Theory Notes p86. In the mid-1930s, Hurewicz showed that the homology groups of an "aspherical space" $X$ depend only on the fundamental group $\pi$ of the space. Thus one could think of the homology groups $H_{r}(X,\mathbb{Z})$ of the space as being the homology groups $H_{r}(\pi,\mathbb{Z})$ of the group $\pi$. It was only in the mid-1940s that Hopf, Eckmann, Eilenberg, MacLane, Freudenthal and others gave purely algebraic definitions of the homology and cohomology groups of a group $G$. |
Aug 23 |
awarded | Nice Answer |