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Oct
18 |
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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 |
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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 |
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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
21 |
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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
1 |
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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
16 |
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Inaccessible cardinals and Andrew Wiles's proof
The Stack Project develops a huge amount of Grothendieck style mathematics, including a lot of etale cohomology, using only ZFC (specifically, NOT using universes). If anyone has any doubt that this can be done, I suggest that they look at it. |
Aug
16 |
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Inaccessible cardinals and Andrew Wiles's proof
Actually, Cosmonut misquoted the article by leaving out the rest of the statement: "But there is a general consensus among mathematicians that this was just a convenient short cut rather than a logical necessity. With a little work, Wiles's proof should be translatable into Peano arithmetic or some slight extension of it." |
Jul
29 |
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How connected are you?
MR collaboration numbers aren't accurate. For example, MR claims my collaboration number with Gauss is an improbable 6. They get that number by claiming Landau and Riemann are coauthors and Riemann and Gauss are coauthors because works by them were included together in reprint collections. Similarly, they incorrectly give my Erdos number as 3 because they count me as a co-author of Gerardin when we only published articles in the same collection. |
Jul
7 |
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Does the Tannaka-Krein theorem come from an equivalence of 2-categories?
@Theo et al. --- well, you could try looking up the definition in Catégories Tannakiennes (Saavedra 1972, Deligne 1990) or Tannakian Categories (Deligne and ... 1982, Breen 1994) or .. A Tannakian category over a field $k$ is neutral if it admits a fibre functor over $k$. In general, it only admits a fibre functor over an extension of $k$. There are various expressions of Tannaka duality in 2-category terms in Saavedra, e.g., III 2.3.2, p180. |
Jun
25 |
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Weight filtration over the integers
algori. Thanks! you have helped clarify things for me. |
Jun
25 |
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Weight filtration over the integers
Actually, I'm still confused by Voisin, since in that section she seems to be talking about complex manifolds, not algebraic varieties. Perhaps the assumption is hidden somewhere. |
Jun
24 |
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Weight filtration over the integers
In 8.36 (p.214 of the English version) of her book, Voisin defines a mixed Hodge structure to have a weight filtration on the integral cohomology. She then says that Deligne's theorem shows the existence a mixed Hodge structure on the integral cohomology of the complement $U$ of a normal divisor crossing, and adds that "One can show that this mixed Hodge structure depends only on $U$ and not on its compactification." This seems to contradict what you are saying. Did Voisin mean to define mixed Hodge structure as Deligne does, with a weight filtration defined on the rational cohomology? |
Jun
21 |
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Least collaborative mathematician
Greg, MathSciNet only lists citations after about 1997. For example, they list only 16 citations from references for Weil's Foundations of Algebraic Geometry, the earliest of which is 1997. |
Jun
21 |
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Least collaborative mathematician
There's also Hooley, Crelle 328 (1981), 161--207, which depends crucially on Milne, Crelle 328 (1981), 208--220. Maybe I should have insisted on a joint paper.... |
Jun
15 |
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Dimension of central simple algebra over a global field “built using class field theory”.
See Theorem 2.6, Chapter VIII, of my class field theory notes, but the proof uses the Grunwald-Wang theorem which is not (yet) proved in the notes. |