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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. |
Jun
14 |
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Are periods of rigid Calabi-Yau threefolds over $Q$ algebraic?
This looks wrong to me. Compute the Mumford-Tate group G of the relevant cohomology group (as a rational Hodge structure); then conjecturally, the transcendence degree of the field generated by the periods is the dimension of G (Grothendieck). The point is that algebraic classes force algebraic relations between the periods; hence conjecturally also Hodge classes do (this is known for abelian varieties by Deligne); and Grothendieck conjectured that these are all the relations. So if you want lots of relations, you need to find lots of Hodge classes. |
Jun
12 |
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Hodge Standard Conjecture in Positive Characteristic
Actually the Hodge standard conjecture is not even known in positive characteristic for abelian varieties --- it is only known that it is implied by the Hodge conjecture for complex CM abelian varieties (see my 2002 Annals paper). Of course, the conjecture is trivial for projective n-space. A good reference for these things is Kleiman's article in the proceedings of the Motives conference, Seattle 1991, published by AMS 1994. |
May
31 |
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What was Galois theory like before Emil Artin?
Charles, the independence of multiplicative characters is usually credited to Dedekind. Galois theory is about separable extensions. |
May
31 |
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What was Galois theory like before Emil Artin?
Actually, it wasn't all that different, except that you first proved the primitive element theorem, and then proved things by choosing a primitive element. Artin disliked having to make a choice, and his main contribution was show that you can do Galois theory without choosing a primitive element. It's not obvious to me that this makes things easier or better. You can find the old approach in A.A. Albert's book on algebra. |
May
29 |
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Points of reductive groups
Actually, "Deligne's result" is already in Saavedra, and the reference given is to an article by Deligne and somebody else. |
May
27 |
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Automorphism of algebraic group preserving a hyperspecial maximal compact
added 482 characters in body |
May
26 |
answered | Automorphism of algebraic group preserving a hyperspecial maximal compact |
May
26 |
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Ext of Tate-modules of abelian varieties
Over a finite field, these Ext groups are studied in one of my 1968 Inventiones papers. The same techniques may work over local fields. |
May
20 |
revised |
Constructing coherent sheaves on Shimura varieties.
added 7 characters in body |
May
20 |
answered | Constructing coherent sheaves on Shimura varieties. |
May
15 |
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Motives versus Motifs
I've quite often seen the English word "motif" (meaning A distinctive, significant, or dominant idea or theme...) written "motive", which is one reason I chose as the title of LNM 900 "Hodge cycles, Motives,..." (at the time both "motive" and "motif" were in use in English). I would prefer to say that the French word "motif" has been borrowed three times into English, and twice anglicized. |
May
14 |
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Quotients of Abelian Varieties by Finite Groups
The group of automorphisms of a polarized abelian variety is finite, and Shimura and Taniyama (in their famous 1961 book, p35) define a Kummer variety to be the quotient of a polarized abelian variety by the full group of automorphisms. For a general polarized abelian variety, the automorphism group is Z/2Z, and so I expect that they have been most studied in that case. |
May
14 |
answered | If Erdős is published as Erdös in a paper, which do I cite? |
May
12 |
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homology of abelian variety ?
Perhaps its the paper that includes: "Recall that the category of abelian varieties up to isogeny is obtained from the category of abelian varieties by taking the same class of objects but replacing $Hom(A,B)$ with $Hom(A;B)\otimes\mathbb{Q}$. We shall always regard an abelian variety as an object in the category of abelian varieties up to isogeny: thus $Hom(A,B)$ is a vector space over $\mathbb{Q}$." If so, $A\otimes E$ means $A\otimes_{\mathbb{Z}}\mathcal{O}_{E}$, which is explained by Torsten. |
May
12 |
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What are examples of mathematical concepts named after the wrong people? (Stigler's law)
It's not so clear. In the historical notes to his book, Nagata says that in the case N=0 and M an ideal, the lemma was given by Krull, and that the general case is in a paper of Azumaya. But Nagata learned it from Nakayama and Azumaya when he was an undergraduate. Since some mathematicians were calling it Nakayama's lemma, he asked Nakayama who had this formulation first, to which Nakayama responded that he did not remember whether Nakayama or Azumaya was the first. Probably it should be considered folklore, and the name "Nakayama's lemma" seems appropriate. |
May
12 |
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Representations of reductive Lie group
There's a complete proof that representations of reductive groups in characteristic zero are semisimple in II, section 5, of the notes on my website. |