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bio website cwru.edu/artsci/phil/…
location Case Western Reserve University
age 63
visits member for 2 years, 11 months
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Mar
2
accepted Authorship of Grothendieck universes
Mar
1
comment Authorship of Grothendieck universes
Yes, I have edited the question to be clear I am talking about Grothendieck's first uses of the term, not about the origin of the term in set theory.
Mar
1
revised Authorship of Grothendieck universes
Clarify that this is about Grothendieck's first uses of universes not the first occurrence of the idea in set theory.
Mar
1
awarded  Necromancer
Mar
1
awarded  Revival
Mar
1
answered Authorship of Grothendieck universes
Feb
28
awarded  Popular Question
Feb
20
awarded  Popular Question
Feb
6
revised How important is Weil's decomposition theorem today?
added 4 characters in body
Feb
6
revised How important is Weil's decomposition theorem today?
added 124 characters in body
Feb
6
accepted How important is Weil's decomposition theorem today?
Feb
6
revised How important is Weil's decomposition theorem today?
Distinguished Weil's motivating statement from the theorem in his dissertation.
Feb
6
comment How important is Weil's decomposition theorem today?
I will get Serre's book. For now, I think Kato, Kurakawa, and Saito Number Theory I do the same thing. They prove many special cases of Weil's theorem using specific inequalities on the simple notion of height for rational numbers. These inequalities measure aspects of what Weil in his autobiography called "almost prime to each other," I believe the version in Lang, or Bombieri and Gubler, is both a sharper estimate and uses a finer version of height, but am I right about that?
Feb
5
comment How important is Weil's decomposition theorem today?
Yes, thanks for cites to uses of the theorem. But many graduate textbooks rely on heights and never mention the decomposition theorem. Even Serre in his Lectures on the Mordell-Weil Theorem never uses "decomposition" in this sense and I really do not think the lectures use any form of the theorem -- they use simpler arithmetic aspects of heights. It seems, so far, heights play a much larger unifying role than the decomposition theorem. I'd like to be wrong! I'd love to find some work stating and using the theorem as an explicit unifying tool for descent arguments.
Feb
5
revised How important is Weil's decomposition theorem today?
Found a further discussion of it by Weil.
Feb
5
comment How important is Weil's decomposition theorem today?
Would it be fair to say that the most broadly used insight from Weil's discovery has been arithmetic uses of the idea of height simple enough that they do not actually require stating (or proving) the decomposition theorem?
Feb
5
awarded  Nice Question
Feb
5
comment How important is Weil's decomposition theorem today?
Extremely helpful. I am much happier to read up to page 63 of Bombieri and Gubler than I was to face page 263 of Lang -- both because it seems more feasible and it seems more up to date in technique. But Weil and Serre write as if this theorem is the right explanation of Fermat's arguments by descent and I am still puzzled that the applications seem to be so few, specific and advanced.
Feb
5
asked How important is Weil's decomposition theorem today?
Feb
3
revised What is known about the reverse mathematics of algebraic number fields?
Added cite to related question from 2013.