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bio website jmilne.org/math
location Ann Arbor, MI, USA, and New Zealand.
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seen Oct 18 '10 at 13:35
Arithmetic geometry (especially Shimura varieties and abelian varieties).

Apr
17
comment Why are modular forms (usually) defined only for congruence subgroups?
Certainly not canonical. For example, the curve for Gamma_N classifies elliptic curves with a level N structure. Such objects have a discrete invariant, namely, an Nth root of 1 given by applying the Weil pairing to the basis. To get a connected family, you need to specify the Nth root, so this only makes sense over a field where you have one. Of course, for nonconnected curves, everything is defined over Q.
Apr
17
revised Why are modular forms (usually) defined only for congruence subgroups?
Minor fixes.
Apr
17
revised What algebraic group does Tannaka-Krein reconstruct when fed the category of modules of a non-algebraic Lie algebra?
Minor fixes.
Apr
16
answered Why are modular forms (usually) defined only for congruence subgroups?
Apr
15
awarded  Nice Answer
Apr
15
comment What algebraic group does Tannaka-Krein reconstruct when fed the category of modules of a non-algebraic Lie algebra?
Thanks --- that looks like the correct way to interpret Iwahori's statements (and, yes, I should have assumed k to be algebraically closed).
Apr
15
answered What algebraic group does Tannaka-Krein reconstruct when fed the category of modules of a non-algebraic Lie algebra?
Apr
15
comment What algebraic group does Tannaka-Krein reconstruct when fed the category of modules of a non-algebraic Lie algebra?
There seems to be an assumption implicit in the question that when you start with an algebraic Lie algebra g, the affine group scheme G you get has Lie algebra g. As @unknown noted, this is far from true (except for semisimple Lie algebras in characteristic zero). So before trying to understand the affine group schemes you get from nonalgebraic Lie algebras, perhaps you should try to understand those you get from algebraic Lie algebras (e.g., the one-dimensional Lie algebra).
Apr
14
awarded  Nice Answer
Apr
14
answered How many groups of size at most n are there? What is the asymptotic growth rate? And what of rings, fields, graphs, partial orders, etc.?
Apr
12
revised Why does the naive definition of compactly supported étale cohomology give the wrong answer?
edited body; added 251 characters in body
Apr
12
answered Why does the naive definition of compactly supported étale cohomology give the wrong answer?
Mar
16
comment The algebraicity of Hodge structure map
The maps $\mathbb{S}\to G_{\mathbb{R}}$ are defined over $\mathbb{R}$,not $\mathbb{Q}$. However, one of the axioms (2.1.1.1 in Deligne's Corvallis articles; SV1 in my articles) for Shimura varieties implies that the weight homomorphism $w$ of each $h$ factors through the centre of $G$ (hence through connected centre, which is a torus). As the $h$'s are all conjugate, this implies that the $w$'s are all equal, and that the $w$ is defined over the algebraic closure of $\mathbb{Q}$ (it is a homomorphism between tori defined over $\mathbb{Q}$).
Mar
16
revised The algebraicity of Hodge structure map
added 316 characters in body
Mar
16
answered The algebraicity of Hodge structure map
Mar
11
awarded  Nice Answer
Mar
11
comment Books on reductive groups using scheme theory
@Brian: I agree that if you try to directly transfer the proof in the smooth case to the nonsmooth case, you can sometimes run into some very heavy scheme theory, but there are also elementary proofs using Hopf algebras. I learnt this from Waterhouse's book. As Serre pointed out, Hopf algebra proofs don't illuminate, but my strategy is to sketch the geometric argument and write out the Hopf algebra argument (when necessary). I'm only doing things over fields (or rings, when it's just as easy).
Mar
11
answered Books on reductive groups using scheme theory
Mar
4
awarded  Enlightened
Mar
4
revised Why do people think that abelian varieties are the hardest case for the Hodge conjecture?
added 377 characters in body