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bio website jmilne.org/math
location Ann Arbor, MI, USA, and New Zealand.
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Arithmetic geometry (especially Shimura varieties and abelian varieties).

May
5
answered Lifting Etale Morphisms
Apr
18
comment Is there a schemetical construction for modular curves over the rationals?
"I agree PVHS is the way to prove analyticity (but not evident to beginners" well, they could try reading [the next version] of my Introduction to Shimura Varieties. More seriously, this argument in the one-dimensional case is pretty trivial, and Katz and Mazur were surely aware of it, so I find the statement in your first comment a bit strong. Perhaps they considered it too obvious to write out (or just forgot).
Apr
18
comment Is there a schemetical construction for modular curves over the rationals?
I'm not sure I understand Brian's objections --- Shimura worked a lot with function fields and really did prove things. However, I would agree that this is completely the wrong way to do things.
Apr
18
comment Is there a schemetical construction for modular curves over the rationals?
From Deligne's point of view (e.g., his Corvallis article), hermitian symmetric domains are exactly the moduli spaces (in the analytic category) of certain rigidified Hodge structures, hence their quotients are also. The algebraic moduli variety carries a variation of Hodge structures of the correct type, so you get an analytic map to the quotient of the HSD (which is algebraic by Borel's theorem). Isn't that all that's going on?
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