2,763 reputation
1123
bio website tau.ac.il/~borovoi
location Tel Aviv, Israel
age
visits member for 5 years, 1 month
seen 8 hours ago

I am a professor at Tel Aviv University, interested in linear algebraic groups, homogeneous spaces, number theory and nonabelian cohomology.


Apr
7
comment Algebraic integer with conjugates on the unit circle
@AndreasThom Could you please give a reference to a book or paper where it is called Kronecker's theorem?
Apr
7
comment Algebraic integer with conjugates on the unit circle
@René Many thanks!
Apr
7
asked Algebraic integer with conjugates on the unit circle
Apr
5
comment Nonabelian topological fundamental group of a conjugate variety
@FrancescoPolizzi: Bauer, Catanese and Grunewald prove that for any $\sigma$ in the absolute Galois group of $\mathbb{Q}$, not conjugate to the complex conjugation, there exists a counterexample (a projective surface).
Apr
5
comment Nonabelian topological fundamental group of a conjugate variety
@AndreasThom Other examples: J. S. Milne and J. Suh, Nonhomeomorphic conjugates of connected Shimura varieties, Amer. J. Math. {\bf 132}(3) (2010), 731--750. C. S. Rajan, An example of non-homeomorphic conjugate varieties,} Math. Res. Lett. {\bf 18} (2011), 937--943. I. Bauer, F. Catanese, and F. Grunewald, Faithful actions of the absolute Galois group on connected components of moduli spaces, Invent. Math. {\bf 199} (2015), no. 3, 859--888.
Apr
5
comment Nonabelian topological fundamental group of a conjugate variety
@FrancescoPolizzi: Right! Try this link: books.google.co.il/…
Apr
5
comment Nonabelian topological fundamental group of a conjugate variety
@FrancescoPolizzi: If $\pi_1^{top}(X)$ is abelian, then it is isomorphic to $F\times \mathbb{Z}^n$ for some finite abelian group $F$ and some $n$. In this case its profinite completion $\pi_1^{et}(X)$ is isomorphic to $F\times\widehat{\mathbb{Z}}^n$. If also $\pi_1^{top}(\sigma X)$ is abelian, then it is isomorphic to $F'\times \mathbb{Z}^{n'}$. Comparing the etale fundamental groups (which are always isomorphic) we see that the topological fundamental groups are isomorphic. In Serre's example both topological fundamental groups are nonabelian. My question is whether one of them can be abelian
Apr
5
comment Nonabelian topological fundamental group of a conjugate variety
@FrancescoPolizzi: If $\sigma$ is the complex conjugation, then the map on $\mathbb{C}$-points $X\to\sigma X$ is a homeomorphism, hence in this case $X$ and $\sigma X$ have isomorphic topological fundamental groups. In Serre's example $\sigma$ is different from complex conjugation.
Apr
3
awarded  Nice Question
Apr
3
asked Nonabelian topological fundamental group of a conjugate variety
Apr
3
comment Subgroups of $Sp_{2g}$ giving rise to Shimura data
Exactly. Noncompact $G_2$ can be embedded into $Sp_{2n}$, but its symmetric space does not admit an invariant complex strusture. Furthermore, the symmetric space of $Spin(11,2)$ admits an invariant complex structure, but if you consider any homomorphism into $Sp_{2n}$ other than the spinor representation, then the corresponding embedding of symmetric spaces will not be holomorphic.
Apr
2
comment Subgroups of $Sp_{2g}$ giving rise to Shimura data
E.g. $G$ cannot be of type $G_2$, or of type $B_n$ $(n\ge 3)$ with any representation different from the spinor representation, etc..
Apr
2
answered Subgroups of $Sp_{2g}$ giving rise to Shimura data
Mar
26
comment Adeles and twisted adeles
Thank you! Unfortunately, I cannot accept two answers...
Mar
24
accepted Adeles and twisted adeles
Mar
24
comment Adeles and twisted adeles
Dear jmc, many thanks! However, could you please add detais? To what is the tensoring functor left adjoint, and why it follows that this functor commutes with colimits? Please give references and, if possible, detailed explanations!
Mar
24
asked Adeles and twisted adeles
Mar
11
comment The action of the center on the extended Dynkin diagram
Thank you, Jim! This is exactly what I needed!
Mar
11
accepted The action of the center on the extended Dynkin diagram
Mar
10
asked The action of the center on the extended Dynkin diagram