42 research outputs found
3-nilpotent obstructions to pi_1 sections for P^1_Q - {0,1,infty}
We study which rational points of the Jacobian of P^1_K -{0,1,infty} can be
lifted to sections of geometrically 3 nilpotent quotients of etale pi_1 over
the absolute Galois group. This is equivalent to evaluating certain triple
Massey products of elements of H^1(G_K). For K=Q_p or R, we give a complete mod
2 calculation. This permits some mod 2 calculations for K = Q. These are
computations of obstructions of Jordan Ellenberg.Comment: 49 page
2-Nilpotent Real Section Conjecture
We show a 2-nilpotent section conjecture over R: for a geometrically
connected curve X over R such that each irreducible component of its
normalization has R-points, pi_0(X(R)) is determined by the maximal 2-nilpotent
quotient of the fundamental group with its Galois action, as the kernel of an
obstruction of Jordan Ellenberg. This implies that for X smooth and proper,
X(R)^{+/-} is determined by the maximal 2-nilpotent quotient of Gal(C(X)) with
its Gal(R)-action, where X(R)^{+/-} denotes the set of real points equipped
with a real tangent direction, showing a 2-nilpotent birational real section
conjecture.Comment: 24 pages, 1 figure. To appear Mathematiche Annale
Desuspensions of S^1 /\ P^1_Q-{0,1,infty}
We use the Galois action on
to show that the homotopy equivalence coming from purity does not desuspend to a map
Massey products <y,x,x,...,x,x,y> in Galois cohomology via rational points
For an element of a field other than or , we compute the order
Massey products of factors of and two factors of by
embedding into its Picard variety and
constructing equivariant maps from applied
to this embedding to unipotent matrix groups. This method produces obstructions
to -sections of , partial computations of
obstructions of Jordan Ellenberg, and also computes the Massey products
Comment: To appear in the Journal of Pure and Applied Algebra's special issue
in honor of Chuck Weibel, edited by Guillermo Corti\~nas, Eric Friedlander,
and Christian Haesemeyer. 26 pages. This paper continues arXiv:1011.0655 and
arXiv:1107.179
Cartier's first theorem for Witt vectors on Z_{>= 0}^n - 0
We show that the dual of the Witt vectors on Z_{\geq 0}^n - 0 as defined by
Angeltveit, Gerhardt, Hill, and Lindenstrauss represent the functor taking a
commutative formal group G to the maps of formal schemes Ahat^n -> G, and that
the Witt vectors are self-dual for Q-algebras or when n=1.Comment: Corrected an error in the previous version of the main theorem, which
is now stated in terms of the dual of the Witt vectors. The original
statement holds under the additional hypothesis that we are working over a
Q-algebra. To appear in Algebraic Topology: Applications and New Directions,
Proceedings of the Stanford Symposium 2012. 7 pages, AIM preprint number AIM
2012-8
Unstable Motivic Homotopy Theory
We give an introduction to unstable motivic homotopy theory of Morel and
Voevodsky, and survey some results.Comment: Prepared for the Handbook of Homotopy Theory, edited by Haynes Mille
The Simplicial EHP Sequence in A1-Algebraic Topology
We give a tool for understanding simplicial desuspension in A1-algebraic
topology: we show that is a fiber sequence up to homotopy in 2-localized A1-algebraic
topology for with . It follows that
there is an EHP sequence spectral sequence Comment: Revisions made at the suggestion of an anonymous refere
Splitting Varieties for Triple Massey Products
We construct splitting varieties for triple Massey products. For a,b,c in F^*
the triple Massey product of the corresponding elements of H^1(F,
mu_2) contains 0 if and only if there is x in F^* and y in F[\sqrt{a},
\sqrt{c}]^* such that b x^2 = N_{F[\sqrt{a}, \sqrt{c}]/F}(y), where
N_{F[\sqrt{a}, \sqrt{c}]/F} denotes the norm, and F is a field of
characteristic different from 2. These varieties satisfy the Hasse principle by
a result of D.B. Lee and A.R. Wadsworth. This shows that triple Massey products
for global fields of characteristic different from 2 always contain 0.Comment: 18 pages. To appear in Journal of Pure and Applied Algebr
An \'etale realization which does not exist
For a global field, local field, or finite field with infinite Galois
group, we show that there can not exist a functor from the Morel--Voevodsky
-homotopy category of schemes over to a genuine Galois
equivariant homotopy category satisfying a list of hypotheses one might expect
from a genuine equivariant category and an \'etale realization functor. For
example, these hypotheses are satisfied by genuine -spaces and
the -realization functor constructed by Morel--Voevodsky. This
result does not contradict the existence of \'etale realization functors to
(pro-)spaces, (pro-)spectra or complexes of modules with actions of the
absolute Galois group when the endomorphisms of the unit is not enriched in a
certain sense. It does restrict enrichments to representation rings of Galois
groups.Comment: 18 pages. To appear in the New Directions in Homotopy Theory volume
of Contemporary Mathematics. V2 has a slightly generalized main theorem, and
stable realization functors are explicitly assumed to be additive, which is
necessary. V3 minor modification
An Arithmetic Count of the Lines on a Smooth Cubic Surface
We give an arithmetic count of the lines on a smooth cubic surface over an
arbitrary field , generalizing the counts that over there are
lines, and over the number of hyperbolic lines minus the
number of elliptic lines is . In general, the lines are defined over a field
extension and have an associated arithmetic type in .
There is an equality in the Grothendieck-Witt group of
where
denotes the trace . Taking the rank and signature recovers the results over
and . To do this, we develop an elementary theory of
the Euler number in -homotopy theory for algebraic vector
bundles. We expect that further arithmetic counts generalizing enumerative
results in complex and real algebraic geometry can be obtained with similar
methods.Comment: 34 pages. Accepted for publication in Compositio Mathematic