scholarly journals Tate module and bad reduction

2020 ◽  
pp. 1
Author(s):  
Tim Dokchitser ◽  
Vladimir Dokchitser ◽  
Adam Morgan
Keyword(s):  
Tate Module

ABELIAN SURFACES WITH SUPERSINGULAR GOOD REDUCTION AND NON-SEMISIMPLE TATE MODULE

2010 ◽  
Vol 06 (04) ◽  
pp. 811-818
Author(s):  
MAJA VOLKOV
Keyword(s):  
Good Reduction ◽  
Galois Module ◽  
Abelian Surfaces ◽  
Tate Module

We show the existence of abelian surfaces [Formula: see text] over ℚp having good reduction with supersingular special fiber whose associated p-adic Galois module [Formula: see text] is not semisimple.

scholarly journals Galois-theoretic characterization of isomorphism classes of monodromically full hyperbolic curves of genus zero

2011 ◽  
Vol 203 ◽  
pp. 47-100 ◽  
Author(s):  
Yuichiro Hoshi
Keyword(s):  
Isomorphism Class ◽  
Galois Representations ◽  
Complex Multiplication ◽  
Galois Representation ◽  
Genus Zero ◽  
Isomorphism Classes ◽  
Tate Module ◽  
Hyperbolic Curve ◽  
Theoretic Characterization

AbstractLet l be a prime number. In this paper, we prove that the isomorphism class of an l-monodromically full hyperbolic curve of genus zero over a finitely generated extension of the field of rational numbers is completely determined by the kernel of the natural pro-l outer Galois representation associated to the hyperbolic curve. This result can be regarded as a genus zero analogue of a result due to Mochizuki which asserts that the isomorphism class of an elliptic curve which does not admit complex multiplication over a number field is completely determined by the kernels of the natural Galois representations on the various finite quotients of its Tate module.

scholarly journals Galois-theoretic characterization of isomorphism classes of monodromically full hyperbolic curves of genus zero

2011 ◽  
Vol 203 ◽  
pp. 47-100 ◽  
Author(s):  
Yuichiro Hoshi
Keyword(s):  
Isomorphism Class ◽  
Galois Representations ◽  
Complex Multiplication ◽  
Galois Representation ◽  
Genus Zero ◽  
Isomorphism Classes ◽  
Tate Module ◽  
Hyperbolic Curve ◽  
Theoretic Characterization

AbstractLetlbe a prime number. In this paper, we prove that the isomorphism class of anl-monodromically full hyperbolic curve of genus zero over a finitely generated extension of the field of rational numbers is completely determined by the kernel of the natural pro-louter Galois representation associated to the hyperbolic curve. This result can be regarded as a genus zero analogue of a result due to Mochizuki which asserts that the isomorphism class of an elliptic curve which does not admit complex multiplication over a number field is completely determined by the kernels of the natural Galois representations on the various finite quotients of its Tate module.

scholarly journals A Deformation of the Tate Module

2000 ◽  
Vol 229 (1) ◽  
pp. 280-313 ◽  
Author(s):  
David E. Rohrlich
Keyword(s):  
Tate Module

scholarly journals Note on linear relations in Galois cohomology and étale K-theory of curves

2021 ◽  
pp. 2150010
Author(s):  
Piotr Krasoń
Keyword(s):  
Abelian Variety ◽  
Galois Cohomology ◽  
Number Fields ◽  
Sufficient Condition ◽  
Linear Relations ◽  
Tate Module ◽  
K Theory ◽  
Weil Type ◽  
Global Principle

In this paper, we investigate a local to global principle for Galois cohomology of number fields with coefficients in the Tate module of an abelian variety. In [G. Banaszak and P. Krasoń, On a local to global principle in étale K-groups of curves, J. K-Theory Appl. Algebra Geom. Topol. 12 (2013) 183–201], G. Banaszak and the author obtained the sufficient condition for the validity of the local to global principle for étale [Formula: see text]-theory of a curve. This condition in fact has been established by means of an analysis of the corresponding problem in the Galois cohomology. We show that in some cases, this result is the best possible i.e. if this condition does not hold we obtain counterexamples. We also give some examples of curves and their Jacobians. Finally, we prove the dynamical version of the local to global principle for étale [Formula: see text]-theory of a curve. The dynamical local to global principle for the groups of Mordell–Weil type has recently been considered by S. Barańczuk in [S. Barańczuk, On a dynamical local-global principle in Mordell-Weil type groups, Expo. Math. 35(2) (2017) 206–211]. We show that all our results remain valid for Quillen [Formula: see text]-theory of [Formula: see text] if the Bass and Quillen–Lichtenbaum conjectures hold true for [Formula: see text]

scholarly journals Lattices in Tate modules

2021 ◽  
Vol 118 (49) ◽  
pp. e2113201118
Author(s):  
Bjorn Poonen ◽  
Sergey Rybakov
Keyword(s):  
Abelian Variety ◽  
Perfect Field ◽  
Tate Module ◽  
Dieudonné Module

Refining a theorem of Zarhin, we prove that, given a g-dimensional abelian variety X and an endomorphism u of X, there exists a matrix A∈M2g(ℤ) such that each Tate module TℓX has a ℤℓ-basis on which the action of u is given by A, and similarly for the covariant Dieudonné module if over a perfect field of characteristic p.

scholarly journals A support theorem for the Hitchin fibration: the case of

2017 ◽  
Vol 153 (6) ◽  
pp. 1316-1347
Author(s):  
Mark Andrea de Cataldo
Keyword(s):  
Analogous Result ◽  
Group Scheme ◽  
Direct Image ◽  
Support Theorem ◽  
Spectral Curves ◽  
Tate Module ◽  
Hitchin Fibration ◽  
Regularity Results ◽  
The Way

We prove that the direct image complex for the $D$-twisted $\text{SL}_{n}$ Hitchin fibration is determined by its restriction to the elliptic locus, where the spectral curves are integral. The analogous result for $\text{GL}_{n}$ is due to Chaudouard and Laumon. Along the way, we prove that the Tate module of the relative Prym group scheme is polarizable, and we also prove $\unicode[STIX]{x1D6FF}$-regularity results for some auxiliary weak abelian fibrations.

scholarly journals INTEGRAL AND ADELIC ASPECTS OF THE MUMFORD–TATE CONJECTURE

2018 ◽  
Vol 19 (3) ◽  
pp. 869-890 ◽  
Author(s):  
Anna Cadoret ◽  
Ben Moonen
Keyword(s):  
Fundamental Group ◽  
Abelian Variety ◽  
Finite Type ◽  
Galois Representation ◽  
Shimura Variety ◽  
General Statement ◽  
Tate Conjecture ◽  
Tate Module ◽  
Special Cases ◽  
Image Theorem

Let $Y$ be an abelian variety over a subfield $k\subset \mathbb{C}$ that is of finite type over $\mathbb{Q}$. We prove that if the Mumford–Tate conjecture for $Y$ is true, then also some refined integral and adelic conjectures due to Serre are true for $Y$. In particular, if a certain Hodge-maximality condition is satisfied, we obtain an adelic open image theorem for the Galois representation on the (full) Tate module of $Y$. We also obtain an (unconditional) adelic open image theorem for K3 surfaces. These results are special cases of a more general statement for the image of a natural adelic representation of the fundamental group of a Shimura variety.

GAUSSIAN LAWS ON DRINFELD MODULES

2009 ◽  
Vol 05 (07) ◽  
pp. 1179-1203 ◽  
Author(s):  
WENTANG KUO ◽  
YU-RU LIU
Keyword(s):  
Finite Field ◽  
Polynomial Ring ◽  
Valuation Ring ◽  
Finite Extension ◽  
Drinfeld Modules ◽  
Good Reduction ◽  
Prime Divisors ◽  
Frobenius Automorphism ◽  
Tate Module ◽  
Rank 2

Let A = 𝔽q[T] be the polynomial ring over the finite field 𝔽q, k = 𝔽q(T) the rational function field, and K a finite extension of k. Let ϕ be a Drinfeld A-module over K of rank r. For a place 𝔓 of K of good reduction, write [Formula: see text], where [Formula: see text] is the valuation ring of 𝔓 and [Formula: see text] its maximal ideal. Let P𝔓, ϕ(X) be the characteristic polynomial of the Frobenius automorphism of 𝔽𝔓acting on a Tate module of ϕ. Let χϕ(𝔓) = P𝔓, ϕ(1), and let ν(χϕ(𝔓)) be the number of distinct primes dividing χϕ(𝔓). If ϕ is of rank 2 with [Formula: see text], we prove that there exists a normal distribution for the quantity [Formula: see text] For r ≥ 3, we show that the same result holds under the open image conjecture for Drinfeld modules. We also study the number of distinct prime divisors of the trace of the Frobenius automorphism of 𝔽𝔓acting on a Tate module of ϕ and obtain similar results.

Sign in / Sign up

Export Citation Format

Share Document