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.

scholarly journals RELATIVE TORSION

2001 ◽  
Vol 03 (01) ◽  
pp. 15-85 ◽  
Author(s):  
DAN BURGHELEA ◽  
LEONID FRIEDLANDER ◽  
THOMAS KAPPELER
Keyword(s):  
Fundamental Group ◽  
Finite Type ◽  
Von Neumann Algebra ◽  
Hilbert Module ◽  
Von Neumann ◽  
Geometric Invariants ◽  
Finite Dimensional ◽  
Finite Von Neumann Algebra ◽  
Special Cases ◽  
Relative Torsion

This paper achieves, among other things, the following: • It frees the main result of [9] from the hypothesis of determinant class and extends this result from unitary to arbitrary representations. • It extends (and at the same times provides a new proof of) the main result of Bismut and Zhang [3] from finite dimensional representations of Γ to representations on an [Formula: see text]-Hilbert module of finite type ([Formula: see text] a finite von Neumann algebra). The result of [3] corresponds to [Formula: see text]. • It provides interesting real valued functions on the space of representations of the fundamental group Γ of a closed manifold M. These functions might be a useful source of topological and geometric invariants of M. These objectives are achieved with the help of the relative torsion ℛ, first introduced by Carey, Mathai and Mishchenko [12] in special cases. The main result of this paper calculates explicitly this relative torsion (cf. Theorem 1.1).

scholarly journals ON SELMER RANK PARITY OF TWISTS

2016 ◽  
Vol 102 (3) ◽  
pp. 316-330 ◽  
Author(s):  
MAJID HADIAN ◽  
MATTHEW WEIDNER
Keyword(s):  
Elliptic Curves ◽  
Abelian Variety ◽  
Number Field ◽  
Arbitrary Number ◽  
Hyperelliptic Curve ◽  
Abelian Varieties ◽  
Hyperelliptic Curves ◽  
Torsion Subgroup ◽  
Tate Conjecture ◽  
Quadratic Twists

In this paper we study the variation of the $p$-Selmer rank parities of $p$-twists of a principally polarized Abelian variety over an arbitrary number field $K$ and show, under certain assumptions, that this parity is periodic with an explicit period. Our result applies in particular to principally polarized Abelian varieties with full $K$-rational $p$-torsion subgroup, arbitrary elliptic curves, and Jacobians of hyperelliptic curves. Assuming the Shafarevich–Tate conjecture, our result allows one to classify the rank parities of all quadratic twists of an elliptic or hyperelliptic curve after a finite calculation.

scholarly journals Frobenius Distribution for Quotients of Fermat Curves of Prime Exponent

2016 ◽  
Vol 68 (2) ◽  
pp. 361-394
Author(s):  
Francesc Fité ◽  
Josep González ◽  
Joan-Carles Lario
Keyword(s):  
Abelian Variety ◽  
Complex Multiplication ◽  
Cyclotomic Field ◽  
Explicit Method ◽  
Fermat Curve ◽  
Tate Conjecture ◽  
Tate Group ◽  
Fermat Curves ◽  
Prime Exponent ◽  
Simple Abelian Variety

AbstractLet denote the Fermat curve over ℚ of prime exponent ℓ. The Jacobian Jac() of splits over ℚ as the product of Jacobians Jac(k), 1 ≤ k ≤ ℓ −2, where k are curves obtained as quotients of by certain subgroups of automorphisms of . It is well known that Jac(k) is the power of an absolutely simple abelian variety Bk with complex multiplication. We call degenerate those pairs (ℓ, k) for which Bk has degenerate CM type. For a non-degenerate pair (ℓ, k), we compute the Sato–Tate group of Jac(Ck), prove the generalized Sato–Tate Conjecture for it, and give an explicit method to compute the moments and measures of the involved distributions. Regardless of whether (ℓ, k) is degenerate, we also obtain Frobenius equidistribution results for primes of certain residue degrees in the ℓ-th cyclotomic field. Key to our results is a detailed study of the rank of certain generalized Demjanenko matrices.

scholarly journals DISTINGUISHED MODELS OF INTERMEDIATE JACOBIANS

2018 ◽  
Vol 19 (3) ◽  
pp. 891-918 ◽  
Author(s):  
Jeffrey D. Achter ◽  
Sebastian Casalaina-Martin ◽  
Charles Vial
Keyword(s):  
Abelian Variety ◽  
Hilbert Scheme ◽  
Projective Variety ◽  
Galois Representation ◽  
Complete Intersections ◽  
Base Field ◽  
Field Of Definition ◽  
Level 1 ◽  
Albanese Variety ◽  
Coniveau Filtration

We show that the image of the Abel–Jacobi map admits functorially a model over the field of definition, with the property that the Abel–Jacobi map is equivariant with respect to this model. The cohomology of this abelian variety over the base field is isomorphic as a Galois representation to the deepest part of the coniveau filtration of the cohomology of the projective variety. Moreover, we show that this model over the base field is dominated by the Albanese variety of a product of components of the Hilbert scheme of the projective variety, and thus we answer a question of Mazur. We also recover a result of Deligne on complete intersections of Hodge level 1.

The fundamental group-scheme of an abelian variety

1983 ◽  
Vol 263 (3) ◽  
pp. 263-266 ◽  
Author(s):  
Madhav V. Nori
Keyword(s):  
Fundamental Group ◽  
Abelian Variety ◽  
Group Scheme

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 Abstract commensurators of surface groups

2019 ◽  
pp. 1-16
Author(s):  
Khalid Bou-Rabee ◽  
Daniel Studenmund
Keyword(s):  
Fundamental Group ◽  
Finite Type ◽  
Computational Methods ◽  
Normal Forms ◽  
Computer Assisted ◽  
Surface Groups ◽  
Finitely Generated ◽  
Residually Finite ◽  
Hnn Extensions ◽  
Concrete Objects

Let [Formula: see text] be the fundamental group of a surface of finite type and [Formula: see text] be its abstract commensurator. Then [Formula: see text] contains the solvable Baumslag–Solitar groups [Formula: see text] for any [Formula: see text]. Moreover, the Baumslag–Solitar group [Formula: see text] has an image in [Formula: see text] that is not residually finite. Our proofs are computer-assisted. Our results also illustrate that finitely-generated subgroups of [Formula: see text] are concrete objects amenable to computational methods. For example, we give a proof that [Formula: see text] is not residually finite without the use of normal forms of HNN extensions.

scholarly journals INVARIANT HYPERSURFACES

2020 ◽  
pp. 1-27
Author(s):  
Jason Bell ◽  
Rahim Moosa ◽  
Adam Topaz
Keyword(s):  
Rational Function ◽  
Finite Type ◽  
Characteristic Zero ◽  
Positive Characteristic ◽  
Inverse Image ◽  
The Other ◽  
Partial Result ◽  
Rational Maps ◽  
Special Cases ◽  
The One

The following theorem, which includes as very special cases results of Jouanolou and Hrushovski on algebraic $D$ -varieties on the one hand, and of Cantat on rational dynamics on the other, is established: Working over a field of characteristic zero, suppose $\unicode[STIX]{x1D719}_{1},\unicode[STIX]{x1D719}_{2}:Z\rightarrow X$ are dominant rational maps from an (possibly nonreduced) irreducible scheme $Z$ of finite type to an algebraic variety $X$ , with the property that there are infinitely many hypersurfaces on  $X$ whose scheme-theoretic inverse images under $\unicode[STIX]{x1D719}_{1}$ and $\unicode[STIX]{x1D719}_{2}$ agree. Then there is a nonconstant rational function $g$ on $X$ such that $g\unicode[STIX]{x1D719}_{1}=g\unicode[STIX]{x1D719}_{2}$ . In the case where $Z$ is also reduced, the scheme-theoretic inverse image can be replaced by the proper transform. A partial result is obtained in positive characteristic. Applications include an extension of the Jouanolou–Hrushovski theorem to generalised algebraic ${\mathcal{D}}$ -varieties and of Cantat’s theorem to self-correspondences.

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