scholarly journals On the Hodge, Tate and Mumford-Tate Conjectures for Fibre Products of Families of Regular Surfaces with Geometric Genus 1

2018 ◽  
Vol 25 (3) ◽  
pp. 312-322
Author(s):  
Olga V. Oreshkina (Nikol’skaya)
Keyword(s):  
Real Field ◽  
Hodge Conjecture ◽  
Geometric Genus ◽  
Smooth Projective Curve ◽  
Tate Conjecture ◽  
Smooth Part ◽  
Fibre Products ◽  
Fibre Product ◽  
Totally Real ◽  
Hodge Groups

The Hodge, Tate and Mumford-Tate conjectures are proved for the fibre product of two non-isotrivial 1-parameter families of regular surfaces with geometric genus 1 under some conditions on degenerated fibres, the ranks of the N\'eron - Severi groups of generic geometric fibres and representations of Hodge groups in transcendental parts of rational cohomology.Let \(\pi_i:X_i\to C\quad (i = 1, 2)\) be a projective non-isotrivial family (possibly with degeneracies) over a smooth projective curve \(C\). Assume that the discriminant loci \(\Delta_i=\{\delta\in C\,\,\vert\,\, Sing(X_{i\delta})\neq\varnothing\} \quad (i = 1, 2)\) are disjoint, \(h^{2,0}(X_{ks})=1,\quad h^{1,0}(X_{ks}) = 0\) for any smooth fibre \(X_{ks}\), and the following conditions hold:\((i)\) for any point \(\delta \in \Delta_i\) and the Picard-Lefschetz transformation \( \gamma \in GL(H^2 (X_{is}, Q)) \), associated with a smooth part \(\pi'_i: X'_i\to C\setminus\Delta_i\) of the morphism \(\pi_i\) and with a loop around the point \(\delta \in C\), we have \((\log(\gamma))^2\neq0\);\((ii)\) the variety \(X_i \, (i = 1, 2)\), the curve \(C\) and the structure morphisms \(\pi_i:X_i\to C\) are defined over a finitely generated subfield \(k \hookrightarrow C\).If for generic geometric fibres \(X_{1s}\) \, and \, \(X_{2s}\) at least one of the following conditions holds: \((a)\) \(b_2(X_{1s})- rank NS(X_{1s})\) is an odd prime number, \(\quad\,\,\) \(b_2(X_{1s})- rank NS(X_{1s})\neq b_2(X_{2s})- rank NS(X_{2s})\); \((b)\) the ring \(End_{ Hg(X_{1s})} NS_ Q(X_{1s})^\perp\) is an imaginary quadratic field, \(\quad\,\, b_2(X_{1s})- rank NS(X_{1s})\neq 4,\) \(\quad\,\, End_{ Hg(X_{2s})} NS_ Q(X_{2s})^\perp\) is a totally real field or \(\,\, b_2(X_{1s})- rank NS(X_{1s})\,>\, b_2(X_{2s})- rank NS(X_{2s})\) ; \((c)\) \([b_2(X_{1s})- rank NS(X_{1s})\neq 4, \, End_{ Hg(X_{1s})} NS_ Q(X_{1s})^\perp= Q\); \(\quad\,\,\) \(b_2(X_{1s})- rank NS(X_{1s})\neq b_2(X_{2s})- rank NS(X_{2s})\),then for the fibre product \(X_1 \times_C X_2\) the Hodge conjecture is true, for any smooth projective \(k\)-variety \(X_0\) with the condition \(X_1 \times_C X_2\) \(\widetilde{\rightarrow}\) \(X_0 \otimes_k C\) the Tate conjecture on algebraic cycles and the Mumford-Tate conjecture for cohomology of even degree are true.

scholarly journals Finite descent obstruction for Hilbert modular varieties

2020 ◽  
pp. 1-22
Author(s):  
Gregorio Baldi ◽  
Giada Grossi
Keyword(s):  
Real Field ◽  
Absolute Galois Group ◽  
Hodge Conjecture ◽  
Hilbert Modular ◽  
Finite Set ◽  
The Absolute ◽  
Hilbert Modular Varieties ◽  
Totally Real ◽  
Modular Varieties ◽  
Special Case

Abstract Let S be a finite set of primes. We prove that a form of finite Galois descent obstruction is the only obstruction to the existence of $\mathbb {Z}_{S}$ -points on integral models of Hilbert modular varieties, extending a result of D. Helm and F. Voloch about modular curves. Let L be a totally real field. Under (a special case of) the absolute Hodge conjecture and a weak Serre’s conjecture for mod $\ell $ representations of the absolute Galois group of L, we prove that the same holds also for the $\mathcal {O}_{L,S}$ -points.

scholarly journals The Hodge conjecture and the Tate conjecture for fermat varieties

1979 ◽  
Vol 55 (3) ◽  
pp. 111-114 ◽  
Author(s):  
Tetsuji Shioda
Keyword(s):  
Hodge Conjecture ◽  
Tate Conjecture

scholarly journals On Goren–Oort stratification for quaternionic Shimura varieties

2016 ◽  
Vol 152 (10) ◽  
pp. 2134-2220 ◽  
Author(s):  
Yichao Tian ◽  
Liang Xiao
Keyword(s):  
Line Bundle ◽  
Real Field ◽  
Necessary Condition ◽  
Shimura Varieties ◽  
Shimura Variety ◽  
Maximal Level ◽  
Totally Real ◽  
Totally Real Field

Let $F$ be a totally real field in which a prime $p$ is unramified. We define the Goren–Oort stratification of the characteristic-$p$ fiber of a quaternionic Shimura variety of maximal level at $p$. We show that each stratum is a $(\mathbb{P}^{1})^{r}$-bundle over other quaternionic Shimura varieties (for an appropriate integer $r$). As an application, we give a necessary condition for the ampleness of a modular line bundle on a quaternionic Shimura variety in characteristic $p$.

scholarly journals Asymptotic generalized Fermat’s last theorem over number fields

2019 ◽  
Vol 16 (05) ◽  
pp. 907-924
Author(s):  
Yasemin Kara ◽  
Ekin Ozman
Keyword(s):  
Number Fields ◽  
Real Field ◽  
Quadratic Number ◽  
Fermat's Last Theorem ◽  
Fermat’S Last Theorem ◽  
Imaginary Quadratic Number ◽  
Fermat Equation ◽  
Totally Real ◽  
Totally Real Fields ◽  
Generalized Fermat Equation

Recent work of Freitas and Siksek showed that an asymptotic version of Fermat’s Last Theorem (FLT) holds for many totally real fields. This result was extended by Deconinck to the generalized Fermat equation of the form [Formula: see text], where [Formula: see text] are odd integers belonging to a totally real field. Later Şengün and Siksek showed that the asymptotic FLT holds over number fields assuming two standard modularity conjectures. In this work, combining their techniques, we show that the generalized Fermat’s Last Theorem (GFLT) holds over number fields asymptotically assuming the standard conjectures. We also give three results which show the existence of families of number fields on which asymptotic versions of FLT or GFLT hold. In particular, we prove that the asymptotic GFLT holds for a set of imaginary quadratic number fields of density 5/6.

scholarly journals Minimal weights of Hilbert modular forms in characteristic

2017 ◽  
Vol 153 (9) ◽  
pp. 1769-1778 ◽  
Author(s):  
Fred Diamond ◽  
Payman L Kassaei
Keyword(s):  
Modular Forms ◽  
Real Field ◽  
Hilbert Modular Forms ◽  
Hilbert Modular ◽  
Hilbert Modular Varieties ◽  
Totally Real ◽  
Modular Varieties ◽  
Totally Real Field

We consider mod $p$ Hilbert modular forms associated to a totally real field of degree $d$ in which $p$ is unramified. We prove that every such form arises by multiplication by partial Hasse invariants from one whose weight (a $d$-tuple of integers) lies in a certain cone contained in the set of non-negative weights, answering a question of Andreatta and Goren. The proof is based on properties of the Goren–Oort stratification on mod $p$ Hilbert modular varieties established by Goren and Oort, and Tian and Xiao.

scholarly journals Stickelberger elements and Kolyvagin systems

2011 ◽  
Vol 203 ◽  
pp. 123-173
Author(s):  
Kâzim Büyükboduk
Keyword(s):  
General Framework ◽  
Euler System ◽  
Real Field ◽  
Ideal Class ◽  
The Core ◽  
Ideal Class Groups ◽  
Kolyvagin Systems ◽  
Totally Real ◽  
Totally Real Fields ◽  
The Ideal

AbstractIn this paper, we construct (many) Kolyvagin systems out of Stickelberger elements utilizing ideas borrowed from our previous work on Kolyvagin systems of Rubin-Stark elements. The applications of our approach are twofold. First, assuming Brumer’s conjecture, we prove results on the odd parts of the ideal class groups of CM fields which are abelian over a totally real field, and we deduce Iwasawa’s main conjecture for totally real fields (for totally odd characters). Although this portion of our results has already been established by Wiles unconditionally (and refined by Kurihara using an Euler system argument, when Wiles’s work is assumed), the approach here fits well in the general framework the author has developed elsewhere to understand Euler/Kolyvagin system machinery when the core Selmer rank isr >1 (in the sense of Mazur and Rubin). As our second application, we establish a rather curious link between the Stickelberger elements and Rubin-Stark elements by using the main constructions of this article hand in hand with the “rigidity” of the collection of Kolyvagin systems proved by Mazur, Rubin, and the author.

scholarly journals SERRE WEIGHTS FOR LOCALLY REDUCIBLE TWO-DIMENSIONAL GALOIS REPRESENTATIONS

2014 ◽  
Vol 14 (3) ◽  
pp. 639-672 ◽  
Author(s):  
Fred Diamond ◽  
David Savitt
Keyword(s):  
Galois Representations ◽  
Real Field ◽  
Two Dimensional ◽  
Serre's Conjecture ◽  
Weight Part ◽  
Totally Real ◽  
Serre’S Conjecture ◽  
Totally Real Field

Let $F$ be a totally real field, and $v$ a place of $F$ dividing an odd prime $p$. We study the weight part of Serre’s conjecture for continuous totally odd representations $\overline{{\it\rho}}:G_{F}\rightarrow \text{GL}_{2}(\overline{\mathbb{F}}_{p})$ that are reducible locally at $v$. Let $W$ be the set of predicted Serre weights for the semisimplification of $\overline{{\it\rho}}|_{G_{F_{v}}}$. We prove that, when $\overline{{\it\rho}}|_{G_{F_{v}}}$ is generic, the Serre weights in $W$ for which $\overline{{\it\rho}}$ is modular are exactly the ones that are predicted (assuming that $\overline{{\it\rho}}$ is modular). We also determine precisely which subsets of $W$ arise as predicted weights when $\overline{{\it\rho}}|_{G_{F_{v}}}$ varies with fixed generic semisimplification.

scholarly journals On the image of complex conjugation in certain Galois representations

2016 ◽  
Vol 152 (7) ◽  
pp. 1476-1488 ◽  
Author(s):  
Ana Caraiani ◽  
Bao V. Le Hung
Keyword(s):  
Symmetric Spaces ◽  
Galois Representations ◽  
Real Field ◽  
Complex Conjugation ◽  
Automorphic Representations ◽  
Locally Symmetric Spaces ◽  
Cuspidal Automorphic Representations ◽  
Totally Real ◽  
Totally Real Field

We compute the image of any choice of complex conjugation on the Galois representations associated to regular algebraic cuspidal automorphic representations and to torsion classes in the cohomology of locally symmetric spaces for $\text{GL}_{n}$ over a totally real field $F$.

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