scholarly journals The effective Shafarevich conjecture for abelian varieties of -type

2021 ◽  
Vol 9 ◽  
Author(s):  
Rafael von Känel
Keyword(s):  
Abelian Varieties ◽  
Integral Points ◽  
Arakelov Theory ◽  
Shafarevich Conjecture ◽  
Higher Dimensional ◽  
Hilbert Modular ◽  
Hilbert Modular Varieties ◽  
Modular Varieties ◽  
Formula Type ◽  
The Way

Abstract In this article we establish the effective Shafarevich conjecture for abelian varieties over ${\mathbb Q}$ of ${\text {GL}_2}$ -type. The proof combines Faltings’ method with Serre’s modularity conjecture, isogeny estimates and results from Arakelov theory. Our result opens the way for the effective study of integral points on certain higher dimensional moduli schemes such as, for example, Hilbert modular varieties.

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 ALGEBRAIC CYCLES AND TATE CLASSES ON HILBERT MODULAR VARIETIES

2014 ◽  
Vol 10 (01) ◽  
pp. 161-176
Author(s):  
JAYCE R. GETZ ◽  
HEEKYOUNG HAHN
Keyword(s):  
Discrete Series ◽  
Algebraic Cycles ◽  
Automorphic Representation ◽  
Abelian Extensions ◽  
Real Place ◽  
Hilbert Modular ◽  
Rank One ◽  
Hilbert Modular Varieties ◽  
Totally Real ◽  
Modular Varieties

Let E/ℚ be a totally real number field that is Galois over ℚ, and let π be a cuspidal, nondihedral automorphic representation of GL 2(𝔸E) that is in the lowest weight discrete series at every real place of E. The representation π cuts out a "motive" M ét (π∞) from the ℓ-adic middle degree intersection cohomology of an appropriate Hilbert modular variety. If ℓ is sufficiently large in a sense that depends on π we compute the dimension of the space of Tate classes in M ét (π∞). Moreover if the space of Tate classes on this motive over all finite abelian extensions k/E is at most of rank one as a Hecke module, we prove that the space of Tate classes in M ét (π∞) is spanned by algebraic cycles.

scholarly journals Stratifying Lie Strata of Hilbert Modular Varieties

2020 ◽  
Vol 24 (6) ◽  
pp. 1307-1352
Author(s):  
Chia-Fu Yu ◽  
Ching-Li Chai ◽  
Frans Oort
Keyword(s):  
Hilbert Modular ◽  
Hilbert Modular Varieties ◽  
Modular Varieties

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.

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