scholarly journals EULER SYSTEMS FOR HILBERT MODULAR SURFACES

2018 ◽  
Vol 6 ◽  
Author(s):  
ANTONIO LEI ◽  
DAVID LOEFFLER ◽  
SARAH LIVIA ZERBES
Keyword(s):  
Galois Representations ◽  
Euler System ◽  
Euler Systems ◽  
System A ◽  
Main Conjecture ◽  
Hilbert Modular Surfaces ◽  
Hilbert Modular

We construct an Euler system—a compatible family of global cohomology classes—for the Galois representations appearing in the geometry of Hilbert modular surfaces. If a conjecture of Bloch and Kato on injectivity of regulator maps holds, this Euler system is nontrivial, and we deduce bounds towards the Iwasawa main conjecture for these Galois representations.

scholarly journals UNRAMIFIEDNESS OF GALOIS REPRESENTATIONS ARISING FROM HILBERT MODULAR SURFACES

2017 ◽  
Vol 5 ◽  
Author(s):  
MATTHEW EMERTON ◽  
DAVIDE REDUZZI ◽  
LIANG XIAO
Keyword(s):  
Real Number ◽  
Prime Number ◽  
Number Field ◽  
Characteristic Zero ◽  
Galois Representations ◽  
Hecke Operator ◽  
Hilbert Modular Surfaces ◽  
Hilbert Modular ◽  
Totally Real ◽  
Real Number Field

Let $p$ be a prime number and $F$ a totally real number field. For each prime $\mathfrak{p}$ of $F$ above $p$ we construct a Hecke operator $T_{\mathfrak{p}}$ acting on $(\text{mod}\,p^{m})$ Katz Hilbert modular classes which agrees with the classical Hecke operator at $\mathfrak{p}$ for global sections that lift to characteristic zero. Using these operators and the techniques of patching complexes of Calegari and Geraghty we prove that the Galois representations arising from torsion Hilbert modular classes of parallel weight $\mathbf{1}$ are unramified at $p$ when $[F:\mathbb{Q}]=2$. Some partial and some conjectural results are obtained when $[F:\mathbb{Q}]>2$.

scholarly journals Families of abelian surfaces with real multiplication over Hilbert modular surfaces

1982 ◽  
Vol 88 ◽  
pp. 17-53 ◽  
Author(s):  
G. van der Geer ◽  
K. Ueno
Keyword(s):  
Endomorphism Ring ◽  
Symplectic Group ◽  
Abelian Surface ◽  
Quadratic Relation ◽  
Holomorphic Map ◽  
Real Multiplication ◽  
Compact Complex Surfaces ◽  
Hilbert Modular Surfaces ◽  
Hilbert Modular ◽  
Abelian Functions

Around the beginning of this century G. Humbert ([9]) made a detailed study of the properties of compact complex surfaces which can be parametrized by singular abelian functions. A surface parametrized by singular abelian functions is the image under a holomorphic map of a singular abelian surface (i.e. an abelian surface whose endomorphism ring is larger than the ring of rational integers). Humbert showed that the periods of a singular abelian surface satisfy a quadratic relation with integral coefficients and he constructed an invariant D of such a relation with respect to the action of the integral symplectic group on the periods.

scholarly journals On the Bloch–Kato conjecture for Hilbert modular forms

2021 ◽  
Author(s):  
Matteo Tamiozzo
Keyword(s):  
Modular Forms ◽  
Automorphic Forms ◽  
Euler System ◽  
Central Point ◽  
Shimura Curves ◽  
Hilbert Modular Forms ◽  
Reciprocity Laws ◽  
Hilbert Modular ◽  
Special Points

AbstractThe aim of this paper is to prove inequalities towards instances of the Bloch–Kato conjecture for Hilbert modular forms of parallel weight two, when the order of vanishing of the L-function at the central point is zero or one. We achieve this implementing an inductive Euler system argument which relies on explicit reciprocity laws for cohomology classes constructed using congruences of automorphic forms and special points on several Shimura curves.

scholarly journals THE IWASAWA MAIN CONJECTURE FOR HILBERT MODULAR FORMS

2015 ◽  
Vol 3 ◽  
Author(s):  
XIN WAN
Keyword(s):  
Modular Forms ◽  
Base Change ◽  
Hilbert Modular Forms ◽  
Selmer Group ◽  
Hilbert Modular Form ◽  
Main Conjecture ◽  
Hilbert Modular ◽  
Trivial Character ◽  
The One ◽  
Ordinary Reduction

Following the ideas and methods of a recent work of Skinner and Urban, we prove the one divisibility of the Iwasawa main conjecture for nearly ordinary Hilbert modular forms under certain local hypotheses. As a consequence, we prove that for a Hilbert modular form of parallel weight, trivial character, and good ordinary reduction at all primes dividing$p$, if the central critical$L$-value is zero then the$p$-adic Selmer group of it has rank at least one. We also prove that one of the local assumptions in the main result of Skinner and Urban can be removed by a base-change trick.

scholarly journals Real multiplication through explicit correspondences

2016 ◽  
Vol 19 (A) ◽  
pp. 29-42 ◽  
Author(s):  
Abhinav Kumar ◽  
Ronen E. Mukamel
Keyword(s):  
Riemann Surfaces ◽  
Algebraic Models ◽  
Modular Surface ◽  
Real Multiplication ◽  
Universal Family ◽  
Hilbert Modular Surfaces ◽  
Hilbert Modular ◽  
Genus 2 ◽  
Genus 2 Curves ◽  
Do So

We compute equations for real multiplication on the divisor classes of genus-2 curves via algebraic correspondences. We do so by implementing van Wamelen’s method for computing equations for endomorphisms of Jacobians on examples drawn from the algebraic models for Hilbert modular surfaces computed by Elkies and Kumar. We also compute a correspondence over the universal family for the Hilbert modular surface of discriminant $5$ and use our equations to prove a conjecture of A. Wright on dynamics over the moduli space of Riemann surfaces.

The Canonical Map for Certain Hilbert Modular Surfaces

1980 ◽  
pp. 75-95
Author(s):  
F. Hirzebruch
Keyword(s):  
Canonical Map ◽  
Hilbert Modular Surfaces ◽  
Hilbert Modular

Period relations and the Tate conjecture for Hilbert modular surfaces

1987 ◽  
Vol 89 (2) ◽  
pp. 319-345 ◽  
Author(s):  
V. Kumar Murty ◽  
Dinakar Ramakrishnan
Keyword(s):  
Tate Conjecture ◽  
Hilbert Modular Surfaces ◽  
Hilbert Modular

Classification of Hilbert modular surfaces

1987 ◽  
pp. 466-500
Author(s):  
D. Zagier
Keyword(s):  
Hilbert Modular Surfaces ◽  
Hilbert Modular

Hilbert modular surfaces

1987 ◽  
pp. 225-323
Author(s):  
Friedrich Hirzebruch
Keyword(s):  
Hilbert Modular Surfaces ◽  
Hilbert Modular
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