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Search: MSC category 11G18 ( Arithmetic aspects of modular and Shimura varieties [See also 14G35] )

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1. CJM Online first

Zhang, Chao
Ekedahl-Oort strata for good reductions of Shimura varieties of Hodge type
For a Shimura variety of Hodge type with hyperspecial level structure at a prime~$p$, Vasiu and Kisin constructed a smooth integral model (namely the integral canonical model) uniquely determined by a certain extension property. We define and study the Ekedahl-Oort stratifications on the special fibers of those integral canonical models when $p\gt 2$. This generalizes Ekedahl-Oort stratifications defined and studied by Oort on moduli spaces of principally polarized abelian varieties and those defined and studied by Moonen, Wedhorn and Viehmann on good reductions of Shimura varieties of PEL type. We show that the Ekedahl-Oort strata are parameterized by certain elements $w$ in the Weyl group of the reductive group in the Shimura datum. We prove that the stratum corresponding to $w$ is smooth of dimension $l(w)$ (i.e. the length of $w$) if it is non-empty. We also determine the closure of each stratum.

Keywords:Shimura variety, F-zip
Categories:14G35, 11G18

2. CJM 2016 (vol 68 pp. 1362)

Papikian, Mihran; Rabinoff, Joseph
Optimal Quotients of Jacobians with Toric Reduction and Component Groups
Let $J$ be a Jacobian variety with toric reduction over a local field $K$. Let $J \to E$ be an optimal quotient defined over $K$, where $E$ is an elliptic curve. We give examples in which the functorially induced map $\Phi_J \to \Phi_E$ on component groups of the Néron models is not surjective. This answers a question of Ribet and Takahashi. We also give various criteria under which $\Phi_J \to \Phi_E$ is surjective, and discuss when these criteria hold for the Jacobians of modular curves.

Keywords:Jacobians with toric reduction, component groups, modular curves
Categories:11G18, 14G22, 14G20

3. CJM 2013 (vol 66 pp. 1305)

Koskivirta, Jean-Stefan
Congruence Relations for Shimura Varieties Associated with $GU(n-1,1)$
We prove the congruence relation for the mod-$p$ reduction of Shimura varieties associated to a unitary similitude group $GU(n-1,1)$ over $\mathbb{Q}$, when $p$ is inert and $n$ odd. The case when $n$ is even was obtained by T. Wedhorn and O. B?ltel, as a special case of a result of B. Moonen, when the $\mu$-ordinary locus of the $p$-isogeny space is dense. This condition fails in our case. We show that every supersingular irreducible component of the special fiber of $p\textrm{-}\mathscr{I}sog$ is annihilated by a degree one polynomial in the Frobenius element $F$, which implies the congruence relation.

Keywords:Shimura varieties, congruence relation
Categories:11G18, 14G35, 14K10

4. CJM 2013 (vol 66 pp. 924)

Stankewicz, James
Twists of Shimura Curves
Consider a Shimura curve $X^D_0(N)$ over the rational numbers. We determine criteria for the twist by an Atkin-Lehner involution to have points over a local field. As a corollary we give a new proof of the theorem of Jordan-Livné on $\mathbf{Q}_p$ points when $p\mid D$ and for the first time give criteria for $\mathbf{Q}_p$ points when $p\mid N$. We also give congruence conditions for roots modulo $p$ of Hilbert class polynomials.

Keywords:Shimura curves, complex multiplication, modular curves, elliptic curves
Categories:11G18, 14G35, 11G15, 11G10

5. CJM 2013 (vol 66 pp. 1167)

Rotger, Victor; de Vera-Piquero, Carlos
Galois Representations Over Fields of Moduli and Rational Points on Shimura Curves
The purpose of this note is introducing a method for proving the existence of no rational points on a coarse moduli space $X$ of abelian varieties over a given number field $K$, in cases where the moduli problem is not fine and points in $X(K)$ may not be represented by an abelian variety (with additional structure) admitting a model over the field $K$. This is typically the case when the abelian varieties that are being classified have even dimension. The main idea, inspired on the work of Ellenberg and Skinner on the modularity of $\mathbb{Q}$-curves, is that to a point $P=[A]\in X(K)$ represented by an abelian variety $A/\bar K$ one may still attach a Galois representation of $\operatorname{Gal}(\bar K/K)$ with values in the quotient group $\operatorname{GL}(T_\ell(A))/\operatorname{Aut}(A)$, provided $\operatorname{Aut}(A)$ lies in the centre of $\operatorname{GL}(T_\ell(A))$. We exemplify our method in the cases where $X$ is a Shimura curve over an imaginary quadratic field or an Atkin-Lehner quotient over $\mathbb{Q}$.

Keywords:Shimura curves, rational points, Galois representations, Hasse principle, Brauer-Manin obstruction
Categories:11G18, 14G35, 14G05

6. CJM 2012 (vol 66 pp. 170)

Guitart, Xavier; Quer, Jordi
Modular Abelian Varieties Over Number Fields
The main result of this paper is a characterization of the abelian varieties $B/K$ defined over Galois number fields with the property that the $L$-function $L(B/K;s)$ is a product of $L$-functions of non-CM newforms over $\mathbb Q$ for congruence subgroups of the form $\Gamma_1(N)$. The characterization involves the structure of $\operatorname{End}(B)$, isogenies between the Galois conjugates of $B$, and a Galois cohomology class attached to $B/K$. We call the varieties having this property strongly modular. The last section is devoted to the study of a family of abelian surfaces with quaternionic multiplication. As an illustration of the ways in which the general results of the paper can be applied we prove the strong modularity of some particular abelian surfaces belonging to that family, and we show how to find nontrivial examples of strongly modular varieties by twisting.

Keywords:Modular abelian varieties, $GL_2$-type varieties, modular forms
Categories:11G10, 11G18, 11F11

7. CJM 2012 (vol 65 pp. 403)

Van Order, Jeanine
On the Dihedral Main Conjectures of Iwasawa Theory for Hilbert Modular Eigenforms
We construct a bipartite Euler system in the sense of Howard for Hilbert modular eigenforms of parallel weight two over totally real fields, generalizing works of Bertolini-Darmon, Longo, Nekovar, Pollack-Weston and others. The construction has direct applications to Iwasawa main conjectures. For instance, it implies in many cases one divisibility of the associated dihedral or anticyclotomic main conjecture, at the same time reducing the other divisibility to a certain nonvanishing criterion for the associated $p$-adic $L$-functions. It also has applications to cyclotomic main conjectures for Hilbert modular forms over CM fields via the technique of Skinner and Urban.

Keywords:Iwasawa theory, Hilbert modular forms, abelian varieties
Categories:11G10, 11G18, 11G40

8. CJM 2011 (vol 64 pp. 588)

Nekovář, Jan
Level Raising and Anticyclotomic Selmer Groups for Hilbert Modular Forms of Weight Two
In this article we refine the method of Bertolini and Darmon and prove several finiteness results for anticyclotomic Selmer groups of Hilbert modular forms of parallel weight two.

Keywords:Hilbert modular forms, Selmer groups, Shimura curves
Categories:11G40, 11F41, 11G18

9. CJM 2011 (vol 63 pp. 826)

Errthum, Eric
Singular Moduli of Shimura Curves
The $j$-function acts as a parametrization of the classical modular curve. Its values at complex multiplication (CM) points are called singular moduli and are algebraic integers. A Shimura curve is a generalization of the modular curve and, if the Shimura curve has genus~$0$, a rational parameterizing function exists and when evaluated at a CM point is again algebraic over~$\mathbf{Q}$. This paper shows that the coordinate maps given by N.~Elkies for the Shimura curves associated to the quaternion algebras with discriminants $6$ and $10$ are Borcherds lifts of vector-valued modular forms. This property is then used to explicitly compute the rational norms of singular moduli on these curves. This method not only verifies conjectural values for the rational CM points, but also provides a way of algebraically calculating the norms of CM points with arbitrarily large negative discriminant.

Categories:11G18, 11F12

10. CJM 2010 (vol 62 pp. 668)

Vollaard, Inken
The Supersingular Locus of the Shimura Variety for GU(1,s)
In this paper we study the supersingular locus of the reduction modulo $p$ of the Shimura variety for $GU(1,s)$ in the case of an inert prime $p$. Using Dieudonné theory we define a stratification of the corresponding moduli space of $p$-divisible groups. We describe the incidence relation of this stratification in terms of the Bruhat--Tits building of a unitary group. In the case of $GU(1,2)$, we show that the supersingular locus is equidimensional of dimension 1 and is of complete intersection. We give an explicit description of the irreducible components and their intersection behaviour.

Categories:14G35, 11G18, 14K10

11. CJM 2009 (vol 61 pp. 828)

Howard, Benjamin
Twisted Gross--Zagier Theorems
The theorems of Gross--Zagier and Zhang relate the N\'eron--Tate heights of complex multiplication points on the modular curve $X_0(N)$ (and on Shimura curve analogues) with the central derivatives of automorphic $L$-function. We extend these results to include certain CM points on modular curves of the form $X(\Gamma_0(M)\cap\Gamma_1(S))$ (and on Shimura curve analogues). These results are motivated by applications to Hida theory that can be found in the companion article "Central derivatives of $L$-functions in Hida families", Math.\ Ann.\ \textbf{399}(2007), 803--818.

Categories:11G18, 14G35

12. CJM 2008 (vol 60 pp. 734)

Baba, Srinath; Granath, H\aa kan
Genus 2 Curves with Quaternionic Multiplication
We explicitly construct the canonical rational models of Shimura curves, both analytically in terms of modular forms and algebraically in terms of coefficients of genus 2 curves, in the cases of quaternion algebras of discriminant 6 and 10. This emulates the classical construction in the elliptic curve case. We also give families of genus 2 QM curves, whose Jacobians are the corresponding abelian surfaces on the Shimura curve, and with coefficients that are modular forms of weight 12. We apply these results to show that our $j$-functions are supported exactly at those primes where the genus 2 curve does not admit potentially good reduction, and construct fields where this potentially good reduction is attained. Finally, using $j$, we construct the fields of moduli and definition for some moduli problems associated to the Atkin--Lehner group actions.

Keywords:Shimura curve, canonical model, quaternionic multiplication, modular form, field of moduli
Categories:11G18, 14G35

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