scholarly journals Towards the Andre–Oort conjecture for mixed Shimura varieties: The Ax–Lindemann theorem and lower bounds for Galois orbits of special points

2017 ◽  
Vol 2017 (732) ◽  
pp. 85-146 ◽  
Author(s):  
Ziyang Gao
Keyword(s):  
Lower Bounds ◽  
Riemann Hypothesis ◽  
Shimura Varieties ◽  
Shimura Variety ◽  
Weierstrass Theorem ◽  
Generalized Riemann Hypothesis ◽  
Galois Orbits ◽  
Special Points

Abstract We prove in this paper the Ax–Lindemann–Weierstraß theorem for all mixed Shimura varieties and discuss the lower bounds for Galois orbits of special points of mixed Shimura varieties. In particular, we reprove a result of Silverberg [57] in a different approach. Then combining these results we prove the André–Oort conjecture unconditionally for any mixed Shimura variety whose pure part is a subvariety of {\mathcal{A}_{6}^{n}} and under the Generalized Riemann Hypothesis for all mixed Shimura varieties of abelian type.

scholarly journals The André–Oort conjecture for Drinfeld modular varieties

2013 ◽  
Vol 149 (4) ◽  
pp. 507-567 ◽  
Author(s):  
Patrik Hubschmid
Keyword(s):  
Riemann Hypothesis ◽  
Classical Case ◽  
Base Field ◽  
Generalized Riemann Hypothesis ◽  
Fixed Set ◽  
Special Points ◽  
Modular Varieties

AbstractWe consider the analogue of the André–Oort conjecture for Drinfeld modular varieties which was formulated by Breuer. We prove this analogue for special points with separable reflex field over the base field by adapting methods which were used by Klingler and Yafaev to prove the André–Oort conjecture under the generalized Riemann hypothesis in the classical case. Our result extends results of Breuer showing the correctness of the analogue for special points lying in a curve and for special points having a certain behaviour at a fixed set of primes.

scholarly journals GALOIS CONJUGATES OF SPECIAL POINTS AND SPECIAL SUBVARIETIES IN SHIMURA VARIETIES

2019 ◽  
pp. 1-15
Author(s):  
Martin Orr
Keyword(s):  
Special Point ◽  
Shimura Varieties ◽  
Shimura Variety ◽  
Canonical Models ◽  
Special Points

Let $S$ be a Shimura variety with reflex field $E$ . We prove that the action of $\text{Gal}(\overline{\mathbb{Q}}/E)$ on $S$ maps special points to special points and special subvarieties to special subvarieties. Furthermore, the Galois conjugates of a special point all have the same complexity (as defined in the theory of unlikely intersections). These results follow from Milne and Shih’s construction of canonical models of Shimura varieties, based on a conjecture of Langlands which was proved by Borovoi and Milne.

scholarly journals UNE MINORATION POUR L'EXPOSANT DU GROUPE DES CLASSES D'UN CORPS ENGENDRÉ PAR UN NOMBRE DE SALEM

2007 ◽  
Vol 03 (02) ◽  
pp. 217-229 ◽  
Author(s):  
FRANCESCO AMOROSO
Keyword(s):  
Zeta Function ◽  
Number Field ◽  
Lower Bounds ◽  
Riemann Hypothesis ◽  
Class Group ◽  
Relative Size ◽  
Generalized Riemann Hypothesis ◽  
Dedekind Zeta Function ◽  
Salem Number

In this article we extend the main result of [2] concerning lower bounds for the exponent of the class group of CM-fields. We consider a number field K generated by a Salem number α. If k denotes the field fixed by α ↦ α-1 we prove, under the generalized Riemann hypothesis for the Dedekind zeta function of K, lower bounds for the relative exponent eK/k and the relative size hK/k of the class group of K with respect to the class group of k.

scholarly journals CATEGORICITY OF MODULAR AND SHIMURA CURVES

2015 ◽  
Vol 16 (5) ◽  
pp. 1075-1101 ◽  
Author(s):  
Christopher Daw ◽  
Adam Harris
Keyword(s):  
Model Theory ◽  
Galois Representations ◽  
Arithmetic Geometry ◽  
Shimura Varieties ◽  
Shimura Curves ◽  
Shimura Variety ◽  
Unique Model ◽  
Infinite Cardinality ◽  
Special Points ◽  
Generic Points

We describe a model-theoretic setting for the study of Shimura varieties, and study the interaction between model theory and arithmetic geometry in this setting. In particular, we show that the model-theoretic statement of a certain${\mathcal{L}}_{\unicode[STIX]{x1D714}_{1},\unicode[STIX]{x1D714}}$-sentence having a unique model of cardinality$\aleph _{1}$is equivalent to a condition regarding certain Galois representations associated with Hodge-generic points. We then show that for modular and Shimura curves this${\mathcal{L}}_{\unicode[STIX]{x1D714}_{1},\unicode[STIX]{x1D714}}$-sentence has a unique model in every infinite cardinality. In the process, we prove a new characterisation of the special points on any Shimura variety.

scholarly journals Applications du théorème d’Ax–Lindemann hyperbolique

2013 ◽  
Vol 150 (2) ◽  
pp. 175-190 ◽  
Author(s):  
Emmanuel Ullmo
Keyword(s):  
Lower Bound ◽  
Shimura Variety ◽  
Arbitrary Integer ◽  
Galois Orbits ◽  
Minimal Structure ◽  
Special Points

AbstractWe explain how the André–Oort conjecture for a general Shimura variety can be deduced from the hyperbolic Ax–Lindemann conjecture, a good lower bound for Galois orbits of special points and the definability, in the $o$-minimal structure ${ \mathbb{R} }_{\mathrm{an} , \mathrm{exp} } $, of the restriction to a fundamental set of the uniformizing map of a Shimura variety. These ingredients are known in some important cases. As a consequence a proof of the André–Oort conjecture for projective special subvarieties of ${ \mathcal{A} }_{6}^{N} $ for an arbitrary integer $N$ is given.

ON THE DIVISORS OF xn – 1 IN 𝔽p[x]

2012 ◽  
Vol 09 (02) ◽  
pp. 421-430 ◽  
Author(s):  
LOLA THOMPSON
Keyword(s):  
Lower Bounds ◽  
Riemann Hypothesis ◽  
Asymptotic Density ◽  
Upper And Lower Bounds ◽  
Generalized Riemann Hypothesis

In a recent paper, we considered integers n for which the polynomial xn – 1 has a divisor in ℤ[x] of every degree up to n, and we gave upper and lower bounds for their distribution. In this paper, we consider those n for which the polynomial xn – 1 has a divisor in 𝔽p[x] of every degree up to n, where p is a rational prime. Assuming the validity of the Generalized Riemann Hypothesis, we show that such integers n have asymptotic density 0.

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 $p$.

Quadratic Non-Residues and Prime-Producing Polynomials

1989 ◽  
Vol 32 (4) ◽  
pp. 474-478 ◽  
Author(s):  
R. A. Mollin ◽  
H. C. Williams
Keyword(s):  
Riemann Hypothesis ◽  
Quadratic Polynomials ◽  
Generalized Riemann Hypothesis ◽  
Initial Values ◽  
Positive Discriminant

AbstractWe will be looking at quadratic polynomials having positive discriminant and having a long string of primes as initial values. We find conditions tantamount to this phenomenon involving another long string of primes for which the discriminant of the polynomial is a quadratic non-residue. Using the generalized Riemann hypothesis (GRH) we are able to determine all discriminants satisfying this connection.

scholarly journals Variant of the truncated Perron formula and primes in polynomial sets

2019 ◽  
Vol 16 (02) ◽  
pp. 309-323
Author(s):  
D. S. Ramana ◽  
O. Ramaré
Keyword(s):  
Dirichlet Series ◽  
Riemann Hypothesis ◽  
Partial Sums ◽  
Generalized Riemann Hypothesis ◽  
Perron’S Formula ◽  
Polynomial Formula

We show under the Generalized Riemann Hypothesis that for every non-constant integer-valued polynomial [Formula: see text], for every [Formula: see text], and almost every prime [Formula: see text] in [Formula: see text], the number of primes from the interval [Formula: see text] that are values of [Formula: see text] modulo [Formula: see text] is the expected one, provided [Formula: see text] is not more than [Formula: see text]. We obtain this via a variant of the classical truncated Perron’s formula for the partial sums of the coefficients of a Dirichlet series.

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