An explicit Andr\'e-Oort type result for P^1(C) x G_m(C)

Using class field theory we prove an explicit result of Andr\'e-Oort type for $\mathbb{P}^1(\mathbb{C}) \times \mathbb{G}_m(\mathbb{C})$. In this variation the special points of $\mathbb{P}^1(\mathbb{C})$ are the singular moduli, while the special points of $\mathbb{G}_m(\mathbb{C})$ are defined to be the roots of unity.Comment: 10 page

Combinatorial Hopf algebra of superclass functions of type DD

We provide a Hopf algebra structure on the space of superclass functions on the unipotent upper triangular group of type D over a finite field based on a supercharacter theory constructed by Andr\'e and Neto. Also, we make further comments with respect to types B and C. Type A was explores by M. Aguiar et. al (2010), thus this paper is a contribution to understand combinatorially the supercharacter theory of the other classical Lie types.Comment: Last section modified. Recent development added and correction with respect to previous version state

Rigidity of Spreadings and Fields of Definition

Varieties without deformations are defined over a number field. Several old and new examples of this phenomenon are discussed such as Bely\u \i\ curves and Shimura varieties. Rigidity is related to maximal Higgs fields which come from variations of Hodge structure. Basic properties for these due to P. Griffiths, W. Schmid, C. Simpson and, on the arithmetic side, to Y. Andr\'e and I. Satake all play a role. This note tries to give a largely self-contained exposition of these manifold ideas and techniques, presenting, where possible, short new proofs for key results.Comment: Accepted for the EMS Surveys in Mathematical Science

Algebraic relations over finite fields that preserve the endomorphism rings of CM jj-invariants

We characterise the integral affine plane curves over a finite field $k$ with the property that all but finitely many of their $\overline{k}$-points have coordinates that are $j$-invariants of elliptic curves with isomorphic endomorphism rings. This settles a finite field variant of the Andr\'e-Oort conjecture for $Y(1)^2_\mathbb{C}$, which is a theorem of Andr\'e. We use our result to solve the modular support problem for function fields of positive characteristic.Comment: 18 page

Special curves and postcritically-finite polynomials

We study the postcritically-finite (PCF) maps in the moduli space of complex polynomials $\mathrm{MP}_d$. For a certain class of rational curves $C$ in $\mathrm{MP}_d$, we characterize the condition that $C$ contains infinitely many PCF maps. In particular, we show that if $C$ is parameterized by polynomials, then there are infinitely many PCF maps in $C$ if and only if there is exactly one active critical point along $C$, up to symmetries; we provide the critical orbit relation satisfied by any pair of active critical points. For the curves $\mathrm{Per}_1(\lambda)$ in the space of cubic polynomials, introduced by Milnor (1992), we show that $\mathrm{Per}_1(\lambda)$ contains infinitely many PCF maps if and only if $\lambda=0$. The proofs involve a combination of number-theoretic methods (specifically, arithmetic equidistribution) and complex-analytic techniques (specifically, univalent function theory). We provide a conjecture about Zariski density of PCF maps in subvarieties of the space of rational maps, in analogy with the Andr\'e-Oort Conjecture from arithmetic geometry.Comment: Final version, appeared in Forum of Math. P