arXiv subjects
Publications and source records attributed to Sebastian Eterović.
We prove that the modular Zilber--Pink conjecture (in Pink's formulation in terms of unlikely intersections) holds for all subvarieties $V$ of $ \mathrm{Y}(1)^n$ for which no projection to any $\dim V + 2$ coordinates is defined over the algebraic numbers.
We characterize the bialgebraic varieties of the $Γ$ function, that is, if $V,W\subseteq\mathbb{C}^n$ are irreducible affine algebraic variety which satisfy $\dim V =\dim W$ and $Γ(V)\subseteq W$, then the equations defining $V$ (and hence also $W$) either give an equality between coordinates, or set some coordinates to be constant. We also classify the case where $V$ and $W$ have the same dimension as varieties over the field of algebraic functions.
We use Hensel minimality, a non-Archimedean analog of o-minimality, to study several questions around transcendental number theory, unlikely intersections, and differential fields in a non-Archimedean setting. In particular, we focus on $p$-adic exponentiation and Tate uniformization on $\mathbb{C}_p$, which we show live in a Hensel minimal structure on $\mathbb{C}_p$. We start by constructing a large collection of derivations on Hensel minimal fields that respect definable functions, which we then apply to the $p$-adic Schanuel conjecture. We also study properties of local definability in analogy to work of Wilkie, and show that $p$-adic Schanuel implies a uniform version of itself. For Tate uniformization we show a strong closure property when blurring, and deduce that $\mathbb{C}_p$ with the blurred Tate uniformization is quasiminimal. Finally, we prove a result on $p$-adic density of likely intersections for powers of elliptic curves.
We show that for any polynomial $F(X,Y_0,Y_1,Y_2) \in \mathbb{C}[X, Y_0, Y_1, Y_2]$, the equation $F(z,j(z),j'(z),j''(z))=0$ has a Zariski dense set of solutions in the hypersurface $F(X,Y_0,Y_1,Y_2)=0$, unless $F$ is in $\mathbb{C}[X]$ or it is divisible by $Y_0$, $Y_0-1728$, or $Y_1$. Our methods establish criteria for finding solutions to more general equations involving periodic functions. Furthermore, they produce a qualitative description of the distribution of these solutions.
Let $n \in \mathbb{Z}_{>0}$. We prove that there exist a finite set $V$ and finitely many algebraic curves $T_1, \ldots, T_k$ with the following property: if $(x_1, \ldots, x_n, y)$ is an $(n+1)$-tuple of pairwise distinct singular moduli such that $\prod_{i=1}^n (x_i - y)^{a_i}=1$ for some $a_1, \ldots, a_n \in \mathbb{Z} \setminus \{0\}$, then $(x_1, \ldots, x_n, y) \in V \cup T_1 \cup \ldots \cup T_k$. Further, the curves $T_1, \ldots, T_k$ may be determined explicitly for a given $n$.
Let $(G, X)$ be a Shimura datum, let $Ω$ be a connected component of $X$, let $Γ$ be a congruence subgroup of $G(\mathbb{Q})^{+}$, and consider the quotient map $q: Ω\to S:=Γ\backslash Ω$. Consider the Harish-Chandra embedding $Ω\subset\mathbb{C}^{N}$, where $N=\dim X$. We prove two results that give geometric conditions which if satisfied by an algebraic variety $V \subset \mathbb{C}^{N} \times S$, ensure that there is a Zariski dense subset of $V$ of points of the form $(x,q(x))$.
Assuming a modular version of Schanuel's conjecture and the modular Zilber-Pink conjecture, we show that the existence of generic solutions of certain families of equations involving the modular $j$ function can be reduced to the problem of finding a Zariski dense set of solutions. By imposing some conditions on the field of definition of the variety, we are also able to obtain unconditional versions of this result.
Let $V\subseteq\mathbb{C}^{2n}$ be an algebraic variety with no constant coordinates and with a dominant projection onto the first $n$ coordinates. We show that the intersection of $V$ with the graph of the $Γ$ function is Zariski dense in $V$.
We prove a general likely intersections theorem, a counterpart to the Zilber-Pink conjectures, under the assumption that the Ax-Schanuel property and some mild additional conditions are known to hold for a given category of complex quotient spaces definable in some fixed o-minimal expansion of the ordered field of real numbers. For an instance of our general result, consider the case of subvarieties of Shimura varieties. Let $S$ be a Shimura variety. Let $π:D \to Γ\backslash D = S$ realize $S$ as a quotient of $D$, a homogeneous space for the action of a real algebraic group $G$, by the action of $Γ< G$, an arithmetic subgroup. Let $S' \subseteq S$ be a special subvariety of $S$ realized as $π(D')$ for $D' \subseteq D$ a homogeneous space for an algebraic subgroup of $G$. Let $X \subseteq S$ be an irreducible subvariety of $S$ not contained in any proper weakly special subvariety of $S$. Assume that the intersection of $X$ with $S'$ is persistently likely meaning that whenever $ζ:S_1 \to S$ and $ξ:S_1 \to S_2$ are maps of Shimura varieties (meaning regular maps of varieties induced by maps of the corresponding Shimura data) with $ζ$ finite, $\dim ξζ^{-1} X + \dim ξζ^{-1} S' \geq \dim ξS_1$. Then $X \cap \bigcup_{g \in G, π(g D') \text{ is special }} π(g D')$ is dense in $X$ for the Euclidean topology.
Given a subfield $F$ of $\mathbb{C}$, we study the linear disjointess of the field $E$ generated by iterated exponentials of elements of $\overline{F}$, and the field $L$ generated by iterated logarithms, in the presence of Schanuel's conjecture. We also obtain similar results replacing $\exp$ by the modular $j$-function, under an appropriate version of Schanuel's conjecture, where linear disjointness is replaced by a notion coming from the action of $\mathrm{GL}_2$ on $\mathbb{C}$. We also show that for certain choices of $F$ we obtain unconditional versions of these statements.
We prove some unconditional cases of the Existential Closedness problem for the modular $j$-function. For this, we show that for any finitely generated field we can find a "convenient" set of generators. This is done by showing that in any field equipped with functions replicating the algebraic behaviour of the modular $j$-function and its derivatives, one can define a natural closure operator in three equivalent different ways.
We prove the Existential Closedness conjecture for the differential equation of the $j$-function and its derivatives. It states that in a differentially closed field certain equations involving the differential equation of the $j$-function have solutions. Its consequences include a complete axiomatisation of $j$-reducts of differentially closed fields, a dichotomy result for strongly minimal sets in those reducts, and a functional analogue of the Modular Zilber-Pink with Derivatives conjecture.
Inspired by work done for systems of polynomial exponential equations, we study systems of equations involving the modular $j$ function. We show general cases in which these systems have solutions, and then we look at certain situations in which the modular Schanuel conjecture implies that these systems have generic solutions. An unconditional result in this direction is proven for certain polynomial equations on $j$ with algebraic coefficients.
This is a short note on various results about the combinatorial properties of line arrangements in terms of the Chern numbers of the corresponding log surfaces. This resembles the study of the geography of surfaces of general type. We prove some new results about the distribution of Chern slopes, we prove a connection between their accumulation points and the accumulation points of linear $H$-constants on the plane, and we present two open problems in relation to geography over $\mathbb Q$ and over $\mathbb C$.
We propose a model-theoretic structure for Shimura varieties and give necessary and sufficient conditions to obtain categoricity. We show that these conditions are directly related to important conjectures in number theory coming from Galois representations attached the points of a Shimura variety. We end by showing that the existing literature is enough to prove categoricity of $\mathcal{A}_{2}$ and $\mathcal{A}_{3}$.
I give a model-theoretic setting for the modular $j$ function and its derivatives. These structures, here called $j$-fields, provide an adequate setting for interpreting the Ax-Schanuel theorem for $j$ (Pila-Tsimerman 2015). Following the ideas of M. Bays, J. Kirby and A.J. Wilkie for exponential fields, I prove a generic transcendence property for the $j$ function.