Related papers: Triples of singular moduli with rational product

Let $G$ be a simple algebraic group over an algebraically closed field $K$ of characteristic $p\geqslant 0$, let $H$ be a proper closed subgroup of $G$ and let $V$ be a nontrivial irreducible $KG$-module, which is $p$-restricted, tensor…

The codimension-three conjecture states that any regular holonomic module extends uniquely beyond an analytic subset with codimension equal to or larger than three. We give a proof of this conjecture.

Let $E$ be a number field and $X$ a smooth geometrically connected variety defined over a characteristic $p$ finite field. Given an $n$-dimensional pure $E$-compatible system of semisimple $\lambda$-adic representations of the \'etale…

We classify order $3$ linear difference operators over $\mathbb{C}(x)$ that are solvable in terms of lower order difference operators. To prove this result, we introduce the notion of absolute irreducibility for difference modules, and…

We prove a conjecture of J. Carlson, N. Mazza and J. Th\'evenaz; namely, we will prove that if $G$ is a finite $p$-nilpotent group which contains a non-cyclic elementary Abelian $p$-subgroup and $k$ is an algebraically closed field of…

Pila and Tsimerman proved in 2017 that for every $k$ there exists at most finitely many $k$-tuples $(x_1,\ldots, x_k)$ of distinct non-zero singular moduli with the property "$x_1, \ldots,x_k$ are multiplicatively dependent, but any proper…

In this short note, we compute the Betti numbers of the moduli stack of flat SU(3)-bundles over a Klein bottle. We also handle the general compact group case over RP^2. In all cases the cohomology is found to be equivariantly formal,…

We treat two different equations involving powers of singular moduli. On the one hand, we show that, with two possible (explicitly specified) exceptions, two distinct singular moduli j(\tau),j(\tau') such that the numbers 1, j(\tau)^m and…

We continue our work (started in ``Multiplicity-free representations of algebraic groups", arXiv:2101.04476), on the program of classifying triples $(X,Y,V)$, where $X,Y$ are simple algebraic groups over an algebraically closed field of…

We prove the following generalization of Severi's Theorem: Let $X$ be a fixed complex variety. Then there exist, up to birational equivalence, only finitely many complex varieties $Y$ of general type of dimension at most three which admit a…

We determine which simple algebraic groups of type $^3D_4$ over arbitrary fields of characteristic different from 2 admit outer automorphisms of order 3, and classify these automorphisms up to conjugation. The criterion is formulated in…

Let $q\geq2$ be a prime power and consider Drinfeld modules of rank 2 over $\mathbb{F}_q[T]$. We prove that there are no points with coordinates being Drinfeld singular moduli, on a family of hyperbolas $XY=\gamma$, where $\gamma$ is a…

Let $k$ be a field, $X$ a variety with tame quotient singularities and $\tilde{X}\to X$ a resolution of singularities. Any smooth rational point $x\in X(k)$ lifts to $\tilde{X}$ by the Lang-Nishimura theorem, but if $x$ is singular this…

Consider a finite-dimensional algebra $A$ and any of its moduli spaces $\mathcal{M}(A,\mathbf{d})^{ss}_{\theta}$ of representations. We prove a decomposition theorem which relates any irreducible component of…

Riffaut (2019) conjectured that a singular modulus of degree $h\ge 3$ cannot be a root of a trinomial with rational coefficients. We show that this conjecture follows from the GRH, and obtain partial unconditional results.

In this note we look at the moduli space $\cR_{3,2}$ of double covers of genus three curves, branched along 4 distinct points. This space was studied by Bardelli, Ciliberto and Verra. It admits a dominating morphism $\cR_{3,2} \to {\mathcal…

We consider small solutions of quadratic congruences of the form $x_1^2+\alpha_2x_2^2+\alpha_3x_3^2\equiv 0 \bmod{q}$, where $q=p^m$ is an odd prime power. Here, $\alpha_2$ is arbitrary but fixed and $\alpha_3$ is variable, and we assume…

Let X be a smooth, connected, projective variety over an algebraically closed field of positive characteristic. In "Flat vector bundles and the fundamental group in non-zero characteristics" (Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2…

Let $(X,x_0)$ be any one--pointed compact connected Riemann surface of genus $g$, with $g\geq 3$. Fix two mutually coprime integers $r>1$ and $d$. Let ${\mathcal M}_X$ denote the moduli space parametrizing all logarithmic…

We prove an Andr\'e--Oort-type result for a family of hypersurfaces in $\mathbb{C}^n$ that is both uniform and effective. Let $K_*$ denote the single exceptional imaginary quadratic field which occurs in the Siegel--Tatuzawa lower bound for…