Related papers: Kobayashi length bounds on bordered surfaces and g…

Let B be a block of a finite group with respect to an algebraically closed field F of characteristic p>0. In a recent paper, Otokita gave an upper bound for the Loewy length LL(ZB) of the center ZB of B in terms of a defect group D of B. We…

We show that a Kleinian surface group, or hyperbolic 3-manifold with a cusp-preserving homotopy-equivalence to a surface, has bounded geometry if and only if there is an upper bound on an associated collection of coefficients that depend…

Let $A$ be a non-isotrivial almost ordinary abelian surface with possibly bad reductions over a global function field of odd characteristic $p$. Suppose $\Delta$ is an infinite set of positive integers, such that…

The main goal of this work is to prove that a very generic surface of degree at least 21 in complex projective 3-dimensional space is hyperbolic in the sense of Kobayashi. This means that every entire holomorphic map $f:{\Bbb C} \to X$ to…

It is shown that the optimal upper and lower bounds for the Kobayashi distance near $\mathcal C^{2,\alpha}$-smooth strongly pseudoconvex boundary points obtained in L. Kosinski, N. Nikolov, A.Y. Okten: "Precise estimates of invariant…

For generalized KdV models with polynomial nonlinearity, we establish nonlinear smoothing property in $H^s$ for $s>\frac{1}{2}$. Such smoothing effect persists globally, provided that the $H^1$ norm does not blow up in finite time. More…

We give upper bounds for $L^p$ norms of eigenfunctions of the Laplacian on compact hyperbolic surfaces in terms of a parameter depending on the growth rate of the number of short geodesic loops passing through a point. When the genus $g \to…

For hyperbolic surfaces with geodesic boundary, we study the orthosystole, i.e. the length of a shortest essential arc from the boundary to the boundary. We recover and extend work by Bavard completely characterizing the surfaces maximizing…

We obtain a lower bound for the normalised height of a non-torsion subvariety $V$ of a C.M. abelian variety. This lower bound is optimal in terms of the geometric degree of $V$, up to a power of a ``log''. We thus extend the results of F.…

Let $S$ be a Shimura variety and let $h$ be a Weil height function on $S$. We conjecture that the heights of special points in $S$ are discriminant negligible. Assuming this conjecture to be true, we prove that the sizes of the Galois…

We prove that the first gap of $\mathbb R$-complementary thresholds of surfaces is $\frac{1}{13}$. More precisely, the largest $\mathbb R$-complementary threshold for surfaces that is strictly less than $1$ is $\frac{12}{13}$. This result…

We show that any polarized abelian variety over a finite field is covered by a Jacobian whose dimension is bounded by an explicit constant. We do this by first proving an effective version of Poonen's Bertini theorem over finite fields,…

Bounding the number of rational points of height at most $H$ on irreducible algebraic plane curves of degree $d$ has been an intense topic of investigation since the work by Bombieri and Pila. In this paper we establish optimal dependence…

We consider Kobayashi geodesics in the moduli space of abelian varieties A_g that is, algebraic curves that are totally geodesic submanifolds for the Kobayashi metric. We show that Kobayashi geodesics can be characterized as those curves…

We count certain abelian surfaces with potential quaternionic multiplication defined over a number field $K$ by counting points of bounded height on some genus zero Shimura curves.

Caro and Pasten gave an explicit upper bound on the number of rational points on a hyperbolic surface that is embedded in an abelian variety of rank at most one. We show how to use their method to produce a refined bound on the number of…

Building on work by Chabauty from 1941, Coleman proved in 1985 an explicit bound for the number of rational points of a curve $C$ of genus $g\ge 2$ defined over a number field $F$, with Jacobian of rank at most $g-1$. Namely, in the case…

We use the fixed point method and toroidal compactifications to establish general lower bounds for the essential dimension of congruence covers $\Gamma' \backslash \mathcal{X}^0 \rightarrow \Gamma \backslash \mathcal{X}^0$ of mixed Shimura…

We study topological lower bounds on the number of small Laplacian eigenvalues on hyperbolic surfaces. We show there exist constants $a,b>0$ such that when $(g+1)<a\frac{n}{\log n}$, any hyperbolic surface of genus-$g$ with $n$ cusps has at…

Combining $2$-descent techniques with Riemann-Roch and B\'ezout's theorems, we give an upper bound on the number of rational points of bounded height on elliptic and hyperelliptic curves over function fields of characteristic $\neq 2$. We…