Related papers: Real regulator maps with finite 0-locus

We introduce and study the maximal-variation locus in families and moduli spaces of projective curves, defined via conductor-level balancing of meromorphic differentials on the normalization. This notion captures precisely when the space of…

The purpose of this paper is to list the refined Humbert invariants for a given automorphism group of a curve $C/K$ of genus 2 over an algebraically closed field $K$ with characteristic $0$. This invariant is an algebraic generalization of…

We give a bound of $k$ for a very ample lattice polytope to be $k$-normal. Equivalently, we give a new combinatorial bound for the Castelnuovo-Mumford regularity of normal projective toric varieties.

Let X be a smooth algebraic variety on which a solvable Lie group acts freely on a dense open orbit. Such varieties include generalized flag varieties, toric varieties, Bott-Samelson varieties, and many spherical varieties, as well as…

A singular curve over a non-perfect field K may not have a smooth model over K. Those are said to "change genus". If K is a global field of positive characteristic and C/K a curve that change genus, then C(K) is known to be finite. The…

I prove the surjectivity of the $p$-adic regulator from Quillen's $K_2$ of Tate curve to the $p$-adic etale cohomology group when the base field is contained in a cyclotomic extension of $Q_p$. This implies the finiteness of torsion part of…

We examine conditions under which there exists a non-constant family of Galois branched covers of curves over an algebraically closed field $k$ of fixed degree and fixed ramification locus, under a notion of equivalence derived from…

This is a continuation of our previous work on the locally analytic vectors of the completed cohomology of modular curves. We construct differential operators on modular curves with infinite level at p in both "holomorphic" and…

We show that the closed convex hull of any one-dimensional semi-algebraic subset of R^n has a semidefinite representation, meaning that it can be written as a linear projection of the solution set of some linear matrix inequality. This is…

For any genuinely ramified morphism $f\, :\, Y\, \longrightarrow\, X$ between irreducible smooth projective curves we prove that $\overline{(Y\times_X Y) \setminus \Delta}$ is connected, where $\Delta\, \subset\, Y\times_X Y$ is the…

Given a real elliptic curve $E$ with non-empty real part and $[D]\in \mbox{Pic}^2 E$ its $g_2^1$, we study the real inflection points of distinguished subseries of the complete real linear series $|\mathcal{L}_\mathbb{R}(kD)|$ for $k\geq…

We show that a function $f : X \to \mathbb R$ defined on a closed uniformly polynomially cuspidal set $X$ in $\mathbb R^n$ is real analytic if and only if $f$ is smooth and all its composites with germs of polynomial curves in $X$ are real…

Quantum cat maps are toy models in quantum chaos associated to hyperbolic symplectic matrices $A\in \operatorname{Sp}(2n,\mathbb{Z})$. The macroscopic limits of sequences of eigenfunctions of a quantum cat map are characterized by…

We call a subset $K$ of $\mathbb C$ \emph{biholomorphically homogeneous} if for any two points $p,q\in K$ there exists a neighborhood $U$ of $p$ and a biholomorphism $\psi:U\to \psi(U)\subset \mathbb C$ such that $\psi(p)=q$ and $\psi(K\cap…

Let $K$ be a field of characteristic $0$ and $E/K$ an elliptic curve over $K$. For a finite extension $L/K$ and a prime~$\ell$, we provide Galois-theoretic sufficient conditions on $L/K$ under which…

We study the regularity of the roots of complex monic polynomials $P(t)$ of fixed degree depending smoothly on a real parameter $t$. We prove that each continuous parameterization of the roots of a generic $C^\infty$ curve $P(t)$ (which…

In this note, we prove that for every ordinary genus-2 curve $X$ over a finite field $\kappa$ of characteristic 2 with $\text{Aut}(X/\kappa)=\db{Z}/2\db{Z} \times S_3$, there exist $\text{SL}(2,\kappa\sembrack{s})$-representations of…

This note defines a family of Laurent polynomials (indexed in the rational projective line) which generalize the Markoff numbers and relate to the character variety of the one-cusped torus. We describe which monomials appear in each…

Let $E$ be an elliptic curve over a number field $K$ defined by a monic irreducible cubic polynomial $F(x)$. When $E$ is \textit{nice} at all finite primes of $K$, we bound its $2$-Selmer rank in terms of the $2$-rank of a modified ideal…

We study families of superelliptic curves with fixed automorphism groups. Such families are parametrized with invariants expressed in terms of the coefficients of the curves. Algebraic relations among such invariants determine the lattice…