18 problems

A Shimura–Teichmüller curve is a Teichmüller curve, equivalently a Veech surface orbit, with totally degenerate Lyapunov spectrum. Möller's conjecture. There are no Shimura–Teichmü…

Let $\mathcal H(4)^{\mathrm{hyp}}$ denote the hyperelliptic connected component of the stratum of translation surfaces with one zero of order $4$. An algebraically primitive Teichm…

Let a Teichmüller curve be a curve generated by the omega^0 differential in a stratum with more than one zero, and let its trace field be the field generated by the traces of the c…

For a translation surface in the stratum $\mathcal{H}(2)$, the associated Veech group is arithmetic when its orbit gives rise to the graphs under consideration. McMullen's expansio…

Let $n$ be a positive integer. Using the generating set $\{P,R\}$ for $\operatorname{SL}(2,\mathbb{Z})$, where … let $\overline{\mathcal{G}}_{n}$, $\overline{\mathcal{G}}_{n}^{A}$,…

Let $X(d)$ be the modular curve equipped with its square-tiling and the metric induced by the quadratic differential $q$. Say that a point $A\in X(d)$ illuminates a point…

Let $d>1$ and $n\geq 1$ be integers. Let $W_{d^2}[n]\subset\mathcal M_2$ be the locus of genus-two Riemann surfaces admitting a primitive degree-$d$ elliptic cover with two critica…

For each genus $g$, consider the hyperelliptic components $\overline{\mathcal{H}}^{\mathrm{hyp}}_g(2g-2)$ and $\overline{\mathcal{H}}^{\mathrm{hyp}}_g(g-1,g-1)$, and let…

Let $C$ be a Teichmüller curve of genus $g$, with Lyapunov exponents $\lambda_1\geq\cdots\geq\lambda_g$ and normalized Harder–Narasimhan numbers $w_1\geq\cdots\geq w_g$. Let $k$ be…

Let $C$ be a Teichmüller curve of genus $g$. Let $boldsymbol{ u}=( u1, u2, u3, u4)=(1,w2,ldots,wg)$ denote the normalized Harder–Narasimhan data of the Hodge bundle, and let…

Let a Teichmüller curve be a curve in a stratum of the moduli space of Abelian differentials, and call it algebraic primitive when its variation of Hodge structures has the decompo…

Twisted Teichmüller curve uniqueness conjecture. Suppose $D\equiv 5\pmod 8$ is a fundamental discriminant with narrow class number $h_D^+=1$. For every $n\ind4aa_D$, all matrices

Let $g\ge 2$, and let a Teichmüller curve lie in a stratum of Abelian differentials in genus $g$. Write its Lyapunov exponents as $\lambda_1\ge\cdots\ge\lambda_g$. The spectrum is…

Fix a stratum $\mathcal H(\mu)$ of Abelian differentials, and let $\mathcal T_{d,\mu,i}$ be an irreducible component of the Teichmüller curve arising from square-tiled surfaces of…

Let $C$ be an algebraically primitive Teichmüller curve in the hyperelliptic stratum ${\Omega\mathcal M}_{3}(4)^{{\rm hyp}}$, and let $(r_1:r_2:r_3)$ be the projectivized triple of…

A Teichmüller curve in ${\mathcal M}_{3}$ is algebraically primitive when its trace field has degree $3$. Finiteness conjecture. There are only finitely many algebraically primitiv…

Let $\overline{C}$ be a smooth projective curve of genus $0$ defined over a number field, and let $({\mathcal E}, \nabla)$ be a flat vector bundle of rank $2$ which is globally nil…

Let $\overline{C}$ be a smooth projective curve defined over a number field. Let $({\mathcal E}, \nabla)$ be an indigenous bundle on $\overline{C}$, and let…