We prove the existence of weak solutions of complex $m$ -Hessian equations on compact Hermitian manifolds for the non-negative right-hand side belonging to $L^{p}$ , $p>n/m$ ( $n$  is the dimension of the manifold). For smooth, positive data the equation has recently been solved by Székelyhidi and Zhang. We also give a stability result for such solutions.

Given $k\geqslant 2$ , we show that there are at most finitely many rational numbers $x$ and $y\neq 0$ and integers $\ell \geqslant 2$ (with $(k,\ell )\neq (2,2)$ ) for which

$$\begin{eqnarray}x(x+1)\cdots (x+k-1)=y^{\ell }.\end{eqnarray}$$

$\ell$

$(x,y,\ell )$

$y=0$

$\ell <\exp (3^{k})$

For an elliptic curve $E$ over a local field $K$ and a separable quadratic extension of $K$ , motivated by connections to the Birch and Swinnerton-Dyer conjecture, Kramer and Tunnell have conjectured a formula for computing the local root number of the base change of $E$ to the quadratic extension in terms of a certain norm index. The formula is known in all cases except some where $K$ is of characteristic $2$ , and we complete its proof by reducing the positive characteristic case to characteristic $0$ . For this reduction, we exploit the principle that local fields of characteristic $p$ can be approximated by finite extensions of $\mathbb{Q}_{p}$ : we find an elliptic curve $E^{\prime }$ defined over a $p$ -adic field such that all the terms in the Kramer–Tunnell formula for $E^{\prime }$ are equal to those for $E$ .

Fuchsian groups with a modular embedding have the richest arithmetic properties among non-arithmetic Fuchsian groups. But they are very rare, all known examples being related either to triangle groups or to Teichmüller curves. In Part I of this paper we study the arithmetic properties of the modular embedding and develop from scratch a theory of twisted modular forms for Fuchsian groups with a modular embedding, proving dimension formulas, coefficient growth estimates and differential equations. In Part II we provide a modular proof for an Apéry-like integrality statement for solutions of Picard–Fuchs equations. We illustrate the theory on a worked example, giving explicit Fourier expansions of twisted modular forms and the equation of a Teichmüller curve in a Hilbert modular surface. In Part III we show that genus two Teichmüller curves are cut out in Hilbert modular surfaces by a product of theta derivatives. We rederive most of the known properties of those Teichmüller curves from this viewpoint, without using the theory of flat surfaces. As a consequence we give the modular embeddings for all genus two Teichmüller curves and prove that the Fourier developments of their twisted modular forms are algebraic up to one transcendental scaling constant. Moreover, we prove that Bainbridge’s compactification of Hilbert modular surfaces is toroidal. The strategy to compactify can be expressed using continued fractions and resembles Hirzebruch’s in form, but every detail is different.

Let $X$ be a compact Kähler manifold, endowed with an effective reduced divisor $B=\sum Y_{k}$ having simple normal crossing support. We consider a closed form of $(1,1)$ -type $\unicode[STIX]{x1D6FC}$ on $X$ whose corresponding class $\{\unicode[STIX]{x1D6FC}\}$ is nef, such that the class $c_{1}(K_{X}+B)+\{\unicode[STIX]{x1D6FC}\}\in H^{1,1}(X,\mathbb{R})$ is pseudo-effective. A particular case of the first result we establish in this short note states the following. Let $m$ be a positive integer, and let $L$ be a line bundle on $X$ , such that there exists a generically injective morphism $L\rightarrow \bigotimes ^{m}T_{X}^{\star }\langle B\rangle$ , where we denote by $T_{X}^{\star }\langle B\rangle$ the logarithmic cotangent bundle associated to the pair $(X,B)$ . Then for any Kähler class $\{\unicode[STIX]{x1D714}\}$ on $X$ , we have the inequality

$$\begin{eqnarray}\displaystyle \int _{X}c_{1}(L)\wedge \{\unicode[STIX]{x1D714}\}^{n-1}\leqslant m\int _{X}(c_{1}(K_{X}+B)+\{\unicode[STIX]{x1D6FC}\})\wedge \{\unicode[STIX]{x1D714}\}^{n-1}.\end{eqnarray}$$

$X$

$Q$

$\bigotimes ^{m}T_{X}^{\star }\langle B\rangle$

$L$

$C\subset X$

$$\begin{eqnarray}\displaystyle \biggl(\frac{n^{m}-1}{n-1}-m\biggr)\int _{C}(c_{1}(K_{X}+B)+\{\unicode[STIX]{x1D6FC}\})-\frac{n^{m}-1}{n-1}\int _{C}\unicode[STIX]{x1D6FC}.\end{eqnarray}$$

We introduce techniques of Suslin, Voevodsky, and others into the study of singular varieties. Our approach is modeled after Goresky–MacPherson intersection homology. We provide a formulation of perversity cycle spaces leading to perversity homology theory and a companion perversity cohomology theory based on generalized cocycle spaces. These theories lead to conditions on pairs of cycles which can be intersected and a suitable equivalence relation on cocycles/cycles enabling pairings on equivalence classes. We establish suspension and splitting theorems, as well as a localization property. Some examples of intersections on singular varieties are computed.

Let $C$ be a smooth, separated and geometrically connected curve over a finitely generated field $k$ of characteristic $p\geqslant 0$ , $\unicode[STIX]{x1D702}$ the generic point of $C$ and $\unicode[STIX]{x1D70B}_{1}(C)$ its étale fundamental group. Let $f:X\rightarrow C$ be a smooth proper morphism, and $i\geqslant 0$ , $j$ integers. To the family of continuous $\mathbb{F}_{\ell }$ -linear representations $\unicode[STIX]{x1D70B}_{1}(C)\rightarrow \text{GL}(R^{i}f_{\ast }\mathbb{F}_{\ell }(j)_{\overline{\unicode[STIX]{x1D702}}})$ (where $\ell$ runs over primes $\neq p$ ), one can attach families of abstract modular curves $C_{0}(\ell )$ and $C_{1}(\ell )$ , which, in this setting, are the analogues of the usual modular curves $Y_{0}(\ell )$ and $Y_{1}(\ell )$ . If $i\not =2j$ , it is conjectured that the geometric and arithmetic gonalities of these abstract modular curves go to infinity with $\ell$ (for the geometric gonality, under a certain necessary condition). We prove the conjecture for the arithmetic gonality of the abstract modular curves $C_{1}(\ell )$ . We also obtain partial results for the growth of the geometric gonality of $C_{0}(\ell )$ and $C_{1}(\ell )$ . The common strategy underlying these results consists in reducing by specialization theory to the case where the base field $k$ is finite in order to apply techniques of counting rational points.