Related papers: A modular Szemeredi-Trotter theorem for hyperbolas

We establish a group-action version of the Szemer\'edi-Trotter theorem over any field, extending Bourgain's result for the group $\mathrm{SL}_2(k)$. As an Elekes-Szab\'o-type application, we obtain quantitative bounds on the number of…

We prove a Morse index theorem for action functionals on paths that are allowed to reflect at a hypersurface (either in the interior or at the boundary of a manifold). Both fixed and periodic boundary conditions are treated.

Let $f$ be a genus two cuspidal Siegel modular eigenform. We prove an adelic open image theorem for the compatible system of Galois representations associated to $f$, generalising the results of Ribet and Momose for elliptic modular forms.…

Let $C$ be an elliptic curve, $w\in C$, and let $S\subset C$ be a finite subset of cardinality at least $3$. We prove a Torelli type theorem for the moduli space of rank two parabolic vector bundles with determinant line bundle $\mathcal…

In this paper, we show Langton's type theorem on separatedness and properness of moduli functor of torsion free semistable sheaves on algebraic orbifolds over an algebraically closed field k

The Hilbert function of a module over a positively graded algebra is of quasi-polynomial type (Hilbert--Serre). We derive an upper bound for its grade, i.e. the index from which on its coefficients are constant. As an application, we give a…

We prove a genuine analogue of Wiener Tauberian theorem for hypergeometric transforms. As an application we prove analogue of Furstenberg theorem on Harmonic functions.

We give a stack-theoretic proof for some results on families of hyperelliptic curves.

We shall prove an extension of the semipositivity theorem for the case of reducible algebraic fiber spaces.

We construct moduli spaces of objects in an abelian category satisfying some finiteness hypotheses. Our approach is based on the work of Artin-Zhang and the intrinsic construction of moduli spaces for stacks developed by…

We develop a theory of modulus triples, for future motivic applications.

We prove non-autonomous maximal $L^p$-regularity results on UMD spaces replacing the common H\"older assumption by a weaker fractional Sobolev regularity in time. This generalizes recent Hilbert space results by Dier and Zacher. In…

In this note, we give a short proof of the Torelli theorem for cubic fourfolds that relies on the global Torelli theorem for irreducible holomorphic symplectic varieties proved by Verbitsky.

We prove an analog of the Szemer\'edi-Trotter theorem in the plane for definable curves and points in any o-minimal structure over an arbitrary real closed field $\mathrm{R}$. One new ingredient in the proof is an extension of the well…

Let $A$ be a subset of a finite field $F := \Z/q\Z$ for some prime $q$. If $|F|^\delta < |A| < |F|^{1-\delta}$ for some $\delta > 0$, then we prove the estimate $|A+A| + |A.A| \geq c(\delta) |A|^{1+\eps}$ for some $\eps = \eps(\delta) > 0$.…

We study polynomials interpolating the (rational) constant terms of certain meromorphic modular forms for Hecke groups. We make observations about the divisibility properties of the constant terms and connect them to several sequences, for…

We prove sharp estimates for incidences involving planar tubes that satisfy packing conditions. We apply them to improve the estimates for the Fourier transform of fractal measures supported on planar curves.

Let $E/\mathbb{Q}$ be an elliptic curve, $p$ a prime and $K_{\infty}/K$ the anticyclotomic $\mathbb{Z}_p$-extension of a quadratic imaginary field $K$ satisfying the Heegner hypothesis. In this paper we give a new proof to a theorem of…

We apply tilting theory over preprojective algebras $Lambda$ to a study of moduli space of $Lambda$-modules. We define the categories of semistable modules and give an equivalence, so-called reflection functors, between them by using…

We establish results with an arithmetic flavor that generalize the polynomial multidimensional Szemeredi theorem and related multiple recurrence and convergence results in ergodic theory. For instance, we show that in all these statements…