Related papers: Igusa class polynomials, embeddings of quartic CM …

Conditionally on a conjecture on the \'etale cohomology of Hilbert modular surfaces and some minor technical assumptions, we establish new instances of the equivariant BSD-conjecture in rank $0$ with applications to the arithmetic of…

Genus 2 curves are useful in cryptography for both discrete-log based and pairing-based systems, but a method is required to compute genus 2 curves such that the Jacobian has a given number of points. Currently, all known methods involve…

We construct a tangent-obstruction theory for Azumaya algebras equipped with a quadratic pair. Under the assumption that either 2 is a global unit or the algebra is of degree 2, we show how the deformation theory of these objects reduces to…

We obtain a combinatorial formula for the Miller-Morita-Mumford classes for the mapping class group of punctured surfaces and prove Witten's conjecture that they are proportional to the dual to the Witten cycles. The proportionality…

The aim of this paper is to show that using some natural curve arrangements in algebraic surfaces and Hirzebruch-Kummer covers one cannot construct new examples of ball-quotients, i.e., minimal smooth complex projective surfaces of general…

Ardila and Brugall\'e conjectured that double tropical Welschinger invariants of Hirzebruch surfaces are piecewise quasipolynomial. In this work, we prove the conjecture holds in full generality, i.e. for toric surfaces corresponding to…

Inspired by piecewise polynomiality results of double Hurwitz numbers, Ardila and Brugall\'e introduced an enumerative problem which they call double Gromov--Witten invariants of Hirzebruch surfaces. These invariants serve as a…

In these lecture notes we review the various relations between intersection theory on the moduli space of Riemann surfaces, integrable hierarchies of KdV type, matrix models, and topological quantum field theories. We explain in particular…

Let S be a nonsingular projective K3 surface. Motivated by the study of the Gromov-Witten theory of the Hilbert scheme of points of S, we conjecture a formula for the Gromov-Witten theory (in all curve classes) of the Calabi-Yau 3-fold S x…

We explore the applications of Lorentzian polynomials to the fields of algebraic geometry, analytic geometry and convex geometry. In particular, we establish a series of intersection theoretic inequalities, which we call rKT property, with…

In this paper, we introduce modular polynomials for the congruence subgroup $\Gamma_0(M)$ when $ X_0(M) $ has genus zero and therefore the polynomials are defined by a Hauptmodul of $ X_0(M) $. We show that the intersection number of two…

Let $M$ be the Shimura variety associated to the group of spinor similitudes of a quadratic space over $\mathbb{Q}$ of signature $(n,2)$. We prove a conjecture of Bruinier and Yang, relating the arithmetic intersection multiplicities of…

Chow rings of toric varieties, which originate in intersection theory, feature a rich combinatorial structure of independent interest. We survey four different ways of computing in these rings, due to Billera, Brion, Fulton--Sturmfels, and…

We prove that there is an algorithm to compute the class of the intersection of the divisor of schemes incident to a fixed line with any other class of a basis of the Chow ring $A^*(\mathbb{P}^{2[N]})$ due to Mallavibarrena and Sols. This…

The properties of the intersection algebra of two principal monomial ideals in a polynomial ring are investigated in detail. Results are obtained regarding the Hilbert series and the canonical ideal of the intersection algebra using methods…

We present a simplified formulation of open intersection numbers, as an alternative to the theory initiated by Pandharipande, Solomon and Tessler. The relevant moduli spaces consist of Riemann surfaces (either with or without boundary) with…

Feng-Yun-Zhang have proved a function field analogue of the arithmetic Siegel-Weil formula, relating special cycles on moduli spaces of unitary shtukas to higher derivatives of Eisenstein series. We prove an extension of this formula in a…

Iizuka's conjecture predicts that, given $m \in \mathbb{N}$ and a prime $p$, there exists infinitely many integers $n$ such that the class numbers of \textit{all} of the following quadratic number fields, \[ \mathbb{Q}(\sqrt{n}),\…

In their recent inspiring paper Mironov and Morozov claim a surprisingly simple expansion formula for the Kontsevich-Witten tau-function in terms of the Schur Q-functions. Here we provide a similar conjecture for the Br\'ezin-Gross-Witten…

Due to the orbifold singularities, the intersection numbers on the moduli space of curves $\bar{\sM}_{g,n}$ are in general rational numbers rather than integers. We study the properties of the denominators of these intersection numbers and…