Related papers: Reduction and lifting problem for differential for…

We explore extensions of tropical methods to arithmetic enumerative problems such as $\mathbb{A}^1$-enumeration with values in the Grothendieck-Witt ring, and rationality over Henselian valued fields, using bitangents to plane quartics as a…

We exploit three classical characterizations of smooth genus two curves to study their tropical and analytic counterparts. First, we provide a combinatorial rule to determine the dual graph of each algebraic curve and the metric structure…

We show that there exists a surjection from the set of effective divisors of degree $g$ on a tropical curve of genus $g$ to its Jacobian by using a tropical version of the Riemann-Roch theorem. We then show that the restriction of the…

In this paper, for any Shimura datum $(G,\mathcal{D})$ satisfying reasonable conditions so that many interesting cases satisfy, we prove some finiteness theorems for any graded vector space consisting of automorphic forms on $\mathcal{D}$…

We give an explicit description and calculate the dimension of the vector space of linear natural liftings of $p$-forms on $n$-dimensional manifolds $M$ to $q$-forms on $T^{{\mathbb D}^r_k}M$, where ${\mathbb D}^r_k$ is the Weil algebra of…

In this paper we consider the theta correspondence over a non-Archimedean local field. Using the homological method and the theory of derivatives, we show that under a mild condition the big theta lift is irreducible.

For a general inelastic Kac-like equation recently proposed, this paper studies the long-time behaviour of its probability-valued solution. In particular, the paper provides necessary and sufficient conditions for the initial datum in order…

Let X be a geometrically split, geometrically irreducible variety over a field F satisfying Rost nilpotence principle. Consider a field extension E/F and a finite field K. We provide in this note a motivic tool giving sufficient conditions…

The classical Shimura correspondence lifts automorphic representations on the double cover of $SL_2$ to automorphic representations on $PGL_2$. Here we take key steps towards establishing a relative trace formula that would give a new…

Let $C$ be a hyperelliptic curve over a local field $K$ with odd residue characteristic, defined by some affine Weierstrass equation $y^2=f(x)$. We assume that $C$ has semistable reduction and denote by $\mathcal{X} \rightarrow…

Given a transitive DG-Lie algebroid $(\mathcal{A}, \rho)$ over a smooth separated scheme $X$ of finite type over a field $\mathbb{K}$ of characteristic $0$ we define a notion of connection $\nabla \colon \mathbf{R}\Gamma(X,\mathrm{Ker}…

In 2010, Hrushovski--Loeser showed that the Berkovich analytification of a quasi-projective variety over a non-Archimedean valued field admits a deformation retraction onto a finite simplicial complex. In this article, we adapt the tools…

In this paper, we study the problem of tropicalizing tame degree three coverings of the projective line. Given any degree three covering $C\longrightarrow{\mathbb{P}^{1}}$, we give an algorithm that produces the Berkovich skeleton of $C$.…

We introduce a gauge-theoretic integer lift of the Rohlin invariant of a smooth 4-manifold X with the homology of $S^1 \times S^3$. The invariant has two terms; one is a count of solutions to the Seiberg-Witten equations on X, and the other…

Let $R$ be a commutative unital ring, $\textsf{X}$ a subshift, and $\widetilde{\mathcal{A}}_R(\textsf{X})$ the corresponding unital subshift algebra. We establish the reduction theorem for $\widetilde{\mathcal{A}}_R(\textsf{X})$. As a…

Let $(G,\tilde{G})$ be a reductive dual pair over a local field ${\Fontauri k}$ of characteristic 0, and denote by $V$ and $\tilde{V}$ the standard modules of $G$ and $\tilde{G}$, respectively. Consider the set $Max Hom(V,\tilde{V})$ of…

This is a follow-up paper of arXiv:1805.00115, where rational curves in surfaces that satisfy general positioned point and cross-ratio conditions were enumerated. A suitable correspondence theorem provided in arXiv:1509.07453 allowed us to…

Let $X$ and $Y$ be finite simplicial sets (e.g. finite simplicial complexes), both equipped with a free simplicial action of a finite group $G$. Assuming that $Y$ is $d$-connected and $\dim X\le 2d$, for some $d\geq 1$, we provide an…

Assume that $X$ is a surface over an algebraically closed field $k$. Let $\tilde{X}$ be obtained from $X$ by blowing up a smooth point and let $L$ be the exceptional curve. Let $\coh(X)$ be the category of coherent sheaves on $X$. In this…

Motivated by the study of periods of automorphic forms and relative trace formulae, we develop the theory of descent necessary to study orbital integrals arising in the fundamental lemma for a general class of symmetric spaces over a…