Related papers: Ax-Schanuel for Shimura varieties

We give a new proof of quantum Shannon-McMillan theorem, extending it to AF $C^*$-systems. Our proof is based on the variational principle, instead of the classical Shannon-McMillan theorem.

In this paper we study the action of complex conjugation on Shimura varieties and the problem of descending these to the maximal totally real field of the reflex field. We prove the existence of such descent for many Shimura varieties whose…

We give a simple proof of a general result describing the action of the Zelevinsky involution on the cohomology of certain basic local Shimura varieties, using the machinery of Fargues-Scholze. As an application, we generalize earlier…

In this paper, we establish a Gauss-Bonnet-Chern theorem for general closed complex Finsler manifolds.

Let A be a Q-linear pseudo-abelian rigid tensor category. A notion of finiteness due to Kimura and (independently) O'Sullivan guarantees that the ideal of numerically trivial endomorphism of an object is nilpotent. We generalize this result…

In this paper we show that certain Shimura varieties, uniformized by the product of complex unit balls, can be p-adically uniformized by the product (of equivariant coverings) of Drinfeld upper half-spaces. We also extend a p-adic…

Building on results of Koll\'ar, we prove Shokurov's ACC Conjecture for log canonical thresholds on smooth varieties, and more generally, on varieties with quotient singularities.

We give a proof of the Morrison--Kawamata cone conjecture for abelian varieties. The proof is a straightforward deduction from well-known results on the real endomorphism algebra of an abelian variety and reduction theory for self-dual…

In this paper we apply Ax-Schanuel's Theorem to the ultraproduct of $p$-adic fields in order to get some results towards algebraic independence of $p$-adic exponentials for almost all primes $p$.

We prove an invariance of plurigenera for some foliated surface pairs of general type.

We prove the $S=T$ conjecture proposed by Xiao--Zhu in \cite{2017arXiv170705700X}, making use of Scholze's theory of diamonds and v-stacks and Fargues--Scholze's geometric Satake equivalence. Following \cite{2018arXiv180205299X}, we deduce…

In this paper we prove the equidistribution of certain families of special subvarieties in Kuga varieties, which is a special case of the general Andre-Oort conjecture formulated for mixed Shimura varieties proposed by R.Pink. Our approach…

We provide a complete enumeration of all quotients of genus $0, 1$ and $2$ of the Shimura curves $X_0^D(N)$ over $\mathbb{Q}$ by non-trivial subgroups of Atkin--Lehner involutions. For all $1270$ genus $1$ quotients $X$ with $N$ squarefree,…

In this article, we extend the van Hamel-Lichtenbaum duality theorem to (not necessarily smooth) proper and geometrically integral varieties defined over a $p$-adic field $k$. More precisely, we prove that for such variety $X$ there exists…

We prove two formulas in the style of the Gross-Zagier theorem, relating derivatives of L-functions to arithmetic intersection pairings on a unitary Shimura variety. We also prove a special case of Colmez's conjecture on the Faltings…

We study the cohomology of various local Shimura varieties for $GL_n$. This provides an explicit description of the spectral action constructed by Fargues-Scholze in certain cases and allows us to prove some strongly generic part of the…

In this note, we discuss several forms of the Eichler-Shimura relation for the compactly supported cohomology of local Shimura varieties, using the work of Fargues-Scholze.

Let X be an orthogonal Shimura variety, and let C_r(X) be the cone generated by the cohomology classes of orthogonal Shimura subvarieties in X of dimension r. We investigate the asymptotic properties of the generating rays of C_r(X) for…

We prove an injective version of Schanuel's lemma from homological algebra in the setting of exact categories.

In this paper we prove Shapiro's 1958 Conjecture on exponential polynomials, assuming Schanuel's Conjecture.