Related papers: Weak Modular Zilber-Pink with Derivatives

Mazur, Tate, and Teitelbaum gave a p-adic analogue of the Birch and Swinnerton-Dyer conjecture for elliptic curves. We provide a generalization of their conjecture in the good ordinary case to higher dimensional modular abelian varieties…

We propose a conjectural semiorthogonal decomposition for the derived category of the moduli space of stable rank 2 bundles with fixed determinant of odd degree, independently formulated by Narasimhan. We discuss some evidence for, and…

We give a construction of quasiminimal fields equipped with pseudo-analytic maps, generalising Zilber's pseudo-exponential function. In particular we construct pseudo-exponential maps of simple abelian varieties, including…

We present a graded modal type theory, a dependent type theory with grades that can be used to enforce various properties of the code. The theory has $\Pi$-types, weak and strong $\Sigma$-types, natural numbers, an empty type, and a…

Let D be a self-adjoint operator on a Hilbert space H and x a bounded operator on H. We say that x is n-times weakly D-differentiable, if for any pair of vectors a, b from H the function < exp(itD)x exp(-itD) a, b> is n-times…

We present two Dialectica-like constructions for models of intensional Martin-L\"of type theory based on G\"odel's original Dialectica interpretation and the Diller-Nahm variant, bringing dependent types to categorical proof theory. We set…

In this note we prove new cases of the Mumford-Tate conjecture by extending a theorem of Richard Pink for abelian varieties without nontrivial endomorphisms and with bad semistable reduction. We use quadratic pairs introduced by…

Let X be a smooth real algebraic variety. Let $\xi$ be a distribution on it. One can define the singular support of $\xi$ to be the singular support of the $D_X$-module generated by $\xi$ (some times it is also called the characteristic…

Let $\R$ be an alternative ring containing a nontrivial idempotent and $\D$ be a multiplicative Lie-type derivation from $\R$ into itself. Under certain assumptions on $\R$, we prove that $\D$ is almost additive. Let $p_n(x_1, x_2, \cdots,…

We discuss the relationships between the Andr\'e-Oort, Andr\'e-Pink-Zannier, and Mordell-Lang conjectures for Shimura varieties. We then combine the latter with the geometric Zilber-Pink conjecture to obtain some new results on unlikely…

A Drinfeld module has a $\mathfrak{p}$-adic Tate module not only for every finite place $\mathfrak{p}$ of the coefficient ring but also for $\mathfrak{p} = \infty$. This was discovered by J.-K. Yu in the form of a representation of the Weil…

In this short paper we discuss a number of effective and/or explicit results of Andre-Oort type for the nonholomorphic function "$\chi^*$", which I have discussed in a number of other papers. After working in a rather ad-hoc manner to get…

Theory of motivic superpolynomials is developed, including its extension to algebraic links colored by rows, relations to $L$-functions of plane curve singularities, the justification of the motivic versions of Weak Riemann Hypothesis, and…

In this paper, we study the action of diamond operators on Hilbert modular forms with coefficients in a general commutative ring. In particular, we generalize a result of Chai on the surjectivity of the constant term map for Hilbert modular…

The paper [GLZ] "L-functions of Carlitz modules, resultantal varieties and rooted binary trees" is devoted to a description of some resultantal varieties related to L-functions of Carlitz modules. It contains a conjecture that some of these…

We say that a function f defined on R or Qp has a well defined weak Mellin transform (or weak zeta integral) if there exists some function $M\_f(s)$ so that we have $Mell(\phi \star f,s) = Mell(\phi,s)M\_f(s)$ for all test functions $\phi$…

The universal Poincar\'e torsor, or more generally the universal semi-abelian scheme, can be viewed as a mixed Shimura variety. We give a classification of special subvarieties of the universal semi-abelian scheme of arbitrary toric rank.…

We complete the proof of a Siegel type statement for finitely generated $\Phi$-submodules of $\mathbb{G}_a$ under the action of a Drinfeld module $\Phi$.

Variational principles are proved for self-adjoint operator functions arising from variational evolution equations of the form \[ \langle\ddot{z}(t),y \rangle + \mathfrak{d}[\dot{z} (t), y] + \mathfrak{a}_0 [z(t),y] = 0. \] Here…

This paper is concerned with an elliptic system of Kirchhoff type, driven by the variable-order fractional $p(x)$-operator. With the help of the direct variational method and Ekeland variational principle, we show the existence of a weak…