Related papers: Arakelov-Green's functions for dynamical systems o…
We establish algebraicity criteria for formal germs of curves in algebraic varieties over number fields and apply them to derive a rationality criterion for formal germs of functions, which extends the classical rationality theorems of…
We show that on any abelian scheme over a complex quasi-projective smooth variety, there is a Green current for the zero-section, which is axiomatically determined up to $\partial$ and $\bar\partial$-exact differential forms. This current…
It follows from the Grothendieck-Ogg-Shafarevich formula that the rank of an abelian variety (with trivial trace) defined over the function field of a curve is bounded by a quantity which depends on the genus of the base curve and on bad…
Using an explicit version of the Mumford isomorphism on the moduli space of hyperelliptic curves we derive a closed formula for the Arakelov-Green function of a hyperelliptic Riemann surface evaluated at its Weierstrass points.
We introduce a new class of adelic heights on the projective line. We estimate their essential minimum and prove a result of equidistribution (at every place) for points of small height with estimates on the speed of convergence. To each…
We consider multiply periodic functions, sometimes called Abelian functions, defined with respect to the period matrices associated with classes of algebraic curves. We realise them as generalisations of the Weierstras P-function using two…
We use the differentiability of the arithmetic volume function and an arithmetic Bertini type theorem to classify when one can find a closed point on the generic fiber of an arithmetic variety, whose heights with respect to some finite…
Higher-abelian gauge theories associated with Cheeger-Simons differential characters are studied on compact manifolds without boundary. The paper consists of two parts: First the functional integral formulation based on zeta function…
We consider elliptic operators in divergence form with lower order terms of the form $Lu=-$div$\nabla u+bu)-c\nabla u-du$, in an open set $\Omega\subset \mathbb{R}^n$, $n\geq 3$, with possibly infinite Lebesgue measure. We assume that the…
By using the $\mathbb R$-filtration approach of Arakelov geometry, one establishes explicit upper bounds for geometric and arithmetic Hilbert-Samuel function for line bundles on projective varieties and hermitian line bundles on arithmetic…
We give new instances where Chabauty--Kim sets can be proved to be finite, by developing a notion of "generalised height functions" on Selmer varieties. We also explain how to compute these generalised heights in terms of iterated integrals…
The purpose of this article is to initiate Arakelov theory in a noncommutative setting. More precisely, we are concerned with Arakelov theory of noncommutative arithmetic curves. Our first main result is an arithmetic Riemann-Roch formula…
In [M. R\"osler and M. Voit. Integral Representation and Uniform Limits for Some Heckman-Opdam Hypergeometric Functions of type BC, Transactions of the American Mathematical Society, Vol. 368, No. 8, 6005-6032, 2016.], R\"osler and Voit…
We prove that every matrix-valued rational function $F$, which is regular on the closure of a bounded domain $\mathcal{D}_\mathbf{P}$ in $\mathbb{C}^d$ and which has the associated Agler norm strictly less than 1, admits a…
We study Hilbert functions of certain non-reduced schemes A supported at finite sets of points in projective space, in particular, fat point schemes. We give combinatorially defined upper and lower bounds for the Hilbert function of A using…
The A-hypergeometric system studied by I.M. Gelfand, M.I. Graev, A.V. Zelevinsky and the author, is defined for a set A of characters of an algebraic torus. In this paper we propose a generalization of the theory where the torus is replaced…
Suppose $\mathcal{E} \to B$ is a non-isotrivial elliptic surface defined over a number field, for smooth projective curve $B$. Let $k$ denote the function field $\overline{\mathbb{Q}}(B)$ and $E$ the associated elliptic curve over $k$. In…
The conical function and its relativistic generalization can be viewed as eigenfunctions of the reduced 2-particle Hamiltonians of the hyperbolic Calogero-Moser system and its relativistic generalization. We prove new product formulas for…
We construct Green functions of Dirichlet boundary value problems for sub-Laplacians on certain unbounded domains of a prototype Heisenberg-type group (prototype H-type group, in short). We also present solutions in an explicit form of the…
We study estimates of the Green's function in $\mathbb{R}^d$ with $d \ge 2$, for the linear second order elliptic equation in divergence form with variable uniformly elliptic coefficients. In the case $d \ge 3$, we obtain estimates on the…