Related papers: On integrality of $p$-adic iterated integrals

We survey our recent work on an extension of the theory of motivic integration, called arithmetic motivic integration. We developed this theory to understand how p-adic integrals of a very general type depend on p.

In this note we investigate the $p$-degree function of elliptic curves over the field $\mathbb{Q}_p$ of $p$-adic numbers. The $p$-degree measures the least complexity of a non-zero $p$-torsion point on an elliptic curve. We prove some…

In this note, basing on a certain functional equation of the dilogarithm function, we establish nontrivial lower bounds for the $p$-adic valuation (where $p$ is a given prime number) of some type of rational numbers involving harmonic…

The purpose of this paper is: 1) to explain the Seiberg-Witten invariants, 2) to show that - on a K\"ahler surface - the solutions of the monopole equations can be interpreted as algebraic objects, namely effective divisors, 3) to give - as…

In this paper, we introduce an algorithm for computing p-adic integrals on bad reduction hyperelliptic curves. For bad reduction curves, there are two notions of p-adic integration: Berkovich-Coleman integrals which can be performed…

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with conductor $N$ and $p\nmid 2N$ a prime. Let $L$ be an imaginary quadratic field with $p$ split. We prove the existence of $p$-adic zeta element for $E$ over $L$, encoding two…

We give a simple proof of the cobordism invariance of the index of an elliptic operator. The proof is based on a study of a Witten-type deformation of an extension of the operator to a complete Riemannian manifold. One of the advantages of…

We describe an algorithm to compute the local Coleman-Gross p-adic height at p on a hyperelliptic curve. Previously, this was only possible using an algorithm due to Balakrishnan and Besser, which was limited to odd degree. While we follow…

We introduce iterated beta integrals, a new class of iterated integrals on the universal abelian covering of the punctured projective line that unifies hyperlogarithms and classical beta integrals while preserving their fundamental…

Let p > 2 be a prime. Let Q(zeta) be the p-cyclotomic field. Let pi be the prime ideal of Q(zeta) lying over p. This article aims to describe some pi-adic congruences characterizing the structure of the p-class group and of the unit group…

The notion of classical $p$-adic integrable system on a $p$-adic symplectic manifold was proposed by Voevodsky, Warren and the second author a decade ago in analogy with the real case. In the present paper we introduce and study, from the…

We give a formula for the component at p of the p-adic height pairing of a divisor of degree 0 on a hyperelliptic curve. We use this to give a Chabauty-like method for finding p-adic approximations to p-integral points on such curves when…

Using the $3D$ mirror symmetry we construct a system of polynomials $T_s(z)$ with integral coefficients which solve the quantum differential equitation of $X=T^{*} Gr(k,n)$ modulo $p^s$, where $p$ is a prime number. We show that the…

We construct the p-adic zeta function for a one-dimensional (as a p-adic Lie extension) non-commutative p-extension of a totally real number field such that the finite part of its Galois group is a pgroup with exponent p. We first calculate…

Let $p$ be an odd prime number. We propose an algorithm for computing rational representations of isogenies between Jacobians of hyperelliptic curves via-adic differential equations with a sharp analysis of the loss of precision.…

We generalize a result of Kostant and Wallach concerning the algebraic integrability of the Gelfand-Zeitlin vector fields to the full set of strongly regular elements in $gl(n,\mathbb{C})$. We use decomposition classes to stratify the…

This paper is a sequel to our earlier paper "Wach modules and Iwasawa theory for modular forms" (arXiv: 0912.1263), where we defined a family of Coleman maps for a crystalline representation of the Galois group of Qp with nonnegative…

Let $r$ be a non-zero rational number. In a paper in the Transactions of the AMS in 2023, O'Desky and Richman gave a construction of a $p$-adic incomplete gamma-function $\Gamma_p(\cdot,r)$ for each prime $p$ for which $|r - 1|_p < 1$.…

We investigate the arithmetic nature of P-recursive sequences through the lens of their D-finite generating functions. Building on classical tools from differential algebra, we revisit the integrality criterion for Motzkin-type sequences…

We generalize Siegel's theorem on integral points on affine curves to integral points of bounded degree, giving a complete characterization of affine curves with infinitely many integral points of degree d or less over some number field.…