arXiv subjects
Publications and source records attributed to Alexandru Buium.
This is the second paper in a series devoted to developing an arithmetic PDE analogue of Riemannian geometry. In Part 1 arithmetic PDE analogues of Levi-Civita and Chern connections were introduced and studied. In this paper arithmetic analogues of curvature and characteristic classes are developed.
Let $p$ be a prime, let $N\geq 3$ be an integer prime to $p$, let $R$ be the ring of $p$-typical Witt vectors with coefficients in an algebraic closure of $\mathbb F_p$, and consider the correspondence $\mathcal A'_{g,1,N,R}\rightrightarrows \mathcal A_{g,1,N,R}$ obtained by taking the union of all prime to $p$ Hecke correspondences on Mumford's moduli scheme of principally polarized abelian schemes of relative dimension $g$ endowed with symplectic similitude level-$N$ structure over $R$-schemes. It is well-known that the coequalizer $\mathcal A_{g,1,N,R}/\mathcal A'_{g,1,N,R}$ of the above correspondence exists and is trivial in the category of schemes, i.e., is $\text{Spec}(R)$. We construct and study in detail such a coequalizer (categorical quotient) in a more refined geometry (category) referred to as {\it $δ$-geometry}. This geometry is in essence obtained from the usual algebraic geometry by equipping all $R$-algebras with {\it $p$-derivations}. In particular, we prove that our substitute of $\mathcal A_{g,1,N,R}/\mathcal A'_{g,1,N,R}$ in $δ$-geometry has the same `dimension' as $\mathcal A_{g,1,N,R}$, thus solving a main open problem in the work of Barcău--Buium. We also give applications to the study of various Zariski dense loci in $\mathcal A_{g,1,N,R}$ such as of isogeny classes and of points with complex multiplication. To prove our results we develop a Serre--Tate expansion theory for {\it Siegel $δ$-modular forms} of arbitrary genus which we then combine with old and new results from the geometric invariant theory of multiple quadratic forms and of multiple endomorphisms.
This is the first in a series on papers developing an arithmetic PDE analogue of Riemannian geometry. The role of partial derivatives is played by Fermat quotient operations with respect to several Frobenius elements in the absolute Galois group of a $p$-adic field. Existence and uniqueness of geodesics and of Levi-Civita and Chern connections are proved in this context. In a sequel to this paper a theory of arithmetic Riemannian curvature and characteristic classes will be developed.
A formalism of arithmetic partial differential equations (PDEs) is being developed in which one considers several arithmetic differentiations at one fixed prime. In this theory solutions can be defined in algebraically closed p-adic fields. As an application we show that for at least two arithmetic directions every elliptic curve possesses a non-zero arithmetic PDE Manin map of order 1; such maps do not exist in the arithmetic ODE case. Similarly we construct and study "genuinely PDE" differential modular forms. As further applications we derive a Theorem of the Kernel and a Reciprocity Theorem for arithmetic PDE Manin maps and also a finiteness Diophantine result for modular parameterizations. We also prove structure results for the spaces of "PDE differential modular forms defined on the ordinary locus." We also produce a system of differential equations satisfied by our PDE modular forms based on Serre and Euler operators.
We prove that the main examples in the theory of algebraic differential equations possess a remarkable total differential overconvergence property. This allows one to consider solutions to these equations with coordinates in algebraically closed fields.
Using arithmetic jet spaces, we attach perfectoid spaces to smooth schemes and to $δ$-morphisms of smooth schemes. We also study perfectoid spaces attached to arithmetic differential equations defined by some of the remarkable $δ$-morphisms appearing in the theory such as the $δ$-characters of elliptic curves and the $δ$-period map on modular curves.
This paper is part of a series of papers where an arithmetic analogue of classical differential geometry is being developed. In this arithmetic differential geometry functions are replaced by integer numbers, derivations are replaced by Fermat quotient operators, and connections (respectively curvature) are replaced by certain adelic (respectively global) objects attached to symmetric matrices with integral coefficients. Previous papers were devoted to an arithmetic analogue of the Chern connection. The present paper is devoted to an arithmetic analogue of the Levi-Civita connection.
We show that the $p$-adic completion of any affine elliptic curve with ordinary reduction possesses Frobenius lifts whose "normalized" action on $1$-forms preserves mod $p$ the space of invariant $1$-forms. We next show that, after removing the $2$-torsion sections, the above situation can be "infinitesimally deformed" in the sense that the above mod $p$ result has a mod $p^2$ analogue. While the "eigenvalues" mod $p$ are given by the reciprocal of the Hasse polynomial, the "eigenvalues" mod $p^2$ are given by an appropriate $\d$-modular function whose reciprocal is a $p$-adic deformation of the Hasse polynomial.
We show that if $G$ is a linear algebraic group over a number field and if $G$ is not a torus then for all but finitely many primes $p$ the $p$-adic completion of $G$ does not possess a Frobenius lift that is "Lie invariant mod $p$" (in the sense of \cite{alie1}). This is in contrast with the situation of elliptic curves studied in \cite{alie1}.
The paper reviews various arithmetic analogues of Hamiltonian systems and presents some new facts suggesting ways to relate/unify these examples.
In this note, we prove a finiteness result for fibers that are canonical lifts in a given elliptic fibration. The question was motivated by the authors' construction of an arithmetic Euler top, and it highlights an interesting discrepancy between the arithmetic and the classical case: in the former, it is impossible to extend the flows to a compactification of the phase space, viewed as an elliptic fibration over the space of action variables.
The theory of differential equations has an arithmetic analogue in which derivatives of functions are replaced by Fermat quotients of numbers. Many classical differential equations (Riccati, Weierstrass, Painlevé, etc.) were previously shown to possess arithmetic analogues. The paper introduces an arithmetic analogue of the Euler differential equations for the rigid body.
Starting with a symmetric/antisymmetric matrix with integer coefficients (which we view as an analogue of a metric/form on a principal bundle over the "manifold" Spec Z) we introduce arithmetic analogues of Chern connections and their curvature (in which usual partial derivative operators acting on functions are replaced by Fermat quotient operators acting on integer numbers); curvature is introduced via the method of "analytic continuation between primes" \cite{laplace}. We prove various non-vanishing, respectively vanishing results for curvature; morally, Spec Z will appear as "intrinsically curved." Along with \cite{adel1, adel2, adel3}, this theory can be viewed as taking first steps in developing a "differential geometry of Spec Z."
In a prequel to this paper \cite{curvature1} a notion of curvature on the integers was introduced, based on the technique of "analytic continuation between primes", introduced in \cite{laplace}. In this paper, which is essentially independent of its prequel, we introduce another notion of curvature on the integers, based on "algebraization of Frobenius lifts by correspondences." Our main results are vanishing/non-vanishing theorems for this new type of curvature in the case of "Chern connections" attached to classical groups.
Following recent work of the author, partly in collaboration with T. Dupuy and M. Barrett, we describe arithmetic analogues of some key concepts from Riemannian geometry such as: metrics, Chern connections, curvature, etc. Theorems are stated to the effect that the spectrum of the integers has a non-vanishing curvature.
Motivated by the search of a concept of linearity in the theory of arithmetic differential equations we introduce here an arithmetic analogue of Lie algebras and a concept of skew arithmetic differential cocycles. We will then construct such skew cocycles, based on certain remarkable lifts of Frobenius for the classical groups $GL_n, SL_n, SO_n, Sp_{n}$. The theory for $GL_n$, especially on the Galois side, will be further developed in a sequel to this paper.
Differential equations have arithmetic analogues in which derivatives are replaced by Fermat quotients; these analogues are called arithmetic differential equations and the present paper is concerned with the "linear" ones. The equations themselves were introduced in a previous paper. In the present paper we deal with the solutions of these equations as well as with the differential Galois groups attached to the solutions.
In a previous paper we attached to classical complex newforms $f$ of weight $2$ certain $δ_p$-modular forms $f^{\sharp}$ of order $2$ and weight $0$; the forms $f^{\sharp}$ can be viewed as "dual" to $f$ and played a key role in some of the applications of the theory. The aim of this paper is to provide a higher weight version of this "$\sharp$-duality," by attaching to classical newforms mod $p$, $\overline{f}$, of weight $κ$ between $3$ and $p$, $δ_π$-modular forms $f^{\sharp}$ of order $2$ and weight $-κ'$, with $κ'$ between $1$ and $p-2$.