arXiv subjects

Misha Gavrilovich

Publications and source records attributed to Misha Gavrilovich.

21 records · Page 2Linked to original sources

arXiv2010

A homotopy approach to set theory

We observe that the notion of two sets being equal up to finitely many elements is a homotopy equivalence relation in a model category, and suggest a homotopy-invariant variant of Generalised Continuum Hypothesis about which more can be proven within ZFC and which first appeared in PCF theory. The formalism allows to draw analogies between notions of set theory and those of homotopy theory, and we indeed observe a similarity between homotopy theory ideology/yoga and that of PCF theory. We also briefly discuss conjectural connections with model theory and arithmetics and geometry.

math.CT↗

arXiv2009

Covers of Abelian varieties as analytic Zariski structures

We use tools of mathematical logic to analyse the notion of a path on an complex algebraic variety, and are led to formulate a "rigidity" property of fundamental groups specific to algebraic varieties, as well as to define a bona fide topology closely related to etale topology. These appear as criteria for uncountable categoricity, or rather stability and homogeneity, of the formal countable language we propose to describe homotopy classes of paths on a variety, or equivalently, its universal covering space. We also show that, with this topology, the universal covering space of the variety is an analytic Zariski structure. Technically, we present a countable $L_{omega_1ω}$-sentence axiomatising a class of analytic Zariski structures containing the universal covering space of an algebraic variety over a number field, under some assumptions on the variety.

math.AG↗

arXiv2005

A remark on transitivity of Galois action on the set of uniquely divisible abelian extensions of the group of algebraic points of an elliptic curve, by $Z^2$

We study Galois action on $\Ext^1(E(\bar \Q),\Z^2)$ and interpret our results as partially showing that the notion of a path on a complex elliptic curve $E$ can be characterised algebraically. The proofs show that our results are just concise reformulations of Kummer theory for $E$ as well as the description of Galois action on the Tate module. Namely, we prove (a),(b) below by showing they are equivalent to (c) which is well-known: (a) Absolute Galois group acts transitively on the set of uniquely divisible abelian \EndE-module extensions of $E(\bar\Q)$ of algebraic points of an elliptic curve, by $Λ\cong\Z2$, (b) natural algebraic properties characterise uniquely the Poincare's fundamental groupoid of a complex elliptic curve, restricted to the algebraic points, (c) (Kummer theory) up to finite index, the image of the Galois action on the sequences $(P_i)_{i>0},jP_{ij}=P_i,i,j>0$ of points $P_i\in E^k(\bar\Q)$ is as large as possible with respect to linear relations between the coordinates of the points $P_i$'s. Our original motivations come from model theory; this paper presents results from the author's thesis.

math.NT↗