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Perturbations, deformations, and variations (and “near-misses”) in geometry, physics, and number theory. (English) Zbl 1057.11033

The paper under review takes the reader to a fascinating realm of mathematical ideas, dubbed as Basic Notions in Mathematics. The central theme of this article is “perturbations, deformations, variations and near-misses” in geometry, physics and number theory. The topics chosen here spread over currently very active fields of research in mathematics and mathematical physics. Reflecting this, the paper covers vast grounds ranging from geometry, number theory, mirror symmetry and mathematical physics. It is written in a very friendly manner with numerous illustrating examples. It is highly recommended for anyone who is interested in these basic notions. The large portion of the article is devoted to the discussion of moduli spaces: parameter spaces describing deformations of mathematical structures, and ultimately, the classification of the mathematical structures in question.
To discuss how a specific geometrical object reacts to perturbations, the notion of stability is introduced. A starting example is taken from the theory of singularities of curves in the plane: The stability of nodal singularities versus the instability of a cuspidal singularity. Another example is Zariski’s discovery that there are two types of plane sextic curves with six cuspidal singularities.
The notion of general-ness in mathematics is a rather elusive notion. In this article, terminologies like general, general member, very general and generic are defined in the context of algebraic geometry (e.g., with respect to subvarieties).
Deformations of Riemann surfaces, and moduli spaces of all Riemann surfaces of given genus are discussed from various points of view. To begin with, Riemann’s results are recalled: the Riemann sphere is stable as a complex manifold, and there are \(6g-6\) real parameters of possible deformations of the complex analytic structure on a Riemann surface \(S\) of genus \(g>1\).
There are a number of different ways of deforming Riemann surfaces, and four different approaches are presented here. Each of these approaches gives rise to different structures on the moduli space, \({\mathcal M}_g\), of all Riemann surfaces of genus \(g\), e.g., on the one hand, \({\mathcal M}_g\) may be viewed as a complex algebraic variety of (complex) dimension \(3g-3\), and on the other hand, the quotient of the complex analytic Teichmüller space \({\mathcal T}_g\) by the properly discontinuous action of the (discrete) mapping class group.
A linearization of the problem is a useful and powerful tool when it is available. One instance discussed here is the Hodge theory of algebraic varieties, in particular, that of Riemann surfaces. Attached to any Riemann surface \(S\) of genus \(g\), there are the complex vector space (of dimension \(g\)) of holomorphic differential \(1\)-forms which is a nondegenerate Hermitian space, and its dual vector space which is endowed with a rank \(2g\) lattice structure. The upshot of this linearlization process is that the isomorphism class of \(S\) is determined by the Hermitian structure on the complex \(g\)-dimensional vector space and the rank \(2g\) lattice in its dual.
This puts the spotlight on the question of classifying lattices in complex vector spaces. First discussion along this line is concentrated on classification of lattices in \({\mathbb R}^N\) such that the volume of their fundamental domain is normalized to be \(1\), up to rotational equivalence. This leads to the homogeneous space \(X_N:=\text{SL}_N({\mathbb R})/ \text{SO}_N({\mathbb R})\), on which \(\Gamma_N:= \text{SL}_N({\mathbb Z})\) acts by left-multiplication. The quotient space \(X_N/\Gamma_N\) is the “moduli space” of the equivalence classes of lattices, which is of dimension \((N^2+N-2)/2\). The case \(N=2\) is discussed in great detail, that there are one-to-one correspondences between
\(\bullet\) isomorphism classes of elliptic curves over \({\mathbb C}\),
\(\bullet\) isomorphism classes of Riemann surfaces of genus \(1\),
\(\bullet\) lattices \(L\subset {\mathbb C}\) up to complex multiplication, and
\(\bullet\) complex numbers \(j(L)\).
The last set is the coarse moduli space for elliptic curves over \({\mathbb C}\). The moduli space for elliptic curves can then be obtained by adding certain auxiliary data. The modular curves \(X_0(N)\), which has a model defined over \({\mathbb Q}\), play a central role in the classification problems of elliptic curves. First, the points on \(Y_0(N)\subset X_0(N)\) classify isomorphism classes of complex elliptic curves with chosen cyclic subgroup of order \(N\). Then the complex structure of \(X_0(N)\) gives parameters for all deformations of elliptic curves over \({\mathbb C}\) with this auxiliary data, and all elliptic curves over \({\mathbb Q}\) are realized as some quotients of \(X_0(N)\) (the modularity theorem for elliptic curves over \({\mathbb Q}\)).
The most amazing application of modular curves is that they describe the universal deformations of Galois representations, namely, the universal parameterizations of certain types of continuous representations of Galois groups of number fields. First the classical results are recalled, aligned with modern treatments. Class Field Theory asserts (a classical result of Kronecker-Weber) that finite abelian extensions of \({\mathbb Q}\) are generated by the coordinates of the points of finite order in the algebraic group \({\mathbb C}^*=\text{GL}_1\). A Kronecker’s generalization of this result to quadratic number fields asserts that a certain large class of abelian extensions of imaginary quadratic fields \(K\) is generated by the values of elliptic modular function \(j(z)\) where \(z\) runs through elements \(\alpha\in K\), or in modern formulation, these abelian extensions are generated by the coordinates of torsion points in certain elliptic curves with complex multiplication by \(K\).
Now take \(K\) to a more general number field, ask the same question of finding all abelian extensions of \(K\). Let \(G_K\) denote the Galois group \(K\). Let \(k={\mathbb Z}/p{\mathbb Z}\) be a finite field, and let \(R\) be a complete local Noetherian ring with residue field \(k\). If \(\bar {\rho}: G_K\to \text{GL}_n(R)\) is a continuous homomorphism (called a residual representation), a deformation of \(\bar{\rho}\) to \(R\) is defined to be a lifting \(\rho: G_K\to \text{GL}_n(R)\) where \(\rho\) is taken only up to conjugation by an element of \(\text{GL}_n(R)\) trivial when reduced to \(k\). A reformulated question then is to describe all deformations of the given residual deformations. For the sake of simplicity, assume \(\bar{\rho}\) is absolutely, and consider only deformations of \(\bar{\rho}\) which are unramified at all primes not in some fixed finite set \(S\) of primes of \(K\). It is a theorem that a universal deformation of \(\bar{\rho}\) with the specified ramification condition indeed exists. Precisely, this means that there is a (unique) complete local Noetherian ring \(R\) with residue field \(k\) with a continuous homomorphism \(\rho: G_K\to \text{GL}_n(R)\) which is a deformation of \(\bar{\rho}\) unramified outside \(S\) with the property that any other deformation \(\rho_0\) of \(\bar{\rho}\) to some complete local Noetherian ring \(R_0\) with residue field \(k\) is obtained from a unique ring homomorphism \(R\to R_0\).
The next natural question is: can one explicitly generate such universal deformation rings \(R\) for the \(\bar{\rho}\)?
There is a conjecture due to Serre where to look for such deformation rings, which involves torsion points of Jacobians of modular curves.
There are some interludes to mathematical physics, e.g., mirror symmetry for elliptic curves, quantizations and non-commutative geometry. Mirror symmetry for elliptic curves is an exposition of R. Dijkgraaf’s article [“Mirror symmetry and elliptic curves”, Dijkgraaf, R. H. (ed.) et al., The moduli space of curves. Proceedings of the conference held on Texel Island, 1994. Basel: Birkhäuser. Prog. Math. 129, 149–163 (1995; Zbl 0913.14007)], and describes the number of simply ramified covers of a marked elliptic curve in terms of quasi-modular forms.
The problem of quantization of a classical algebra of observables may viewed as the problem of associating to a commutative Poisson algebra \(A_o\) over \({\mathbb C}\) a deformation of \(A_o\). Which Poisson algebra can be quantized in this sense? Inspired by Quantum Field Theory, Kontsevich provides such quantum deformation to any Poisson manifold.
The final section discusses perturbational questions appearing in algebra, that is, “near-misses” questions.
\(\bullet\) How close can you get to \(\sqrt{d}\) by rationals whose denominators have restricted size? Pell’s equation is visited here in this connection.
\(\bullet\) Given a curve in the plane cut out by a polynomial, how near to it can a lattice point get and still miss? The Hall problem, the conjectures of Vojta, and the ABC conjecture are treated here.
\(\bullet\) Given a smooth curve in the plane, how near to it can a point with rational coordinates get and still miss?
Here lattice reduction algorithms and its applications are discussed focusing on N. D. Elkies’ article [Rational points near curves and small nonzero \(| x^3-y^2| \) via lattice reduction, Bosma, Wieb (ed.), Algorithmic number theory. 4th international symposium. ANTS-IV, Leiden, 2000. Proceedings. Berlin: Springer. Lect. Notes Comput. Sci. 1838, 33–63 (2000; Zbl 0991.11072)].

MSC:

11G18 Arithmetic aspects of modular and Shimura varieties
11-02 Research exposition (monographs, survey articles) pertaining to number theory
11F23 Relations with algebraic geometry and topology
11J68 Approximation to algebraic numbers
14G05 Rational points
53D55 Deformation quantization, star products
58K40 Classification; finite determinacy of map germs
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