Moduli interpretations for noncongruence modular curves

In this paper, we discuss Bagger–Witten line bundles over moduli spaces of SCFTs. We review how in general they are “fractional” line bundles, not honest line bundles, twisted on triple overlaps. We discuss the special case of moduli spaces of elliptic curves in detail. There, the Bagger–Witten line bundle does not exist as an ordinary line bundle, but rather is necessarily fractional. As a fractional line bundle, it is nontrivial (though torsion) over the uncompactified moduli stack, and its restriction to the interior, excising corners with enhanced stabilizers, is also fractional. It becomes an honest line bundle on a moduli stack defined by a quotient of the upper half plane by a metaplectic group, rather than [Formula: see text]. We review and compare to results of recent work arguing that well-definedness of the worldsheet metric implies that the Bagger–Witten line bundle admits a flat connection (which includes torsion bundles as special cases), and gives general arguments on the existence of universal structures on moduli spaces of SCFTs, in which superconformal deformation parameters are promoted to nondynamical fields ranging over the SCFT moduli space.

Each elliptic curve can be embedded uniquely in the projective plane, up to projective equivalence. The hessian curve of the embedding is generically a new elliptic curve, whose isomorphism type depends only on that of the initial elliptic curve. One gets like this a rational map from the moduli space of elliptic curves to itself. We call it the hessian dynamical system. We compute it in terms of the $j$-invariant of elliptic curves. We deduce that, seen as a map from a projective line to itself, it has 3 critical values, which correspond to the point at infinity of the moduli space and to the two elliptic curves with special symmetries. Moreover, it sends the set of critical values into itself, which shows that all its iterates have the same set of critical values. One gets like this a sequence of dessins d'enfants. We describe an algorithm allowing to construct this sequence.

The spin moduli spaceSg is the parameter space of theta characteristics (spin structures) on stable curves of genus g. It has two connected components, S g andS + g , depending on the parity of the spin structure. We establish a complete birational classication by Kodaira dimension of the odd componentS g of the spin moduli space. We show thatS g is uniruled for g < 12 and even unirational for g 8. In this range, introducing the concept of cluster for the Mukai variety whose one-dimensional linear sections are general canonical curves of genus g, we construct new birational models ofS g . These we then use to explicitly describe the birational structure of S g . For instance, S 8 is birational to a locally trivial P 7 -bundle over the moduli space of elliptic curves with seven pairs of marked points. For g 12, we prove thatS g is a variety of general type. In genus 12, this requires the construction of a counterexample to the Slope Conjecture on eective divisors on the moduli space of stable curves of genus 12.

The concept of moduli space was introduced by Riemann in his study of the conformal (or equivalently, complex) structures on a Riemann surface. Let us consider the simplest non-trivial case, namely, that of a Riemann surface of genus 1 or the torus T 2. The set of all complex structures \(\mathcal{C}({T}^{2})\) on the torus is an infinite-dimensional space acted on by the infinite-dimensional group Diff(T 2). The quotient space $$\mathcal{M}({T}^{2}) := \mathcal{C}({T}^{2})/\mathrm{Diff}({T}^{2})$$ is the moduli space of complex structures on T 2. Since T 2 with a given complex structure defines an elliptic curve, \(\mathcal{M}({T}^{2})\) is, in fact, the moduli space of elliptic curves. It is well known that a point ω = ω1 + iω2 in the upper half-plane H (ω2 > 0) determines a complex structure and is called the modulus of the corresponding elliptic curve. The modular group SL(2, Z) acts on H by modular transformations and we can identify \(\mathcal{M}({T}^{2})\) with H ∕ SL(2, Z). This is the reason for calling \(\mathcal{M}({T}^{2})\) the space of moduli of elliptic curves or simply the moduli space. The topology and geometry of the moduli space has rich structure. The natural boundary ω2 = 0 of the upper half plane corresponds to singular structures. Several important aspects of this classical example are also found in the moduli spaces of other geometric structures. Typically, there is an infinite-dimensional group acting on an infinite-dimensional space of geometric structures with quotient a “nice space” (for example, a finite-dimensional manifold with singularities). For a general discussion of moduli spaces arising in various applications see, for example, [193].

Given an odd prime p and a representation ϱ of the absolute Galois group of a number field k onto PGL 2 (𝔽 p ) with cyclotomic determinant, the moduli space of elliptic curves defined over k with p-torsion giving rise to ϱ consists of two twists of the modular curve X(p). We make here explicit the only genus-zero cases p=3 and p=5, which are also the only symmetric cases: PGL 2 (𝔽 p )≃𝒮 n for n=4 or n=5, respectively. This is done by studying the corresponding twisted Galois actions on the function field of the curve, for which a description in terms of modular units is given. As a consequence of this twisting process, we recover an equivalence between the ellipticity of ϱ and its principality, that is, the existence in its fixed field of an element α of degree n over k such that α and α 2 have both trace zero over k.

We study the essential minimum of the (stable) Faltings’ height on the moduli space of elliptic curves. We prove that, in contrast to the Weil height on a projective space and the Néron-Tate height of an abelian variety, Faltings’ height takes at least two values that are smaller than its essential minimum. We also provide upper and lower bounds for this quantity that allow us to compute it up to five decimal places. In addition, we give numerical evidence that there are at least four isolated values before the essential minimum. One of the main ingredients in our analysis is a good approximation of the hyperbolic Green function associated to the cusp of the modular curve of level one. To establish this approximation, we make an intensive use of distortion theorems for univalent functions. Our results have been motivated and guided by numerical experiments that are described in detail in the companion files.

We study the cobordism conjecture of McNamara and Vafa which asserts that the bordism group of quantum gravity is trivial. In the context of type IIB string theory compactified on a circle, this predicts the presence of D7-branes. On the other hand, the non-Abelian structure of the IIB duality group $SL(2,\mathbb{Z})$ implies the existence of additional $[p,q]$ 7-branes. We find that this additional information is instead captured by the space of closed paths on the moduli space of elliptic curves parameterizing distinct values of the type IIB axio-dilaton. This description allows to recover the full structure of non-Abelian braid statistics for 7-branes. Combining the cobordism conjecture with an earlier Swampland conjecture by Ooguri and Vafa, we argue that only certain congruence subgroups $\mathrm{\ensuremath{\Gamma}}\ensuremath{\subset}SL(2,\mathbb{Z})$ specifying genus zero modular curves can appear in 8D F-theory vacua. This leads to a successful prediction for the allowed Mordell--Weil torsion groups for 8D F-theory vacua.

We define a universal version of the Knizhnik–Zamolodchikov–Bernard (KZB) connection in genus 1. This is a flat connection over a principal bundle on the moduli space of elliptic curves with marked points. It restricts to a flat connection on configuration spaces of points on elliptic curves, which can be used for proving the formality of the pure braid groups on genus 1 surfaces. We study the monodromy of this connection and show that it gives rise to a relation between the KZ associator and a generating series for iterated integrals of Eisenstein forms. We show that the universal KZB connection is realized as the usual KZB connection for simple Lie algebras, and that in the $$\mathfrak {sl}_n$$ case this realization factors through the Cherednik algebras. This leads us to define a functor from the category of equivariant D-modules on $$\mathfrak {sl}_n$$ to that of modules over the Cherednik algebra, and to compute the character of irreducible equivariant D-modules over $$\mathfrak {sl}_n$$ that are supported on the nilpotent cone.

The universal elliptic KZB equation is the integrable connection on the pro-vector bundle over M_{1,2} whose fiber over the point corresponding to the elliptic curve E and a non-zero point x of E is the unipotent completion of \pi_1(E-{0},x). This was written down independently by Calaque, Enriquez and Etingof (arXiv:math/0702670), and by Levin and Racinet (arXiv:math/0703237). It generalizes the KZ-equation in genus 0. These notes are in four parts. The first two parts provide a detailed exposition of this connection (following Levin-Racinet); the third is a leisurely exploration of the connection in which, for example, we compute the limit mixed Hodge structure on the unipotent fundamental group of the Tate curve minus its identity. In the fourth part we elaborate on ideas of Levin and Racinet and explicitly compute the connection over the moduli space of elliptic curves with a non-zero abelian differential, showing that it is defined over Q.

We show an algorithm to draw the famous picture of Kleinian modular group PSL(2, ), which appears in describing the moduli space of elliptic curves.

On extra components in the functorial compactification of Ag.- On Mumford's uniformization and Neron models of Jacobians of semistable curves over complete rings.- Torelli theorem via Fourier-Mukai transform.- On the Andre-Oort conjecture for Hilbert modular surfaces.- Toroidal resolutions for some matrix singularities.- Formal Brauer groups and moduli of abelian surfaces.- Isogeny classes of abelian varieties with no principal polarizations.- Igusa's modular form and the classification of Siegel modular threefolds.- Mirror symmetry and quantization of abelian varieties.- Group schemes with additional structures and Weyl group cosets.- Moduli space of elliptic curves with Heisenberg level structure.- Singularities of the height strata in the moduli of K3 surfaces.- A stratification of a moduli space of abelian varieties.- Newton polygon strata in the moduli space of abelian varieties.- The dimension of Oort strata of Shimura varieties of PEL-type.- Hyperelliptic Jacobians and modular representations.- Windows for displays of p-divisible groups.

For each algebraic number [Formula: see text], a result of Habegger [P. Habegger, Singular moduli that are algebraic units, Algebra Number Theory 9(7) (2015) 1515–1524] shows that there are only finitely many singular moduli [Formula: see text] such that [Formula: see text] is an algebraic unit. His result uses Duke’s Equidistribution Theorem and is thus not effective. In this paper, we give an effective proof of Habegger’s result assuming that [Formula: see text] is not a singular modulus itself. We give an explicit bound, which depends only on [Formula: see text], on the discriminant [Formula: see text] associated with a singular modulus [Formula: see text] such that [Formula: see text] is a unit. This implies explicit bounds on the number of these singular moduli.

We establish equidistribution with respect to the bifurcation measure of postcritically finite (PCF) maps in any one-dimensional algebraic family of unicritical polynomials. Using this equidistribution result, together with a combinatorial analysis of certain algebraic correspondences on the complement of the Mandelbrot set M2 (or generalized Mandelbrot set Md for degree d>2), we classify all curves C⊂A2 defined over C with Zariski-dense subsets of points (a,b)∈C, such that both zd+a and zd+b are simultaneously PCF for a fixed degree d≥2. Our result is analogous to the famous result of André regarding plane curves which contain infinitely many points with both coordinates being complex multiplication parameters in the moduli space of elliptic curves and is the first complete case of the dynamical André–Oort phenomenon studied by Baker and DeMarco.

The moduli space of principally polarized abelian varieties of dimensiongadmits in positive characteristicpa stratification by the p-rank of the abelian variety, i.e. by the rank of the p-torsion. In the case of the moduli space of elliptic curves this is simply the stratification given by the (open and dense) ordinary locus and the zero-dimensional supersingular locus. But forg >2 this stratification is too coarse for many purposes. A refinement is provided by stratifying the moduli space according to the Newton polygon of an abelian variety. This stratification has been studied thoroughly by Oort ([OO1], [Oo2], see also [dJO]). He also suggested to study another stratification given by the isomorphism type of the p-torsion which we will call the Oort stratification in the sequel (often it is also called Ekedahl-Oort stratification). This stratification is also a refinement of the p-rank stratification.KeywordsModulus SpaceConjugacy ClassIsomorphism ClassAbelian VarietyNewton PolygonThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Once one has models for Hilbert modular varieties over (rings of integers of) number fields one can start investigating the arithmetic properties of Hilbert modular varieties. The wealth of results (and conjectures) for the moduli spaces of elliptic curves suggests that unexpected treasures lie buried in a largely unexplored territory.