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51. CJM 2011 (vol 64 pp. 409)

Rainer, Armin
Lifting Quasianalytic Mappings over Invariants
Let $\rho \colon G \to \operatorname{GL}(V)$ be a rational finite dimensional complex representation of a reductive linear algebraic group $G$, and let $\sigma_1,\dots,\sigma_n$ be a system of generators of the algebra of invariant polynomials $\mathbb C[V]^G$. We study the problem of lifting mappings $f\colon \mathbb R^q \supseteq U \to \sigma(V) \subseteq \mathbb C^n$ over the mapping of invariants $\sigma=(\sigma_1,\dots,\sigma_n) \colon V \to \sigma(V)$. Note that $\sigma(V)$ can be identified with the categorical quotient $V /\!\!/ G$ and its points correspond bijectively to the closed orbits in $V$. We prove that if $f$ belongs to a quasianalytic subclass $\mathcal C \subseteq C^\infty$ satisfying some mild closedness properties that guarantee resolution of singularities in $\mathcal C$, e.g., the real analytic class, then $f$ admits a lift of the same class $\mathcal C$ after desingularization by local blow-ups and local power substitutions. As a consequence we show that $f$ itself allows for a lift that belongs to $\operatorname{SBV}_{\operatorname{loc}}$, i.e., special functions of bounded variation. If $\rho$ is a real representation of a compact Lie group, we obtain stronger versions.

Keywords:lifting over invariants, reductive group representation, quasianalytic mappings, desingularization, bounded variation
Categories:14L24, 14L30, 20G20, 22E45

52. CJM 2011 (vol 64 pp. 81)

David, C.; Wu, J.
Pseudoprime Reductions of Elliptic Curves
Let $E$ be an elliptic curve over $\mathbb Q$ without complex multiplication, and for each prime $p$ of good reduction, let $n_E(p) = | E(\mathbb F_p) |$. For any integer $b$, we consider elliptic pseudoprimes to the base $b$. More precisely, let $Q_{E,b}(x)$ be the number of primes $p \leq x$ such that $b^{n_E(p)} \equiv b\,({\rm mod}\,n_E(p))$, and let $\pi_{E, b}^{\operatorname{pseu}}(x)$ be the number of compositive $n_E(p)$ such that $b^{n_E(p)} \equiv b\,({\rm mod}\,n_E(p))$ (also called elliptic curve pseudoprimes). Motivated by cryptography applications, we address the problem of finding upper bounds for $Q_{E,b}(x)$ and $\pi_{E, b}^{\operatorname{pseu}}(x)$, generalising some of the literature for the classical pseudoprimes to this new setting.

Keywords:Rosser-Iwaniec sieve, group order of elliptic curves over finite fields, pseudoprimes
Categories:11N36, 14H52

53. CJM 2011 (vol 63 pp. 1058)

Easton, Robert W.
$S_3$-covers of Schemes
We analyze flat $S_3$-covers of schemes, attempting to create structures parallel to those found in the abelian and triple cover theories. We use an initial local analysis as a guide in finding a global description.

Keywords:nonabelian groups, permutation group, group covers, schemes
Category:14L30

54. CJM 2011 (vol 63 pp. 992)

Bruin, Nils; Doerksen, Kevin
The Arithmetic of Genus Two Curves with (4,4)-Split Jacobians
In this paper we study genus $2$ curves whose Jacobians admit a polarized $(4,4)$-isogeny to a product of elliptic curves. We consider base fields of characteristic different from $2$ and $3$, which we do not assume to be algebraically closed. We obtain a full classification of all principally polarized abelian surfaces that can arise from gluing two elliptic curves along their $4$-torsion, and we derive the relation their absolute invariants satisfy. As an intermediate step, we give a general description of Richelot isogenies between Jacobians of genus $2$ curves, where previously only Richelot isogenies with kernels that are pointwise defined over the base field were considered. Our main tool is a Galois theoretic characterization of genus $2$ curves admitting multiple Richelot isogenies.

Keywords:Genus 2 curves, isogenies, split Jacobians, elliptic curves
Categories:11G30, 14H40

55. CJM 2011 (vol 63 pp. 1388)

Misamore, Michael D.
Nonabelian $H^1$ and the Étale Van Kampen Theorem
Generalized étale homotopy pro-groups $\pi_1^{\operatorname{ét}}(ċ{C}, x)$ associated with pointed, connected, small Grothendieck sites $(\mathcal{C}, x)$ are defined, and their relationship to Galois theory and the theory of pointed torsors for discrete groups is explained.
Applications include new rigorous proofs of some folklore results around $\pi_1^{\operatorname{ét}}(ét(X), x)$, a description of Grothendieck's short exact sequence for Galois descent in terms of pointed torsor trivializations, and a new étale van Kampen theorem that gives a simple statement about a pushout square of pro-groups that works for covering families that do not necessarily consist exclusively of monomorphisms. A corresponding van Kampen result for Grothendieck's profinite groups $\pi_1^{\mathrm{Gal}}$ immediately follows.

Keywords:étale homotopy theory, simplicial sheaves
Categories:18G30, 14F35

56. CJM 2011 (vol 63 pp. 755)

Chu, Kenneth C. K.
On the Geometry of the Moduli Space of Real Binary Octics
The moduli space of smooth real binary octics has five connected components. They parametrize the real binary octics whose defining equations have $0,\dots,4$ complex-conjugate pairs of roots respectively. We show that each of these five components has a real hyperbolic structure in the sense that each is isomorphic as a real-analytic manifold to the quotient of an open dense subset of $5$-dimensional real hyperbolic space $\mathbb{RH}^5$ by the action of an arithmetic subgroup of $\operatorname{Isom}(\mathbb{RH}^5)$. These subgroups are commensurable to discrete hyperbolic reflection groups, and the Vinberg diagrams of the latter are computed.

Keywords:real binary octics, moduli space, complex hyperbolic geometry, Vinberg algorithm
Categories:32G13, 32G20, 14D05, 14D20

57. CJM 2011 (vol 63 pp. 878)

Howard, Benjamin; Manon, Christopher; Millson, John
The Toric Geometry of Triangulated Polygons in Euclidean Spac
Speyer and Sturmfels associated Gröbner toric degenerations $\mathrm{Gr}_2(\mathbb{C}^n)^{\mathcal{T}}$ of $\mathrm{Gr}_2(\mathbb{C}^n)$ with each trivalent tree $\mathcal{T}$ having $n$ leaves. These degenerations induce toric degenerations $M_{\mathbf{r}}^{\mathcal{T}}$ of $M_{\mathbf{r}}$, the space of $n$ ordered, weighted (by $\mathbf{r}$) points on the projective line. Our goal in this paper is to give a geometric (Euclidean polygon) description of the toric fibers and describe the action of the compact part of the torus as "bendings of polygons". We prove the conjecture of Foth and Hu that the toric fibers are homeomorphic to the spaces defined by Kamiyama and Yoshida.

Categories:14L24, 53D20

58. CJM 2011 (vol 63 pp. 616)

Lee, Edward
A Modular Quintic Calabi-Yau Threefold of Level 55
In this note we search the parameter space of Horrocks-Mumford quintic threefolds and locate a Calabi-Yau threefold that is modular, in the sense that the $L$-function of its middle-dimensional cohomology is associated with a classical modular form of weight 4 and level 55.

Keywords: Calabi-Yau threefold, non-rigid Calabi-Yau threefold, two-dimensional Galois representation, modular variety, Horrocks-Mumford vector bundle
Categories:14J15, 11F23, 14J32, 11G40

59. CJM 2011 (vol 63 pp. 481)

Baragar, Arthur
The Ample Cone for a K3 Surface
In this paper, we give several pictorial fractal representations of the ample or Kähler cone for surfaces in a certain class of $K3$ surfaces. The class includes surfaces described by smooth $(2,2,2)$ forms in ${\mathbb P^1\times\mathbb P^1\times \mathbb P^1}$ defined over a sufficiently large number field $K$ that have a line parallel to one of the axes and have Picard number four. We relate the Hausdorff dimension of this fractal to the asymptotic growth of orbits of curves under the action of the surface's group of automorphisms. We experimentally estimate the Hausdorff dimension of the fractal to be $1.296 \pm .010$.

Keywords:Fractal, Hausdorff dimension, K3 surface, Kleinian groups, dynamics
Categories:14J28, , , , 14J50, 11D41, 11D72, 11H56, 11G10, 37F35, 37D05

60. CJM 2010 (vol 63 pp. 86)

Chen, Xi
On Vojta's $1+\varepsilon$ Conjecture
We give another proof of Vojta's $1+\varepsilon$ conjecture.

Keywords:Vojta, 1+epsilon
Categories:14G40, 14H15

61. CJM 2010 (vol 62 pp. 1293)

Kasprzyk, Alexander M.
Canonical Toric Fano Threefolds
An inductive approach to classifying all toric Fano varieties is given. As an application of this technique, we present a classification of the toric Fano threefolds with at worst canonical singularities. Up to isomorphism, there are $674,\!688$ such varieties.

Keywords:toric, Fano, threefold, canonical singularities, convex polytopes
Categories:14J30, 14J30, 14M25, 52B20

62. CJM 2010 (vol 62 pp. 1201)

Alzati, Alberto; Besana, Gian Mario
Criteria for Very Ampleness of Rank Two Vector Bundles over Ruled Surfaces
Very ampleness criteria for rank $2$ vector bundles over smooth, ruled surfaces over rational and elliptic curves are given. The criteria are then used to settle open existence questions for some special threefolds of low degree.

Keywords:vector bundles, very ampleness, ruled surfaces
Categories:14E05, 14J30

63. CJM 2010 (vol 62 pp. 1131)

Kleppe, Jan O.
Moduli Spaces of Reflexive Sheaves of Rank 2
Let $\mathcal{F}$ be a coherent rank $2$ sheaf on a scheme $Y \subset \mathbb{P}^{n}$ of dimension at least two and let $X \subset Y$ be the zero set of a section $\sigma \in H^0(\mathcal{F})$. In this paper, we study the relationship between the functor that deforms the pair $(\mathcal{F},\sigma)$ and the two functors that deform $\mathcal{F}$ on $Y$, and $X$ in $Y$, respectively. By imposing some conditions on two forgetful maps between the functors, we prove that the scheme structure of \emph{e.g.,} the moduli scheme ${\rm M_Y}(P)$ of stable sheaves on a threefold $Y$ at $(\mathcal{F})$, and the scheme structure at $(X)$ of the Hilbert scheme of curves on $Y$ become closely related. Using this relationship, we get criteria for the dimension and smoothness of $ {\rm M_{Y}}(P)$ at $(\mathcal{F})$, without assuming $ {\textrm{Ext}^2}(\mathcal{F} ,\mathcal{F} ) = 0$. For reflexive sheaves on $Y=\mathbb{P}^{3}$ whose deficiency module $M = H_{*}^1(\mathcal{F})$ satisfies $ {_{0}\! \textrm{Ext}^2}(M ,M ) = 0 $ (\emph{e.g.,} of diameter at most 2), we get necessary and sufficient conditions of unobstructedness that coincide in the diameter one case. The conditions are further equivalent to the vanishing of certain graded Betti numbers of the free graded minimal resolution of $H_{*}^0(\mathcal{F})$. Moreover, we show that every irreducible component of ${\rm M}_{\mathbb{P}^{3}}(P)$ containing a reflexive sheaf of diameter one is reduced (generically smooth) and we compute its dimension. We also determine a good lower bound for the dimension of any component of ${\rm M}_{\mathbb{P}^{3}}(P)$ that contains a reflexive stable sheaf with ``small'' deficiency module $M$.

Keywords:moduli space, reflexive sheaf, Hilbert scheme, space curve, Buchsbaum sheaf, unobstructedness, cup product, graded Betti numbers.xdvi
Categories:14C05, qqqqq14D22, 14F05, 14J10, 14H50, 14B10, 13D02, 13D07

64. CJM 2010 (vol 62 pp. 1246)

Chaput, P. E.; Manivel, L.; Perrin, N.
Quantum Cohomology of Minuscule Homogeneous Spaces III. Semi-Simplicity and Consequences
We prove that the quantum cohomology ring of any minuscule or cominuscule homogeneous space, specialized at $q=1$, is semisimple. This implies that complex conjugation defines an algebra automorphism of the quantum cohomology ring localized at the quantum parameter. We check that this involution coincides with the strange duality defined in our previous article. We deduce Vafa--Intriligator type formulas for the Gromov--Witten invariants.

Keywords:quantum cohomology, minuscule homogeneous spaces, Schubert calculus, quantum Euler class
Categories:14M15, 14N35

65. CJM 2010 (vol 62 pp. 870)

Valdimarsson, Stefán Ingi
The Brascamp-Lieb Polyhedron
A set of necessary and sufficient conditions for the Brascamp--Lieb inequality to hold has recently been found by Bennett, Carbery, Christ, and Tao. We present an analysis of these conditions. This analysis allows us to give a concise description of the set where the inequality holds in the case where each of the linear maps involved has co-rank $1$. This complements the result of Barthe concerning the case where the linear maps all have rank $1$. Pushing our analysis further, we describe the case where the maps have either rank $1$ or rank $2$. A separate but related problem is to give a list of the finite number of conditions necessary and sufficient for the Brascamp--Lieb inequality to hold. We present an algorithm which generates such a list.

Keywords:Brascamp-Lieb inequality, Loomis-Whitney inequality, lattice, flag
Categories:44A35, 14M15, 26D20

66. CJM 2010 (vol 62 pp. 668)

Vollaard, Inken
The Supersingular Locus of the Shimura Variety for GU(1,s)
In this paper we study the supersingular locus of the reduction modulo $p$ of the Shimura variety for $GU(1,s)$ in the case of an inert prime $p$. Using Dieudonné theory we define a stratification of the corresponding moduli space of $p$-divisible groups. We describe the incidence relation of this stratification in terms of the Bruhat--Tits building of a unitary group. In the case of $GU(1,2)$, we show that the supersingular locus is equidimensional of dimension 1 and is of complete intersection. We give an explicit description of the irreducible components and their intersection behaviour.

Categories:14G35, 11G18, 14K10

67. CJM 2010 (vol 62 pp. 787)

Landquist, E.; Rozenhart, P.; Scheidler, R.; Webster, J.; Wu, Q.
An Explicit Treatment of Cubic Function Fields with Applications
We give an explicit treatment of cubic function fields of characteristic at least five. This includes an efficient technique for converting such a field into standard form, formulae for the field discriminant and the genus, simple necessary and sufficient criteria for non-singularity of the defining curve, and a characterization of all triangular integral bases. Our main result is a description of the signature of any rational place in a cubic extension that involves only the defining curve and the order of the base field. All these quantities only require simple polynomial arithmetic as well as a few square-free polynomial factorizations and, in some cases, square and cube root extraction modulo an irreducible polynomial. We also illustrate why and how signature computation plays an important role in computing the class number of the function field. This in turn has applications to the study of zeros of zeta functions of function fields.

Keywords:cubic function field, discriminant, non-singularity, integral basis, genus, signature of a place, class number
Categories:14H05, 11R58, 14H45, 11G20, 11G30, 11R16, 11R29

68. CJM 2009 (vol 62 pp. 262)

Goresky, Mark; MacPherson, Robert
On the Spectrum of the Equivariant Cohomology Ring
If an algebraic torus $T$ acts on a complex projective algebraic variety $X$, then the affine scheme $\operatorname{Spec} H^*_T(X;\mathbb C)$ associated with the equivariant cohomology is often an arrangement of linear subspaces of the vector space $H_2^T(X;\mathbb C).$ In many situations the ordinary cohomology ring of $X$ can be described in terms of this arrangement.

Categories:14L30, 54H15

69. CJM 2009 (vol 62 pp. 473)

Yun, Zhiwei
Goresky—MacPherson Calculus for the Affine Flag Varieties
We use the fixed point arrangement technique developed by Goresky and MacPherson to calculate the part of the equivariant cohomology of the affine flag variety $\mathcal{F}\ell_G$ generated by degree 2. We use this result to show that the vertices of the moment map image of $\mathcal{F}\ell_G$ lie on a paraboloid.

Categories:14L30, 55N91

70. CJM 2009 (vol 62 pp. 456)

Yang, Tonghai
The Chowla—Selberg Formula and The Colmez Conjecture
In this paper, we reinterpret the Colmez conjecture on the Faltings height of CM abelian varieties in terms of Hilbert (and Siegel) modular forms. We construct an elliptic modular form involving the Faltings height of a CM abelian surface and arithmetic intersection numbers, and prove that the Colmez conjecture for CM abelian surfaces is equivalent to the cuspidality of this modular form.

Categories:11G15, 11F41, 14K22

71. CJM 2009 (vol 61 pp. 1407)

Will, Pierre
Traces, Cross-Ratios and 2-Generator Subgroups of $\SU(2,1)$
In this work, we investigate how to decompose a pair $(A,B)$ of loxodromic isometries of the complex hyperbolic plane $\mathbf H^{2}_{\mathbb C}$ under the form $A=I_1I_2$ and $B=I_3I_2$, where the $I_k$'s are involutions. The main result is a decomposability criterion, which is expressed in terms of traces of elements of the group $\langle A,B\rangle$.

Categories:14L24, 22E40, 32M15, 51M10

72. CJM 2009 (vol 61 pp. 1118)

Pontreau, Corentin
Petits points d'une surface
Pour toute sous-vari\'et\'e g\'eom\'etriquement irr\'eductible $V$ du grou\-pe multiplicatif $\mathbb{G}_m^n$, on sait qu'en dehors d'un nombre fini de translat\'es de tores exceptionnels inclus dans $V$, tous les points sont de hauteur minor\'ee par une certaine quantit\'e $q(V)^{-1}>0$. On conna\^it de plus une borne sup\'erieure pour la somme des degr\'es de ces translat\'es de tores pour des valeurs de $q(V)$ polynomiales en le degr\'e de $V$. Ceci n'est pas le cas si l'on exige une minoration quasi-optimale pour la hauteur des points de $V$, essentiellement lin\'eaire en l'inverse du degr\'e. Nous apportons ici une r\'eponse partielle \`a ce probl\`eme\,: nous donnons une majoration de la somme des degr\'es de ces translat\'es de sous-tores de codimension $1$ d'une hypersurface $V$. Les r\'esultats, obtenus dans le cas de $\mathbb{G}_m^3$, mais compl\`etement explicites, peuvent toutefois s'\'etendre \`a $\mathbb{G}_m^n$, moyennant quelques petites complications inh\'erentes \`a la dimension $n$.

Keywords:Hauteur normalisée, groupe multiplicatif, problème de Lehmer, petits points
Categories:11G50, 11J81, 14G40

73. CJM 2009 (vol 61 pp. 1050)

Bertin, Marie-Amélie
Examples of Calabi--Yau 3-Folds of $\mathbb{P}^{7}$ with $\rho=1$
We give some examples of Calabi--Yau $3$-folds with $\rho=1$ and $\rho=2$, defined over $\mathbb{Q}$ and constructed as $4$-codimensional subvarieties of $\mathbb{P}^7$ via commutative algebra methods. We explain how to deduce their Hodge diamond and top Chern classes from computer based computations over some finite field $\mathbb{F}_{p}$. Three of our examples (of degree $17$ and $20$) are new. The two others (degree $15$ and $18$) are known, and we recover their well-known invariants with our method. These examples are built out of Gulliksen--Neg{\aa}rd and Kustin--Miller complexes of locally free sheaves. Finally, we give two new examples of Calabi--Yau $3$-folds of $\mathbb{P}^6$ of degree $14$ and $15$ (defined over $\mathbb{Q}$). We show that they are not deformation equivalent to Tonoli's examples of the same degree, despite the fact that they have the same invariants $(H^3,c_2\cdot H, c_3)$ and $\rho=1$.

Categories:14J32, 14Q15

74. CJM 2009 (vol 61 pp. 828)

Howard, Benjamin
Twisted Gross--Zagier Theorems
The theorems of Gross--Zagier and Zhang relate the N\'eron--Tate heights of complex multiplication points on the modular curve $X_0(N)$ (and on Shimura curve analogues) with the central derivatives of automorphic $L$-function. We extend these results to include certain CM points on modular curves of the form $X(\Gamma_0(M)\cap\Gamma_1(S))$ (and on Shimura curve analogues). These results are motivated by applications to Hida theory that can be found in the companion article "Central derivatives of $L$-functions in Hida families", Math.\ Ann.\ \textbf{399}(2007), 803--818.

Categories:11G18, 14G35

75. CJM 2009 (vol 61 pp. 930)

Sidman, Jessica; Sullivant, Seth
Prolongations and Computational Algebra
We explore the geometric notion of prolongations in the setting of computational algebra, extending results of Landsberg and Manivel which relate prolongations to equations for secant varieties. We also develop methods for computing prolongations that are combinatorial in nature. As an application, we use prolongations to derive a new family of secant equations for the binary symmetric model in phylogenetics.

Categories:13P10, 14M99
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