Expositiones Mathematicae最新文献

In this paper we introduce several new cohomologies of almost complex manifolds, among which stands a generalization of Bott–Chern and Aeppli cohomologies defined using the operators $d$, $d^c$. We explain how they are connected to already existing cohomologies of almost complex manifolds and we study the spaces of harmonic forms associated to $d$, $d^c$, showing their relation with Bott–Chern and Aeppli cohomologies and to other well-studied spaces of harmonic forms. Notably, Bott–Chern cohomology of 1-forms is finite-dimensional on compact manifolds and provides an almost complex invariant $h_{d+d^c}^1$ that distinguishes between almost complex structures. On almost Kähler 4-manifolds, the spaces of harmonic forms we consider are particularly well-behaved and are linked to harmonic forms considered by Tseng and Yau in the study of symplectic cohomology.

We study the splitting fields of the family of polynomials $f_n\left(X\right)=X^n-X-1$. This family of polynomials has been much studied in the literature and has some remarkable properties. In Serre (2003), Serre related the function on primes $N_p\left(f_n\right)$, for a fixed $n\le 4$ and $p$ a varying prime, which counts the number of roots of $f_n\left(X\right)$ in $F_p$ to coefficients of modular forms. We study the case $n=5$, and relate $N_p\left(f_5\right)$ to mod 5 modular forms over $Q$, and to characteristic 0, parallel weight 1 Hilbert modular forms over $Q\left(\sqrt{19\cdot 151}\right)$.

Let $X$ be a curve of genus $g>1$ over $Q$ whose Jacobian $J$ has Mordell–Weil rank $r$ and Néron–Severi rank $\rho$. When $r<g+\rho -1$, the geometric quadratic Chabauty method determines a finite set of $p$-adic points containing the rational points of $X$. We describe algorithms for geometric quadratic Chabauty that translate the geometric quadratic Chabauty method into the language of $p$-adic heights and $p$-adic (Coleman) integrals. This translation also allows us to give a comparison to the (original) cohomological method for quadratic Chabauty. We show that the finite set of $p$-adic points produced by the geometric method is contained in the finite set produced by the cohomological method, and give a description of their difference.

We introduce a conjecture on the arithmetic sparcity of the degeneration locus of a $p$-adic local system on a smooth variety over a number field and, modulo the Bombieri–Lang conjecture, show that it follows from a conjecture on the geometry of the level varieties attached to the local system. We present a few applications of our conjecture to classical problems in arithmetic geometry. Eventually, we give some evidences and discuss a few perspectives to attack it, in particular for $p$-adic local systems arising from geometry.

We use the theory of logarithmic line bundles to construct compactifications of spaces of roots of a line bundle on a family of curves, generalising work of a number of authors. This runs via a study of the torsion in the tropical and logarithmic jacobians (recently constructed by Molcho and Wise). Our moduli space carries a ‘double ramification cycle’ measuring the locus where the given root is isomorphic to the trivial bundle, and we give a tautological formula for this class in the language of piecewise polynomial functions (as recently developed by Molcho–Pandharipande–Schmitt and Holmes–Schwarz).

This note recalls an early 13th century result on congruent numbers by Leonardo Pisano (“Fibonacci”), and shows how it relates to a specific much studied K3 surface and to an elliptic fibration on this surface. As an aside, the discussion reveals how, via explicit maps of degree two, the surface is covered by the Fermat quartic surface and also covers one of the two famous ‘most algebraic K3 surfaces’.

We give an explicit description of the category of central extensions of a group scheme by a sheaf of Abelian groups. Based on this, we describe a framework for computing with central extensions of finite locally free commutative group schemes, torsors under such group schemes and groups of isomorphism classes of these objects.

This short survey is part of a minicourse I gave during the CMI-HIMR Summer School “Unlikely Intersections in Diophantine Geometry” on the Zilber–Pink conjecture, formulated independently by Zilber (2002), Bombieri, Masser and Zannier (1999) in the case of tori and by Pink (2005) in the more general setting of mixed Shimura varieties. This conjecture, which includes in its general formulation many important results in number theory, has been intensively studied by several mathematicians in the past 20 years. We will mainly focus on these problems in the special setting of semiabelian varieties and families of abelian varieties.

We present a simple and efficient algorithm to compute the sum of the algebraic conjugates of a point on an elliptic curve.

Nous proposons une “Conjecture d’André-Oort en pinceau arithmétique”.

C’est une extension de la conjecture d’André-Oort, disons “classique”, formulée à l’origine par Y. André et F. Oort. La conjecture fait intervenir les modèles entiers des variétés de Shimura.