Expositiones Mathematicae最新文献

This note presents an elementary iterative construction of the generators for the complex $K$-groups $K_i\left(C\left(S^d\right)\right)$ of the $d$-dimensional spheres. These generators are explicitly given as the restrictions of Dirac or Weyl Hamiltonians to the unit sphere. Connections to solid state physics are briefly elaborated on.

We prove that for a given unmarked trace spectrum with multiplicity, there are only a finite number of convex real projective surfaces with that spectrum up to remarking.

We define and study homotopy groups of cubical sets. To this end, we give four definitions of homotopy groups of a cubical set, prove that they are equivalent, and further that they agree with their topological analogues via the geometric realization functor. We also provide purely combinatorial proofs of several classical theorems, including: product preservation, commutativity of higher homotopy groups, the long exact sequence of a fibration, and Whitehead’s theorem.

This is a companion paper to our “Cubical setting for discrete homotopy theory, revisited” in which we apply these results to study the homotopy theory of simple graphs.

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 $Q\left(\sqrt{19\cdot 151}\right)$.

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).

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.

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’.

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.