Galois representations and Galois groups over Q

In this paper we generalize results of P. Le Duff to genus n hyperelliptic curves. More precisely, let C/Q be a hyperelliptic genus n curve, let J(C) be the associated Jacobian variety and let ¯ρℓ : GQ → GSp(J(C)[ℓ]) be the Galois representation attached to the ℓ-torsion of J(C). Assume that there exists a prime p such that J(C) has semistable reduction with toric dimension 1 at p. We provide an algorithm to compute a list of primes ℓ (if they exist) such that ¯ρℓ is surjective. In particular we realize GSp6 (Fℓ) as a Galois group over Q for all primes ℓ ∈ [11, 500000]

Attitudes toward intuition in calculus textbooks

Intuition was long held in high regard by mathematicians, who considered it all but synonymous with clarity and illumination. But in the 20th century there was a strong tendency to vilify intuition and cast it as the opposite of rigorous reasoning. Calculus in particular became a battleground for these opposing views. By systematically surveying references to intuition in historical and modern calculus textbooks, I look at how its status has changed across the centuries. In particular, I argue against the veracity of the self-fashioned origin story of the modern anti-intuition movement, which relies heavily on a particular historical narrative to portray the demise of intuition as an inexorable triumph of logic and reason

Note On Coisotropic Floer Homology And Leafwise Fixed Points

For an adiscal or monotone regular coisotropic submanifold N of a symplectic manifold I define its Floer homology to be the Floer homology of a certain Lagrangian embedding of N. Given a Hamiltonian isotopy φ=(φt) and a suitable almost complex structure, the corresponding Floer chain complex is generated by the (N,φ)-contractible leafwise fixed points. I also outline the construction of a local Floer homology for an arbitrary closed coisotropic submanifold. Results by Floer and Albers about Lagrangian Floer homology imply lower bounds on the number of leafwise fixed points. This reproduces earlier results of mine. The first construction also gives rise to a Floer homology for a Boothby-Wang fibration, by applying it to the circle bundle inside the associated complex line bundle. This can be used to show that translated points exist

Two treatises by Ibn al-Haytham on the meridian line, translated from the edition by Fuat Sezgin

This paper contains an annotated English translation of the two surviving treatises by Ibn al-Haytham (ca. 965-1041) on the determination of the meridian line. These very technical treatises survive in an Arabic manuscript in Berlin, and they were published by Fuat Sezgin in an excellent Arabic edition which appeared in this journal in 1986. The edition has not been translated before, but it has served as a source for a reconstruction of Ibn al-Haytham's instrument for the determination of the meridian line. A study of Sezgin's editions, which are translated in this paper, shows that the reconstructed instrument gives an incorrect representation of Ibn al-Haytham's ideas

The cohomology of the elliptic tangent bundle

In this note we compute the cohomology of the elliptic tangent bundle, a Lie algebroid introduced in Cavalcanti and Gualtieri (2018), Cavalcanti et al. (2020) used to describe singular symplectic forms arising from generalised complex geometry

Realising πe r–algebras by global ring spectra

We approach a problem of realising algebraic objects in a certain universal equivariant stable homotopy theory, the global homotopy theory of Schwede (2018). Specifically, for a global ring spectrum R, we consider which classes of ring homomorphisms ηe W πe eR ! Se can be realised by a map ηW R ! S in the category of global R– modules, and what multiplicative structures can be placed on S. If ηe witnesses Se as a projective πe eR–module, then such an η exists as a map between homotopy commutative global R–algebras. If ηe is in addition étale or S0 is a Q–algebra, then η can be upgraded to a map of E1–global R–algebras or a map of G1–R–algebras, respectively. Various global spectra and E1–global ring spectra are then obtained from classical homotopy-theoretic and algebraic constructions, with a controllable global homotopy type

Rigorous Purposes of Analysis in Greek Geometry

Analyses in Greek geometry are traditionally seen as heuristic devices. However, many occurrences of analysis in formal treatises are difficult to justify in such terms. I show that Greek analysies of geometrics can also serve formal mathematical purposes, which are arguably incomplete without which their associated syntheses are arguably incomplete. Firstly, when the solution of a problem is preceded by an analysis, the analysis latter proves rigorously that there are no other solutions to the problem than those offered in the synthesis. Secondly, whenever some construction assumption beyond ruler and compass is made, the problem is not only solvable by that assumption but is in fact equivalent to that assumption in a rigorous sense