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 Compositio MathematicaJournal Prestige (SJR): 3.139 Citation Impact (citeScore): 1Number of Followers: 0      Subscription journal ISSN (Print) 0010-437X - ISSN (Online) 1570-5846 Published by Cambridge University Press  [386 journals]
• Ordinary primes in Hilbert modular varieties
• Authors: Junecue Suh
Pages: 647 - 678
Abstract: A well-known conjecture, often attributed to Serre, asserts that any motive over any number field has infinitely many ordinary reductions (in the sense that the Newton polygon coincides with the Hodge polygon). In the case of Hilbert modular cuspforms $f$ of parallel weight $(2,\ldots ,2)$ , we show how to produce more ordinary primes by using the Sato–Tate equidistribution and combining it with the Galois theory of the Hecke field. Under the assumption of stronger forms of Sato–Tate equidistribution, we get stronger (but conditional) results. In the case of higher weights, we formulate the ordinariness conjecture for submotives of the intersection cohomology of proper algebraic varieties with motivic coefficients, and verify it for the motives whose $\ell$ -adic Galois realisations are abelian on a finite-index subgroup. We get some results for Hilbert cuspforms of weight $(3,\ldots ,3)$ , weaker than those for $(2,\ldots ,2)$ .
PubDate: 2020-04-01T00:00:00.000Z
DOI: 10.1112/S0010437X19007826
Issue No: Vol. 156, No. 4 (2020)

• The algebraic dimension of compact complex threefolds with vanishing
second Betti number
• Authors: Frédéric Campana; Jean-Pierre Demailly, Thomas Peternell
Pages: 679 - 696
Abstract: We study compact complex three-dimensional manifolds with vanishing second Betti number. In particular, we show that a compact complex manifold homeomorphic to the six-dimensional sphere does carry any non-constant meromorphic function.
PubDate: 2020-04-01T00:00:00.000Z
DOI: 10.1112/S0010437X19007802
Issue No: Vol. 156, No. 4 (2020)

• Deformation spaces and normal forms around transversals
• Authors: Francis Bischoff; Henrique Bursztyn, Hudson Lima, Eckhard Meinrenken
Pages: 697 - 732
Abstract: Given a manifold $M$ with a submanifold $N$ , the deformation space ${\mathcal{D}}(M,N)$ is a manifold with a submersion to $\mathbb{R}$ whose zero fiber is the normal bundle $\unicode[STIX]{x1D708}(M,N)$ , and all other fibers are equal to $M$ . This article uses deformation spaces to study the local behavior of various geometric structures associated with singular foliations, with $N$ a submanifold transverse to the foliation. New examples include $L_{\infty }$ -algebroids, Courant algebroids, and Lie bialgebroids. In each case, we obtain a normal form theorem around $N$ , in terms of a model structure over $\unicode[STIX]{x1D708}(M,N)$ .
PubDate: 2020-04-01T00:00:00.000Z
DOI: 10.1112/S0010437X1900784X
Issue No: Vol. 156, No. 4 (2020)

• Gonality of dynatomic curves and strong uniform boundedness of preperiodic
points
• Authors: John R. Doyle; Bjorn Poonen
Pages: 733 - 743
Abstract: Fix $d\geqslant 2$ and a field $k$ such that $\operatorname{char}k\nmid d$ . Assume that $k$ contains the $d$ th roots of $1$ . Then the irreducible components of the curves over $k$ parameterizing preperiodic points of polynomials of the form $z^{d}+c$ are geometrically irreducible and have gonality tending to $\infty$ . This implies the function field analogue of the strong uniform boundedness conjecture for preperiodic points of $z^{d}+c$ . It also has consequences over number fields: it implies strong uniform boundedness for preperiodic points of bounded eventual period, which in turn reduces the full conjecture for preperiodic points to the conjecture for periodic points. Our proofs involve a novel argument specific to finite fields, in addition to more standard tools such as the Castelnuovo–Severi inequality.
PubDate: 2020-04-01T00:00:00.000Z
DOI: 10.1112/S0010437X20007022
Issue No: Vol. 156, No. 4 (2020)

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