Annals of Mathematics

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• Good Locally Testable Codes AI翻译
  作者：Dinur, Irit; Evra, Shai; Livne, Ron; Lubotzky, Alexander; Mozes, Shahar
  作者单位：Weizmann Institute of Science; Institute for Advanced Study - USA; Hebrew University of Jerusalem; Weizmann Institute of Science
  摘要：An explicit construction of locally testable codes of constant rate, constant distance and constant number of queries is given. Hence answering affirmatively the c3-problem.
• Naked singularity censoring with anisotropic apparent horizon AI翻译
  作者：An, Xinliang
  作者单位：National University of Singapore
  摘要：Employing the Einstein-scalar field system, we demonstrate an approach for proving high co-dimensional nonlinear instability of naked-singularity solutions as constructed by Christodoulou in 1994. We further investigate the censorship of Christodoulou's naked singularity and show that a tiny anisotropic perturbation arising from the outgoing characteristic initial data would lead to the emergence of an anisotropic apparent horizon, which covers and censors the naked singularity. Our approach a...
• Galois groups of random integer polynomials and van der Waerden's Conjecture AI翻译
  作者：Bhargava, Manjul
  作者单位：Princeton University
  摘要：Of the (2H+1)(n) monic integer polynomials f(x)=x(n)+a1x(n-1)+...+a(n) with max{|a(1)|,...,|a(n)|} <= H, how many have associated Galois group that is not the full symmetric group S-n?There are clearly >> Hn-1 such polynomials, as may be obtained by setting a(n)=0. In 1936, van der Waerden conjectured that O(Hn-1) should in fact also be the correct upper bound for the count of such polynomials. The conjecture has been known previously for degrees n <= 4, due to work of van der Waerden and Chow...
• On approximation to a real number by algebraic numbers of bounded degree AI翻译
  作者：Poels, Anthony
  作者单位：Institut National des Sciences Appliquees de Lyon - INSA Lyon; Universite Jean Monnet; Universite Lyon 1; Ecole Centrale de Lyon; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI)
  摘要：In his seminal 1961 paper, Wirsing studied how well a given transcendental real number xi can be approximated by algebraic numbers alpha of degree at most n for a given positive integer n , in terms of the so-called naive height H(alpha) of alpha . He showed that the supremum omega & lowast;n(xi) of all omega for which infinitely many such alpha have |xi- alpha| <= H(alpha)-omega-1 is at least (n + 1)/2. He also asked if we could even have omega & lowast;n(xi) >= nas it is generally expected. ...
• Rational approximations to linear subspaces AI翻译
  作者：De Saxce, Nicolas
  作者单位：Centre National de la Recherche Scientifique (CNRS)
  摘要：We study intrinsic diophantine approximation on Grassmann varieties. Using a new correspondence between the diophantine properties of a linear subspace x in Rd and certain diagonal orbits in the space of lattices, we are able to solve some problems suggested by Schmidt in 1967. In particular, we obtain a version of Dirichlet's principle in this setting with an optimal exponent.
• Independence of ℓ for Frobenius conjugacy classes attached to abelian varieties AI翻译
  作者：Kisin, Mark; Zhou, Bong
  作者单位：Harvard University; University of Cambridge
  摘要：Let A be an abelian variety over a number field E subset of C, and let G denote the Mumford-Tate group of A. After replacing E by a finite extension, the action of the absolute Galois group Gal(E/E) on the & ell;-adic cohomology H1et(AE, Q & ell;) factors through G(Q & ell;). We show that for v an odd prime of E where A has good reduction, the conjugacy class of Frobenius Frobvin G(Q & ell;) is independent of & ell;. Along the way, we prove that under certain hypotheses, every point in the mu ...
• The McKay Conjecture on character degrees AI翻译
  作者：Cabanes, MARc; Spaeth, Britta
  作者单位：Sorbonne Universite; Universite Paris Cite; University of Wuppertal
  摘要：We prove that for any prime P, any finite group has as many irreducible complex characters of degree prime to P as the normalizers of its Sylow P-subgroups. This equality was conjectured by John McKay in 1971. The conjecture was reduced by Isaacs-Malle-Navarro (2007) to a conjecture on representations, linear and projective, of finite simple groups that we finish proving here using the classification of those groups. We study mainly characters of normalizers N-G(S)(F )of Sylow d-tori S (d >= 3...
• Fractal uncertainty in higher dimensions AI翻译
  作者：Cohen, Alex
  作者单位：Massachusetts Institute of Technology (MIT)
  摘要：We prove that if a fractal set in Rd avoids lines in a certain quantitative sense, which we call line porosity, then it has a fractal uncertainty principle. The main ingredient is a new higher-dimensional Beurling-Malliavin multiplier theorem.
• The endoscopic fundamental lemma for unitary Friedberg-Jacquet periods AI翻译
  作者：Leslie, Spencer
  作者单位：Boston College
  摘要：We prove the endoscopic fundamental lemma for the Lie algebra of the symmetric variety U(2n)/U(n) x U(n), where U(n) denotes a unitary group of rank n. This is the first major step in the stabilization of the relative trace formula associated to the U(n) x U(n)-periods of automorphic forms on U(2n).
• The Shafarevich conjecture for hypersurfaces in abelian varieties AI翻译
  作者：Lawrence, Brian; Sawin, Will
  作者单位：Princeton University
  摘要：Faltings proved that there are finitely many abelian varieties of genus g over a number field K, with good reduction outside a finite set of primes S. Fixing one of these abelian varieties A, we prove that there are finitely many smooth hypersurfaces in A, with good reduction outside S, representing a given ample class in the Neron-Severi group of A, up to translation, as long as the dimension of A is at least four. Our approach builds on the approach of Lawrence and Venkatesh, which studies p...