Our main theorem shows that the regularity of non-degenerate homogeneous prime ideals is not bounded by any polynomial function of the degree; this holds over any field k. In particular, we provide counterexamples to the longstanding Regularity Conjecture, also known as the Eisenbud-Goto Conjecture (1984). We introduce a method which, starting from a homogeneous ideal I, produces a prime ideal whose projective dimension, regularity, degree, dimension, depth, and codimension are expressed in terms of numerical invariants of I. The method is also related to producing bounds in the spirit of Stillman’s Conjecture, recently solved by Ananyan-Hochster. Mathematics Department, Iowa State University, Ames, IA 50011, USA Mathematics Department, Cornell University, Ithaca, NY 14853, USA
{"title":"Counterexamples to the Eisenbud–Goto regularity conjecture","authors":"J. McCullough, I. Peeva","doi":"10.1090/JAMS/891","DOIUrl":"https://doi.org/10.1090/JAMS/891","url":null,"abstract":"Our main theorem shows that the regularity of non-degenerate homogeneous prime ideals is not bounded by any polynomial function of the degree; this holds over any field k. In particular, we provide counterexamples to the longstanding Regularity Conjecture, also known as the Eisenbud-Goto Conjecture (1984). We introduce a method which, starting from a homogeneous ideal I, produces a prime ideal whose projective dimension, regularity, degree, dimension, depth, and codimension are expressed in terms of numerical invariants of I. The method is also related to producing bounds in the spirit of Stillman’s Conjecture, recently solved by Ananyan-Hochster. Mathematics Department, Iowa State University, Ames, IA 50011, USA Mathematics Department, Cornell University, Ithaca, NY 14853, USA","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"65 1","pages":"473-496"},"PeriodicalIF":3.9,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/JAMS/891","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60551981","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Small subalgebras of polynomial rings and Stillman’s Conjecture","authors":"Tigran Ananyan, M. Hochster","doi":"10.1090/JAMS/932","DOIUrl":"https://doi.org/10.1090/JAMS/932","url":null,"abstract":"<p>Let <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n comma d comma eta\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>d</mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>η<!-- η --></mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">n, d, eta</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> be positive integers. We show that in a polynomial ring <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper R\">\u0000 <mml:semantics>\u0000 <mml:mi>R</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">R</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> in <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper N\">\u0000 <mml:semantics>\u0000 <mml:mi>N</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">N</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> variables over an algebraically closed field <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\">\u0000 <mml:semantics>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">K</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> of arbitrary characteristic, any <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\">\u0000 <mml:semantics>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">K</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-subalgebra of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper R\">\u0000 <mml:semantics>\u0000 <mml:mi>R</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">R</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> generated over <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\">\u0000 <mml:semantics>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">K</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> by at most <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\">\u0000 <mml:semantics>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">n</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> forms of degree at most <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"d\">\u0000 <mml:semantics>\u0000 <mml:mi>d</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">d</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is contained in a <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\">\u0000 ","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"1 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2016-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/JAMS/932","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60552005","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Gregory R. Chambers, Dominic Dotterrer, Fedor Manin, S. Weinberger
<p>For a given null-cobordant Riemannian <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-manifold, how does the minimal geometric complexity of a null-cobordism depend on the geometric complexity of the manifold? Gromov has conjectured that this dependence should be linear. We show that it is at most a polynomial whose degree depends on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In the appendix the bound is improved to one that is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O left-parenthesis upper L Superscript 1 plus epsilon Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mi>ε<!-- ε --></mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">O(L^{1+varepsilon })</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for every <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="epsilon greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi>ε<!-- ε --></mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">varepsilon >0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.</p> <p>This construction relies on another of independent interest. Take <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y"> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding="application/x-tex">Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to be sufficiently nice compact metric spaces, such as Riemannian manifolds or simplicial complexes. Suppose <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y"> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding="application/x-tex">Y</mml:annotation> </mml:semanti
对于给定的零协黎曼nn流形，零协的最小几何复杂度如何依赖于流形的几何复杂度?Gromov推测这种相关性应该是线性的。我们证明了它最多是一个多项式，它的次数依赖于n。在附录中，对每一个ε >0 varepsilon >0，将界改进为O(L 1+ ε) O(L^{1+varepsilon})。这个构造依赖于另一个独立的构造。假设X X和Y Y是足够好的紧化度量空间，比如黎曼流形或者简单复形。假设Y是单连通且理性同伦等价于Eilenberg-MacLane空间的乘积，例如，任意单连通李群。然后两个L L -Lipschitz映射f,g:X→Y f,g:X 到Y通过CL L L -Lipschitz同伦是同伦的。我们给出了一个反例来证明这对于更大的空间Y Y类是不成立的。
{"title":"Quantitative null-cobordism","authors":"Gregory R. Chambers, Dominic Dotterrer, Fedor Manin, S. Weinberger","doi":"10.1090/jams/903","DOIUrl":"https://doi.org/10.1090/jams/903","url":null,"abstract":"<p>For a given null-cobordant Riemannian <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\">\u0000 <mml:semantics>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">n</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-manifold, how does the minimal geometric complexity of a null-cobordism depend on the geometric complexity of the manifold? Gromov has conjectured that this dependence should be linear. We show that it is at most a polynomial whose degree depends on <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\">\u0000 <mml:semantics>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">n</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. In the appendix the bound is improved to one that is <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper O left-parenthesis upper L Superscript 1 plus epsilon Baseline right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>O</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:msup>\u0000 <mml:mi>L</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mn>1</mml:mn>\u0000 <mml:mo>+</mml:mo>\u0000 <mml:mi>ε<!-- ε --></mml:mi>\u0000 </mml:mrow>\u0000 </mml:msup>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">O(L^{1+varepsilon })</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> for every <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"epsilon greater-than 0\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>ε<!-- ε --></mml:mi>\u0000 <mml:mo>></mml:mo>\u0000 <mml:mn>0</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">varepsilon >0</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>.</p>\u0000\u0000<p>This construction relies on another of independent interest. Take <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\">\u0000 <mml:semantics>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">X</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper Y\">\u0000 <mml:semantics>\u0000 <mml:mi>Y</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">Y</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> to be sufficiently nice compact metric spaces, such as Riemannian manifolds or simplicial complexes. Suppose <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper Y\">\u0000 <mml:semantics>\u0000 <mml:mi>Y</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">Y</mml:annotation>\u0000 </mml:semanti","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"1 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2016-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/jams/903","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60552059","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
<p>Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a locally finite irreducible affine building of dimension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="greater-than-or-equal-to 2"> <mml:semantics> <mml:mrow> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">geq 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma less-than-or-equal-to upper A u t left-parenthesis upper X right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal">Γ<!-- Γ --></mml:mi> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>Aut</mml:mi> <mml:mo><!-- --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">Gamma leq operatorname {Aut}(X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a discrete group acting cocompactly. The goal of this paper is to address the following question: When is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma"> <mml:semantics> <mml:mi mathvariant="normal">Γ<!-- Γ --></mml:mi> <mml:annotation encoding="application/x-tex">Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> linear? More generally, when does <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma"> <mml:semantics> <mml:mi mathvariant="normal">Γ<!-- Γ --></mml:mi> <mml:annotation encoding="application/x-tex">Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> admit a finite-dimensional representation with infinite image over a commutative unital ring? If <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the Bruhat–Tits building of a simple algebraic group over a local field and if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma"> <mml:semantics> <mml:mi mathvariant="normal">Γ<!-- Γ --></mml:mi> <mml:annotation encoding="application/x-tex">Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an arithmetic lattice, then <inline
{"title":"On the linearity of lattices in affine buildings and ergodicity of the singular Cartan flow","authors":"U. Bader, P. Caprace, Jean L'ecureux","doi":"10.1090/JAMS/914","DOIUrl":"https://doi.org/10.1090/JAMS/914","url":null,"abstract":"<p>Let <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\">\u0000 <mml:semantics>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">X</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> be a locally finite irreducible affine building of dimension <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"greater-than-or-equal-to 2\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo>≥<!-- ≥ --></mml:mo>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">geq 2</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, and let <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Gamma less-than-or-equal-to upper A u t left-parenthesis upper X right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi mathvariant=\"normal\">Γ<!-- Γ --></mml:mi>\u0000 <mml:mo>≤<!-- ≤ --></mml:mo>\u0000 <mml:mi>Aut</mml:mi>\u0000 <mml:mo><!-- --></mml:mo>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">Gamma leq operatorname {Aut}(X)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> be a discrete group acting cocompactly. The goal of this paper is to address the following question: When is <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Gamma\">\u0000 <mml:semantics>\u0000 <mml:mi mathvariant=\"normal\">Γ<!-- Γ --></mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">Gamma</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> linear? More generally, when does <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Gamma\">\u0000 <mml:semantics>\u0000 <mml:mi mathvariant=\"normal\">Γ<!-- Γ --></mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">Gamma</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> admit a finite-dimensional representation with infinite image over a commutative unital ring? If <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\">\u0000 <mml:semantics>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">X</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is the Bruhat–Tits building of a simple algebraic group over a local field and if <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Gamma\">\u0000 <mml:semantics>\u0000 <mml:mi mathvariant=\"normal\">Γ<!-- Γ --></mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">Gamma</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is an arithmetic lattice, then <inline","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"1 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2016-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/JAMS/914","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60552203","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article establishes new cases of the Birch and Swinnerton-Dyer conjecture in analytic rank 0, for elliptic curves over $ mathbb{Q}$ viewed over the fields cut out by certain self-dual Artin representations of dimension at most $ 4$. When the associated $ L$-function vanishes (to even order $ ge 2$) at its central point, two canonical classes in the corresponding Selmer group are constructed and shown to be linearly independent assuming the non-vanishing of a Garrett-Hida $ p$-adic $ L$-function at a point lying outside its range of classical interpolation. The key tool for both results is the study of certain $ p$-adic families of global Galois cohomology classes arising from Gross-Kudla-Schoen diagonal cycles in a tower of triple products of modular curves.
{"title":"Diagonal cycles and Euler systems II: the Birch and Swinnerton-Dyer conjecture for Hasse-Weil-Artin L-functions","authors":"H. Darmon, V. Rotger","doi":"10.1090/JAMS/861","DOIUrl":"https://doi.org/10.1090/JAMS/861","url":null,"abstract":"This article establishes new cases of the Birch and Swinnerton-Dyer conjecture in analytic rank 0, for elliptic curves over $ mathbb{Q}$ viewed over the fields cut out by certain self-dual Artin representations of dimension at most $ 4$. When the associated $ L$-function vanishes (to even order $ ge 2$) at its central point, two canonical classes in the corresponding Selmer group are constructed and shown to be linearly independent assuming the non-vanishing of a Garrett-Hida $ p$-adic $ L$-function at a point lying outside its range of classical interpolation. The key tool for both results is the study of certain $ p$-adic families of global Galois cohomology classes arising from Gross-Kudla-Schoen diagonal cycles in a tower of triple products of modular curves.","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"30 1","pages":"601-672"},"PeriodicalIF":3.9,"publicationDate":"2016-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/JAMS/861","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60551579","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Tsirelson’s problem asks whether the commuting operator model for two-party quantum correlations is equivalent to the tensor-product model. We give a negative answer to this question by showing that there are non-local games which have perfect commuting-operator strategies, but do not have perfect tensor-product strategies. The weak Tsirelson problem, which is known to be equivalent to the Connes embedding problem, remains open. The examples we construct are instances of (binary) linear system games. For such games, previous results state that the existence of perfect strategies is controlled by the solution group of the linear system. Our main result is that every finitely-presented group embeds in some solution group. As an additional consequence, we show that the problem of determining whether a linear system game has a perfect commuting-operator strategy is undecidable.
{"title":"Tsirelson’s problem and an embedding theorem for groups arising from non-local games","authors":"William Slofstra","doi":"10.1090/JAMS/929","DOIUrl":"https://doi.org/10.1090/JAMS/929","url":null,"abstract":"Tsirelson’s problem asks whether the commuting operator model for two-party quantum correlations is equivalent to the tensor-product model. We give a negative answer to this question by showing that there are non-local games which have perfect commuting-operator strategies, but do not have perfect tensor-product strategies. The weak Tsirelson problem, which is known to be equivalent to the Connes embedding problem, remains open.\u0000\u0000The examples we construct are instances of (binary) linear system games. For such games, previous results state that the existence of perfect strategies is controlled by the solution group of the linear system. Our main result is that every finitely-presented group embeds in some solution group. As an additional consequence, we show that the problem of determining whether a linear system game has a perfect commuting-operator strategy is undecidable.","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"6 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2016-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/JAMS/929","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60551883","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Remodeling Conjecture proposed by Bouchard-Klemm-Mariño-Pasquetti (BKMP) relates the A-model open and closed topological string amplitudes (the all genus open and closed Gromov-Witten invariants) of a semiprojective toric Calabi-Yau 3-manifold/3-orbifold to the Eynard-Orantin invariants of its mirror curve. It is an all genus open-closed mirror symmetry for toric Calabi-Yau 3-manifolds/3-orbifolds. In this paper, we present a proof of the BKMP Remodeling Conjecture for all genus open-closed orbifold Gromov-Witten invariants of an arbitrary semiprojective toric Calabi-Yau 3-orbifold relative to an outer framed Aganagic-Vafa Lagrangian brane. We also prove the conjecture in the closed string sector at all genera.
{"title":"On the remodeling conjecture for toric Calabi-Yau 3-orbifolds","authors":"Bohan Fang, Chiu-Chu Melissa Liu, Zhengyu Zong","doi":"10.1090/JAMS/934","DOIUrl":"https://doi.org/10.1090/JAMS/934","url":null,"abstract":"The Remodeling Conjecture proposed by Bouchard-Klemm-Mariño-Pasquetti (BKMP) relates the A-model open and closed topological string amplitudes (the all genus open and closed Gromov-Witten invariants) of a semiprojective toric Calabi-Yau 3-manifold/3-orbifold to the Eynard-Orantin invariants of its mirror curve. It is an all genus open-closed mirror symmetry for toric Calabi-Yau 3-manifolds/3-orbifolds. In this paper, we present a proof of the BKMP Remodeling Conjecture for all genus open-closed orbifold Gromov-Witten invariants of an arbitrary semiprojective toric Calabi-Yau 3-orbifold relative to an outer framed Aganagic-Vafa Lagrangian brane. We also prove the conjecture in the closed string sector at all genera.","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"477 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2016-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/JAMS/934","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60552014","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We construct stable envelopes in equivariant elliptic cohomology of Nakajima quiver varieties. In particular, this gives an elliptic generalization of the results of Maulik and Okounkov [Astérisque 408 (2019), ix+209]. We apply them to the computation of the monodromy of q q -difference equations arising in the enumerative K-theory of rational curves in Nakajima varieties, including the quantum Knizhnik–Zamolodchikov equations.
{"title":"Elliptic stable envelopes","authors":"Mina Aganagic, A. Okounkov","doi":"10.1090/jams/954","DOIUrl":"https://doi.org/10.1090/jams/954","url":null,"abstract":"We construct stable envelopes in equivariant elliptic cohomology of Nakajima quiver varieties. In particular, this gives an elliptic generalization of the results of Maulik and Okounkov [Astérisque 408 (2019), ix+209]. We apply them to the computation of the monodromy of \u0000\u0000 \u0000 q\u0000 q\u0000 \u0000\u0000-difference equations arising in the enumerative K-theory of rational curves in Nakajima varieties, including the quantum Knizhnik–Zamolodchikov equations.","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"1 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2016-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60552032","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that, in good residual characteristic, most supercuspidal representations of a tamely ramified reductive p p -adic group G G arise from pairs ( S , θ ) (S,theta ) , where S S is a tame elliptic maximal torus of G G , and θ theta is a character of S S satisfying a simple root-theoretic property. We then give a new expression for the roots of unity that appear in the Adler-DeBacker-Spice character formula for these supercuspidal representations and use it to show that this formula bears a striking resemblance to the character formula for discrete series representations of real reductive groups. Led by this, we explicitly construct the local Langlands correspondence for these supercuspidal representations and prove stability and endoscopic transfer in the case of toral representations. In large residual characteristic this gives a construction of the local Langlands correspondence for almost all supercuspidal representations of reductive p p -adic groups.
{"title":"Regular supercuspidal representations","authors":"Tasho Kaletha","doi":"10.1090/JAMS/925","DOIUrl":"https://doi.org/10.1090/JAMS/925","url":null,"abstract":"We show that, in good residual characteristic, most supercuspidal representations of a tamely ramified reductive \u0000\u0000 \u0000 p\u0000 p\u0000 \u0000\u0000-adic group \u0000\u0000 \u0000 G\u0000 G\u0000 \u0000\u0000 arise from pairs \u0000\u0000 \u0000 \u0000 (\u0000 S\u0000 ,\u0000 θ\u0000 )\u0000 \u0000 (S,theta )\u0000 \u0000\u0000, where \u0000\u0000 \u0000 S\u0000 S\u0000 \u0000\u0000 is a tame elliptic maximal torus of \u0000\u0000 \u0000 G\u0000 G\u0000 \u0000\u0000, and \u0000\u0000 \u0000 θ\u0000 theta\u0000 \u0000\u0000 is a character of \u0000\u0000 \u0000 S\u0000 S\u0000 \u0000\u0000 satisfying a simple root-theoretic property. We then give a new expression for the roots of unity that appear in the Adler-DeBacker-Spice character formula for these supercuspidal representations and use it to show that this formula bears a striking resemblance to the character formula for discrete series representations of real reductive groups. Led by this, we explicitly construct the local Langlands correspondence for these supercuspidal representations and prove stability and endoscopic transfer in the case of toral representations. In large residual characteristic this gives a construction of the local Langlands correspondence for almost all supercuspidal representations of reductive \u0000\u0000 \u0000 p\u0000 p\u0000 \u0000\u0000-adic groups.","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"42 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2016-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/JAMS/925","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60551875","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that Bernoulli convolutions μλmu _lambda are absolutely continuous provided the parameter λlambda is an algebraic number sufficiently close to 11 depending on the Mahler measure of λlambda.
我们证明了伯努利卷积μ λ mu ＿ lambda是绝对连续的，只要参数λ lambda是一个足够接近于11的代数数，依赖于λ lambda的马勒测度。
{"title":"Absolute continuity of Bernoulli convolutions for algebraic parameters","authors":"P. P. Varj'u","doi":"10.1090/jams/916","DOIUrl":"https://doi.org/10.1090/jams/916","url":null,"abstract":"<p>We prove that Bernoulli convolutions <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"mu Subscript lamda\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mi>μ<!-- μ --></mml:mi>\u0000 <mml:mi>λ<!-- λ --></mml:mi>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">mu _lambda</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> are absolutely continuous provided the parameter <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"lamda\">\u0000 <mml:semantics>\u0000 <mml:mi>λ<!-- λ --></mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">lambda</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is an algebraic number sufficiently close to <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"1\">\u0000 <mml:semantics>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:annotation encoding=\"application/x-tex\">1</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> depending on the Mahler measure of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"lamda\">\u0000 <mml:semantics>\u0000 <mml:mi>λ<!-- λ --></mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">lambda</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>.</p>","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"1 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2016-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/jams/916","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60551747","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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