Proceedings of the American Mathematical Society, Series B最新文献

{"title":"Reduction of quad-equations consistent around a cuboctahedron I: Additive case","authors":"N. Joshi, N. Nakazono","doi":"10.1090/bproc/96","DOIUrl":"https://doi.org/10.1090/bproc/96","url":null,"abstract":"In this paper, we consider a reduction of a new system of partial difference equations, which was obtained in our previous paper (Joshi and Nakazono, arXiv:1906.06650) and shown to be consistent around a cuboctahedron. We show that this system reduces to $A_2^{(1)ast}$-type discrete Painleve equations by considering a periodic reduction of a three-dimensional lattice constructed from overlapping cuboctahedra.","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"52 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129213798","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Quantum trees which maximize higher eigenvalues are unbalanced","authors":"Jonathan Rohleder","doi":"10.1090/bproc/60","DOIUrl":"https://doi.org/10.1090/bproc/60","url":null,"abstract":"The isoperimetric problem of maximizing all eigenvalues of the Laplacian on a metric tree graph within the class of trees of a given average edge length is studied. It turns out that, up to rescaling, the unique maximizer of the \u0000\u0000 \u0000 k\u0000 k\u0000 \u0000\u0000-th positive eigenvalue is the star graph with three edges of lengths \u0000\u0000 \u0000 \u0000 2\u0000 k\u0000 −\u0000 1\u0000 \u0000 2 k - 1\u0000 \u0000\u0000, \u0000\u0000 \u0000 1\u0000 1\u0000 \u0000\u0000 and \u0000\u0000 \u0000 1\u0000 1\u0000 \u0000\u0000. This complements the previously known result that the first nonzero eigenvalue is maximized by all equilateral star graphs and indicates that optimizers of isoperimetric problems for higher eigenvalues may be less balanced in their shape—an observation which is known from numerical results on the optimization of higher eigenvalues of Laplacians on Euclidean domains.","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"73 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130306829","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Zeros of a one-parameter family of harmonic trinomials","authors":"Michael A. Brilleslyper, J. Brooks, M. Dorff, Russell W. Howell, Lisbeth E. Schaubroeck","doi":"10.1090/bproc/51","DOIUrl":"https://doi.org/10.1090/bproc/51","url":null,"abstract":"It is well known that complex harmonic polynomials of degree \u0000\u0000 \u0000 n\u0000 n\u0000 \u0000\u0000 may have more than \u0000\u0000 \u0000 n\u0000 n\u0000 \u0000\u0000 zeros. In this paper, we examine a one-parameter family of harmonic trinomials and determine how the number of zeros depends on the parameter. Our proof heavily utilizes the Argument Principle for Harmonic Functions and involves finding the winding numbers about the origin for a family of hypocycloids.","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"63 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114210575","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Boshernitzan’s condition, factor complexity, and an application","authors":"Van Cyr, Bryna Kra","doi":"10.1090/bproc/90","DOIUrl":"https://doi.org/10.1090/bproc/90","url":null,"abstract":"Boshernitzan gave a decay condition on the measure of cylinder sets that implies unique ergodicity for minimal subshifts. Interest in the properties of subshifts satisfying this condition has grown recently, due to a connection with discrete Schrödinger operators, and of particular interest is how restrictive the Boshernitzan condition is. While it implies zero topological entropy, our main theorem shows how to construct minimal subshifts satisfying the condition, and whose factor complexity grows faster than any pre-assigned subexponential rate. As an application, via a theorem of Damanik and Lenz, we show that there is no subexponentially growing sequence for which the spectra of all discrete Schrödinger operators associated with subshifts whose complexity grows faster than the given sequence have only finitely many gaps.","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122335432","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Global regularity for solutions of the Navier–Stokes equation sufficiently close to being eigenfunctions of the Laplacian","authors":"E. Miller","doi":"10.1090/bproc/62","DOIUrl":"https://doi.org/10.1090/bproc/62","url":null,"abstract":"<p>In this paper, we will prove a new, scale critical regularity criterion for solutions of the Navier–Stokes equation that are sufficiently close to being eigenfunctions of the Laplacian. This estimate improves previous regularity criteria requiring control on the <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"ModifyingAbove upper H With dot Superscript alpha\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mover>\u0000 <mml:mi>H</mml:mi>\u0000 <mml:mo>˙<!-- ˙ --></mml:mo>\u0000 </mml:mover>\u0000 </mml:mrow>\u0000 <mml:mi>α<!-- α --></mml:mi>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">dot {H}^alpha</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> norm of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"u comma\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>u</mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">u,</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> with <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"2 less-than-or-equal-to alpha greater-than five halves comma\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mn>2</mml:mn>\u0000 <mml:mo>≤<!-- ≤ --></mml:mo>\u0000 <mml:mi>α<!-- α --></mml:mi>\u0000 <mml:mo>></mml:mo>\u0000 <mml:mfrac>\u0000 <mml:mn>5</mml:mn>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:mfrac>\u0000 <mml:mo>,</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">2leq alpha >frac {5}{2},</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> to a regularity criterion requiring control on the <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"ModifyingAbove upper H With dot Superscript alpha\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mover>\u0000 <mml:mi>H</mml:mi>\u0000 <mml:mo>˙<!-- ˙ --></mml:mo>\u0000 </mml:mover>\u0000 </mml:mrow>\u0000 <mml:mi>α<!-- α --></mml:mi>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">dot {H}^alpha</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> norm multiplied by the deficit in the interpolation inequality for the embedding of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"ModifyingAbove upper H With dot Superscript alpha minus 2 Baseline intersection ModifyingAbove upper H With dot Superscript alpha Baseline right-arrow with hook ModifyingAbove upper H With dot Superscript alpha minus 1 Baseline period\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mover>\u0000 <mml:mi>H</mml:mi>\u0000 <mml:mo>˙<!-- ˙ --></mml:mo>\u0000 </mml:mover>\u0000 ","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117106214","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Transchromatic extensions in motivic bordism","authors":"A. Beaudry, M. Hill, Xiaolin Shi, Mingcong Zeng","doi":"10.1090/bproc/108","DOIUrl":"https://doi.org/10.1090/bproc/108","url":null,"abstract":"<p>We show a number of Toda brackets in the homotopy of the motivic bordism spectrum <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M upper G upper L\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>M</mml:mi>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:mi>L</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">MGL</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and of the Real bordism spectrum <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M upper U Subscript double-struck upper R\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>M</mml:mi>\u0000 <mml:msub>\u0000 <mml:mi>U</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">R</mml:mi>\u0000 </mml:mrow>\u0000 </mml:mrow>\u0000 </mml:msub>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">MU_{mathbb R}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. These brackets are “red-shifting” in the sense that while the terms in the bracket will be of some chromatic height <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>, the bracket itself will be of chromatic height <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis n plus 1 right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:mo>+</mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">(n+1)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. Using these, we deduce a family of exotic multiplications in the <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"pi Subscript left-parenthesis asterisk comma asterisk right-parenthesis Baseline upper M upper G upper L\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msub>\u0000 <mml:mi>π<!-- π --></mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mo>∗<!-- ∗ --></mml:mo>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mo>∗<!-- ∗ --></mml:mo>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 </mml:msub>\u0000 <mml:mi>M</mml:mi>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:mi>L</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">pi _{(ast ,ast )}MGL</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-module structure of the motivic Morava <inline-formula content-type=\"math/mathml\">\u0000<mml:ma","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"97 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116100155","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Big fields that are not large","authors":"B. Mazur, K. Rubin","doi":"10.1090/bproc/57","DOIUrl":"https://doi.org/10.1090/bproc/57","url":null,"abstract":"<p>A subfield <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 <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Q overbar\">\u0000 <mml:semantics>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mover>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">Q</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mo stretchy=\"false\">¯<!-- ¯ --></mml:mo>\u0000 </mml:mover>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">bar {mathbb {Q}}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is <italic>large</italic> if every smooth curve <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C\">\u0000 <mml:semantics>\u0000 <mml:mi>C</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">C</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> 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> with a <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>-rational point has infinitely many <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>-rational points. A subfield <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 <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Q overbar\">\u0000 <mml:semantics>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mover>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">Q</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mo stretchy=\"false\">¯<!-- ¯ --></mml:mo>\u0000 </mml:mover>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">bar {mathbb {Q}}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is <italic>big</italic> if for every positive integer <inline-formula con","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133524467","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A locally anisotropic regularity criterion for the Navier–Stokes equation in terms of vorticity","authors":"E. Miller","doi":"10.1090/bproc/74","DOIUrl":"https://doi.org/10.1090/bproc/74","url":null,"abstract":"In this paper, we will prove a regularity criterion that guarantees solutions of the Navier–Stokes equation must remain smooth so long as the vorticity restricted to a plane remains bounded in the scale critical space \u0000\u0000 \u0000 \u0000 \u0000 L\u0000 t\u0000 4\u0000 \u0000 \u0000 L\u0000 x\u0000 2\u0000 \u0000 \u0000 L^4_t L^2_x\u0000 \u0000\u0000, where the plane may vary in space and time as long as the gradient of the unit vector orthogonal to the plane remains bounded. This extends previous work by Chae and Choe that guaranteed that solutions of the Navier–Stokes equation must remain smooth as long as the vorticity restricted to a fixed plane remains bounded in a family of scale critical spaces. This regularity criterion also can be seen as interpolating between Chae and Choe’s regularity criterion in terms of two vorticity components and Beirão da Veiga and Berselli’s regularity criterion in terms of the gradient of vorticity direction. In physical terms, this regularity criterion is consistent with key aspects of the Kolmogorov theory of turbulence, because it requires that finite-time blowup for solutions of the Navier–Stokes equation must be fully three dimensional at all length scales.","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128271436","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A bound for the image conductor of a principally polarized abelian variety with open Galois image","authors":"Jacob Mayle","doi":"10.1090/bproc/131","DOIUrl":"https://doi.org/10.1090/bproc/131","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 A\">\u0000 <mml:semantics>\u0000 <mml:mi>A</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">A</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> be a principally polarized abelian variety of dimension <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"g\">\u0000 <mml:semantics>\u0000 <mml:mi>g</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">g</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> over a number 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>. Assume that the image of the adelic Galois representation of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A\">\u0000 <mml:semantics>\u0000 <mml:mi>A</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">A</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is an open subgroup of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G upper S p Subscript 2 g Baseline left-parenthesis ModifyingAbove double-struck upper Z With caret right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:mi>S</mml:mi>\u0000 <mml:msub>\u0000 <mml:mi>p</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mn>2</mml:mn>\u0000 <mml:mi>g</mml:mi>\u0000 </mml:mrow>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mover>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">Z</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mo stretchy=\"false\">^<!-- ^ --></mml:mo>\u0000 </mml:mover>\u0000 </mml:mrow>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">GSp_{2g}(hat {mathbb {Z}})</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. Then there exists a positive integer <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"m\">\u0000 <mml:semantics>\u0000 <mml:mi>m</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">m</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> so that the Galois image of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A\">\u0000 <mml:semantics>\u0000 <mml:mi>A</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">A</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is the full preimage of its reduction modulo <inline-formula content-type=\"math/math","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123155889","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Cusp types of quotients of hyperbolic knot complements","authors":"Neil R. Hoffman","doi":"10.1090/bproc/104","DOIUrl":"https://doi.org/10.1090/bproc/104","url":null,"abstract":"<p>This paper completes a classification of the types of orientable and non-orientable cusps that can arise in the quotients of hyperbolic knot complements. In particular, <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper S squared left-parenthesis 2 comma 4 comma 4 right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msup>\u0000 <mml:mi>S</mml:mi>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:msup>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mn>2</mml:mn>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mn>4</mml:mn>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mn>4</mml:mn>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">S^2(2,4,4)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> cannot be the cusp cross-section of any orbifold quotient of a hyperbolic knot complement. Furthermore, if a knot complement covers an orbifold with a <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper S squared left-parenthesis 2 comma 3 comma 6 right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msup>\u0000 <mml:mi>S</mml:mi>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:msup>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mn>2</mml:mn>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mn>3</mml:mn>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mn>6</mml:mn>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">S^2(2,3,6)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> cusp, it also covers an orbifold with a <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper S squared left-parenthesis 3 comma 3 comma 3 right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msup>\u0000 <mml:mi>S</mml:mi>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:msup>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mn>3</mml:mn>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mn>3</mml:mn>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mn>3</mml:mn>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">S^2(3,3,3)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> cusp. We end with a discussion that shows all cusp types arise in the quotients of link complements.</p>","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126407385","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}