Mathematika最新文献

{"title":"M0, 5: Toward the Chabauty–Kim method in higher dimensions","authors":"Ishai Dan-Cohen,&nbsp;David Jarossay","doi":"10.1112/mtk.12215","DOIUrl":"10.1112/mtk.12215","url":null,"abstract":"<p>If <i>Z</i> is an open subscheme of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>Spec</mo>\u0000 <mi>Z</mi>\u0000 </mrow>\u0000 <annotation>$operatorname{Spec}mathbb {Z}$</annotation>\u0000 </semantics></math>, <i>X</i> is a sufficiently nice <i>Z</i>-model of a smooth curve over <math>\u0000 <semantics>\u0000 <mi>Q</mi>\u0000 <annotation>$mathbb {Q}$</annotation>\u0000 </semantics></math>, and <i>p</i> is a closed point of <i>Z</i>, the Chabauty–Kim method leads to the construction of locally analytic functions on <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>X</mi>\u0000 <mo>(</mo>\u0000 <msub>\u0000 <mi>Z</mi>\u0000 <mi>p</mi>\u0000 </msub>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$X({mathbb {Z}_p})$</annotation>\u0000 </semantics></math> which vanish on <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>X</mi>\u0000 <mo>(</mo>\u0000 <mi>Z</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$X(Z)$</annotation>\u0000 </semantics></math>; we call such functions “Kim functions”. At least in broad outline, the method generalizes readily to higher dimensions. In fact, in some sense, the surface <i>M</i>_{0, 5} should be easier than the previously studied curve <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>M</mi>\u0000 <mrow>\u0000 <mn>0</mn>\u0000 <mo>,</mo>\u0000 <mn>4</mn>\u0000 </mrow>\u0000 </msub>\u0000 <mo>=</mo>\u0000 <msup>\u0000 <mi>P</mi>\u0000 <mn>1</mn>\u0000 </msup>\u0000 <mo>∖</mo>\u0000 <mrow>\u0000 <mo>{</mo>\u0000 <mn>0</mn>\u0000 <mo>,</mo>\u0000 <mn>1</mn>\u0000 <mo>,</mo>\u0000 <mi>∞</mi>\u0000 <mo>}</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$M_{0,4} = mathbb {P}^1 setminus lbrace 0,1,infty rbrace$</annotation>\u0000 </semantics></math> since its points are closely related to those of <i>M</i>_{0, 4}, yet they face a further condition to integrality. This is mirrored by a certain <i>weight advantage</i> we encounter, because of which, <i>M</i>_{0, 5} possesses <i>new Kim functions</i> not coming from <i>M</i>_{0, 4}. Here we focus on the case “<math>\u0000 <semantics>\u0000 <mrow>\u0000 ","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"69 4","pages":"1011-1059"},"PeriodicalIF":0.8,"publicationDate":"2023-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12215","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41296649","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Banach spaces of continuous functions without norming Markushevich bases","authors":"Tommaso Russo,&nbsp;Jacopo Somaglia","doi":"10.1112/mtk.12217","DOIUrl":"10.1112/mtk.12217","url":null,"abstract":"<p>We investigate the question whether a scattered compact topological space <i>K</i> such that <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>C</mi>\u0000 <mo>(</mo>\u0000 <mi>K</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$C(K)$</annotation>\u0000 </semantics></math> has a norming Markushevich basis (M-basis, for short) must be Eberlein. This question originates from the recent solution, due to Hájek, Todorčević and the authors, to an open problem from the 1990s, due to Godefroy. Our prime tool consists in proving that <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>C</mi>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mo>[</mo>\u0000 <mn>0</mn>\u0000 <mo>,</mo>\u0000 <msub>\u0000 <mi>ω</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 <mo>]</mo>\u0000 </mrow>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$C([0,omega _1])$</annotation>\u0000 </semantics></math> does not embed in a Banach space with a norming M-basis, thereby generalising a result due to Alexandrov and Plichko. Subsequently, we give sufficient conditions on a compact <i>K</i> for <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>C</mi>\u0000 <mo>(</mo>\u0000 <mi>K</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$C(K)$</annotation>\u0000 </semantics></math> not to embed in a Banach space with a norming M-basis. Examples of such conditions are that <i>K</i> is a zero-dimensional compact space with a P-point, or a compact tree of height at least <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>ω</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 <mo>+</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation>$omega _1 +1$</annotation>\u0000 </semantics></math>. In particular, this allows us to answer the said question in the case when <i>K</i> is a tree and to obtain a rather general result for Valdivia compacta. Finally, we give some structural results for scattered compact trees; in particular, we prove that scattered trees of height less than ω₂ are Valdivia.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"69 4","pages":"992-1010"},"PeriodicalIF":0.8,"publicationDate":"2023-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12217","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47910696","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On simply normal numbers with digit dependencies","authors":"Verónica Becher,&nbsp;Agustín Marchionna,&nbsp;Gérald Tenenbaum","doi":"10.1112/mtk.12216","DOIUrl":"10.1112/mtk.12216","url":null,"abstract":"<p>Given an integer <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>b</mi>\u0000 <mo>⩾</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation>$bgeqslant 2$</annotation>\u0000 </semantics></math> and a set <math>\u0000 <semantics>\u0000 <mi>P</mi>\u0000 <annotation>${EuScript P}$</annotation>\u0000 </semantics></math> of prime numbers, the set <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>T</mi>\u0000 <mi>P</mi>\u0000 </msub>\u0000 <annotation>${EuScript T}_{EuScript P}$</annotation>\u0000 </semantics></math> of Toeplitz numbers comprises all elements of [0, <i>b</i>[ whose digits <math>\u0000 <semantics>\u0000 <msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msub>\u0000 <mi>a</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>⩾</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msub>\u0000 <annotation>$(a_n)_{ngeqslant 1}$</annotation>\u0000 </semantics></math> in the base-<i>b</i> expansion satisfy <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>a</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <mo>=</mo>\u0000 <msub>\u0000 <mi>a</mi>\u0000 <mrow>\u0000 <mi>p</mi>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$a_n=a_{pn}$</annotation>\u0000 </semantics></math> for all <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>p</mi>\u0000 <mo>∈</mo>\u0000 <mi>P</mi>\u0000 </mrow>\u0000 <annotation>$pin {EuScript P}$</annotation>\u0000 </semantics></math> and <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>⩾</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation>$ngeqslant 1$</annotation>\u0000 </semantics></math>. Using a completely additive arithmetical function, we construct a number in <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>T</mi>\u0000 <mi>P</mi>\u0000 </msub>\u0000 <annotation>${EuScript T}_{EuScript P}$</annotation>\u0000 </semantics></math> that is simply Borel normal if, and only if, <math>\u0000 <semantics>\u0000 <mstyle>\u0000 <mrow>\u0000 <msub>\u0000 <mo>∑</mo>\u0000 <mrow>\u0000 <mi>p</mi>\u0000 <mo>∉</mo>\u0000 ","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"69 4","pages":"988-991"},"PeriodicalIF":0.8,"publicationDate":"2023-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46557821","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Rational points close to non-singular algebraic curves","authors":"Faustin Adiceam,&nbsp;Oscar Marmon","doi":"10.1112/mtk.12214","DOIUrl":"10.1112/mtk.12214","url":null,"abstract":"<p>We study the density of solutions to Diophantine inequalities involving non-singular ternary forms, or equivalently, the density of rational points close to non-singular plane algebraic curves.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"69 4","pages":"957-987"},"PeriodicalIF":0.8,"publicationDate":"2023-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12214","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49630476","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The functional Orlicz–Brunn–Minkowski inequality for q-torsional rigidity","authors":"Jinrong Hu,&nbsp;Ping Zhang","doi":"10.1112/mtk.12213","DOIUrl":"10.1112/mtk.12213","url":null,"abstract":"<p>In this paper, we obtain the functional Orlicz–Brunn–Minkowski inequality and the functional Orlicz–Minkowski inequality for <i>q</i>-torsional rigidity in the smooth category. Furthermore, using an approximation method, we give the general functional Orlicz–Brunn–Minkowski inequality for <i>q</i>-torsional rigidity. As a corollary, we reveal that the functional Orlicz–Brunn–Minkowski inequality is equivalent to the functional Orlicz–Minkowski inequality for <i>q</i>-torsional rigidity in the smooth category. We also give some applications with respect to these two inequalities.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"69 4","pages":"934-956"},"PeriodicalIF":0.8,"publicationDate":"2023-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45701642","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the distribution of equivalence classes of random symmetric p-adic matrices","authors":"Valeriya Kovaleva","doi":"10.1112/mtk.12212","DOIUrl":"https://doi.org/10.1112/mtk.12212","url":null,"abstract":"<p>We consider random symmetric matrices with independent entries distributed according to the Haar measure on <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>Z</mi>\u0000 <mi>p</mi>\u0000 </msub>\u0000 <annotation>$mathbb {Z}_p$</annotation>\u0000 </semantics></math> for odd primes <i>p</i> and derive the distribution of their canonical form with respect to several equivalence relations. We give a few examples of applications including an alternative proof for the result of Bhargava, Cremona, Fisher, Jones and Keating on the probability that a random quadratic form over <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>Z</mi>\u0000 <mi>p</mi>\u0000 </msub>\u0000 <annotation>$mathbb {Z}_p$</annotation>\u0000 </semantics></math> has a non-trivial zero.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"69 4","pages":"903-933"},"PeriodicalIF":0.8,"publicationDate":"2023-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12212","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50152399","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On polynomials with only rational roots","authors":"Lajos Hajdu,&nbsp;Robert Tijdeman,&nbsp;Nóra Varga","doi":"10.1112/mtk.12209","DOIUrl":"10.1112/mtk.12209","url":null,"abstract":"<p>In this paper, we study upper bounds for the degrees of polynomials with only rational roots. First, we assume that the coefficients are bounded. In the second theorem, we suppose that the primes 2 and 3 do not divide any coefficient. The third theorem concerns the case that all coefficients are composed of primes from a fixed finite set.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"69 3","pages":"867-878"},"PeriodicalIF":0.8,"publicationDate":"2023-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12209","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49035118","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A pretentious proof of Linnik's estimate for primes in arithmetic progressions","authors":"Stelios Sachpazis","doi":"10.1112/mtk.12211","DOIUrl":"10.1112/mtk.12211","url":null,"abstract":"<p>In the present paper, the author adopts a pretentious approach and recovers an estimate obtained by Linnik for the sums of the von Mangoldt function Λ on arithmetic progressions. It is the analogue of an estimate that Linnik established in his attempt to prove his celebrated theorem concerning the size of the smallest prime number of an arithmetic progression. Our work builds on ideas coming from the pretentious large sieve of Granville, Harper, and Soundararajan and it also borrows insights from the treatment of Koukoulopoulos on multiplicative functions with small averages.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"69 3","pages":"879-902"},"PeriodicalIF":0.8,"publicationDate":"2023-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44477461","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Entropic exercises around the Kneser–Poulsen conjecture","authors":"Gautam Aishwarya,&nbsp;Irfan Alam,&nbsp;Dongbin Li,&nbsp;Sergii Myroshnychenko,&nbsp;Oscar Zatarain-Vera","doi":"10.1112/mtk.12210","DOIUrl":"10.1112/mtk.12210","url":null,"abstract":"<p>We develop an information-theoretic approach to study the Kneser–Poulsen conjecture in discrete geometry. This leads us to a broad question regarding whether Rényi entropies of independent sums decrease when one of the summands is contracted by a 1-Lipschitz map. We answer this question affirmatively in various cases.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"69 3","pages":"841-866"},"PeriodicalIF":0.8,"publicationDate":"2023-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12210","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49598006","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Dimension formulas for Siegel modular forms of level 4","authors":"Manami Roy,&nbsp;Ralf Schmidt,&nbsp;Shaoyun Yi","doi":"10.1112/mtk.12207","DOIUrl":"10.1112/mtk.12207","url":null,"abstract":"<p>We prove several dimension formulas for spaces of scalar-valued Siegel modular forms of degree 2 with respect to certain congruence subgroups of level 4. In case of cusp forms, all modular forms considered originate from cuspidal automorphic representations of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>GSp</mi>\u0000 <mo>(</mo>\u0000 <mn>4</mn>\u0000 <mo>,</mo>\u0000 <mi>A</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>${rm GSp}(4,{mathbb {A}})$</annotation>\u0000 </semantics></math> whose local component at <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>p</mi>\u0000 <mo>=</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation>$p=2$</annotation>\u0000 </semantics></math> admits nonzero fixed vectors under the principal congruence subgroup of level 2. Using known dimension formulas combined with dimensions of spaces of fixed vectors in local representations at <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>p</mi>\u0000 <mo>=</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation>$p=2$</annotation>\u0000 </semantics></math>, we obtain formulas for the number of relevant automorphic representations. These, in turn, lead to new dimension formulas, in particular for Siegel modular forms with respect to the Klingen congruence subgroup of level 4.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"69 3","pages":"795-840"},"PeriodicalIF":0.8,"publicationDate":"2023-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44248425","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}