Indagationes Mathematicae-New Series最新文献

{"title":"Ranks of elliptic curves in cyclic sextic extensions of Q","authors":"Hershy Kisilevsky ,&nbsp;Masato Kuwata","doi":"10.1016/j.indag.2024.01.004","DOIUrl":"10.1016/j.indag.2024.01.004","url":null,"abstract":"<div><p><span>For an elliptic curve </span><span><math><mrow><mi>E</mi><mo>/</mo><mi>Q</mi></mrow></math></span> we show that there are infinitely many cyclic sextic extensions <span><math><mrow><mi>K</mi><mo>/</mo><mi>Q</mi></mrow></math></span> such that the Mordell–Weil group <span><math><mrow><mi>E</mi><mrow><mo>(</mo><mi>K</mi><mo>)</mo></mrow></mrow></math></span> has rank greater than the subgroup of <span><math><mrow><mi>E</mi><mrow><mo>(</mo><mi>K</mi><mo>)</mo></mrow></mrow></math></span> generated by all the <span><math><mrow><mi>E</mi><mrow><mo>(</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span> for the proper subfields <span><math><mrow><mi>F</mi><mo>⊂</mo><mi>K</mi></mrow></math></span>. For certain curves <span><math><mrow><mi>E</mi><mo>/</mo><mi>Q</mi></mrow></math></span> we show that the number of such fields <span><math><mi>K</mi></math></span> of conductor less than <span><math><mi>X</mi></math></span> is <span><math><mrow><mo>≫</mo><msqrt><mrow><mi>X</mi></mrow></msqrt></mrow></math></span>.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 4","pages":"Pages 728-743"},"PeriodicalIF":0.5,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139551961","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Computing the Weil representation of a superelliptic curve","authors":"Irene I. Bouw,&nbsp;Duc Khoi Do,&nbsp;Stefan Wewers","doi":"10.1016/j.indag.2024.01.002","DOIUrl":"10.1016/j.indag.2024.01.002","url":null,"abstract":"<div><p>We study the Weil representation <span><math><mi>ρ</mi></math></span> of a curve over a <span><math><mi>p</mi></math></span>-adic field with potential reduction of compact type. We show that <span><math><mi>ρ</mi></math></span> can be reconstructed from its stable reduction. For superelliptic curves of the form <span><math><mrow><msup><mrow><mi>y</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> at primes <span><math><mi>p</mi></math></span> whose residue characteristic is prime to the exponent <span><math><mi>n</mi></math></span> we make this explicit.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 4","pages":"Pages 708-727"},"PeriodicalIF":0.5,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0019357724000028/pdfft?md5=a98632bbf32b4580b4c64c774c1f6a96&pid=1-s2.0-S0019357724000028-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139506875","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Regular models of hyperelliptic curves","authors":"Simone Muselli","doi":"10.1016/j.indag.2023.12.001","DOIUrl":"10.1016/j.indag.2023.12.001","url":null,"abstract":"<div><p>Let <span><math><mi>K</mi></math></span> be a complete discretely valued field of residue characteristic not 2 and <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>K</mi></mrow></msub></math></span> its ring of integers. We explicitly construct a regular model over <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>K</mi></mrow></msub></math></span> with strict normal crossings of any hyperelliptic curve <span><math><mrow><mi>C</mi><mo>/</mo><mi>K</mi><mo>:</mo><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span>. For this purpose, we introduce the new notion of <em>MacLane cluster picture</em>, that aims to be a link between clusters and MacLane valuations.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 4","pages":"Pages 646-697"},"PeriodicalIF":0.5,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0019357723001040/pdfft?md5=04ca296b6016027d47af6c7c64f21d09&pid=1-s2.0-S0019357723001040-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138555115","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Curves over finite fields and arithmetic and geometry of K3 surfaces: Celebrating Jaap Top’s 60th anniversary","authors":"Remke Kloosterman (Managing Editor),&nbsp;Steffen Müller,&nbsp;Cecília Salgado,&nbsp;Lenny Taelman","doi":"10.1016/j.indag.2024.06.006","DOIUrl":"https://doi.org/10.1016/j.indag.2024.06.006","url":null,"abstract":"","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 4","pages":"Page 609"},"PeriodicalIF":0.5,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141582268","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Counterexamples to the Hasse Principle among the twists of the Klein quartic","authors":"Elisa Lorenzo García ,&nbsp;Michaël Vullers","doi":"10.1016/j.indag.2023.08.007","DOIUrl":"10.1016/j.indag.2023.08.007","url":null,"abstract":"<div><p>In this paper we inspect from closer the local and global points of the twists of the Klein quartic. For the local ones we use geometric arguments, while for the global ones we strongly use the modular interpretation of the twists. The main result is providing families with (conjecturally infinitely many) twists of the Klein quartic that are counterexamples to the Hasse Principle.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 4","pages":"Pages 638-645"},"PeriodicalIF":0.5,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0019357723000848/pdfft?md5=03e46a5dbb56004e38e7926d976cb7c3&pid=1-s2.0-S0019357723000848-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135944511","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the Galois-invariant part of the Weyl group of the Picard lattice of a K3 surface","authors":"Wim Nijgh,&nbsp;Ronald van Luijk","doi":"10.1016/j.indag.2023.08.004","DOIUrl":"10.1016/j.indag.2023.08.004","url":null,"abstract":"<div><p>Let <span><math><mi>X</mi></math></span> denote a K3 surface over an arbitrary field <span><math><mi>k</mi></math></span>. Let <span><math><msup><mrow><mi>k</mi></mrow><mrow><mtext>s</mtext></mrow></msup></math></span> denote a separable closure of <span><math><mi>k</mi></math></span> and let <span><math><msup><mrow><mi>X</mi></mrow><mrow><mtext>s</mtext></mrow></msup></math></span> denote the base change of <span><math><mi>X</mi></math></span> to <span><math><msup><mrow><mi>k</mi></mrow><mrow><mtext>s</mtext></mrow></msup></math></span>. Let <span><math><mrow><mo>O</mo><mrow><mo>(</mo><mo>Pic</mo><mi>X</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mo>O</mo><mrow><mo>(</mo><mo>Pic</mo><msup><mrow><mi>X</mi></mrow><mrow><mtext>s</mtext></mrow></msup><mo>)</mo></mrow></mrow></math></span> denote the group of isometries of the lattices <span><math><mrow><mo>Pic</mo><mi>X</mi></mrow></math></span> and <span><math><mrow><mo>Pic</mo><msup><mrow><mi>X</mi></mrow><mrow><mtext>s</mtext></mrow></msup></mrow></math></span>, respectively. Let <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>X</mi></mrow></msub></math></span> denote the Galois invariant part of the Weyl group of <span><math><mrow><mo>Pic</mo><msup><mrow><mi>X</mi></mrow><mrow><mtext>s</mtext></mrow></msup></mrow></math></span>. One can show that each element in <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>X</mi></mrow></msub></math></span> can be restricted to an element of <span><math><mrow><mo>O</mo><mrow><mo>(</mo><mo>Pic</mo><mi>X</mi><mo>)</mo></mrow></mrow></math></span>. The following question arises: <em>Is the image of the restriction map</em> <span><math><mrow><msub><mrow><mi>R</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>→</mo><mo>O</mo><mrow><mo>(</mo><mo>Pic</mo><mi>X</mi><mo>)</mo></mrow></mrow></math></span> <em>a normal subgroup of</em> <span><math><mrow><mo>O</mo><mrow><mo>(</mo><mo>Pic</mo><mi>X</mi><mo>)</mo></mrow></mrow></math></span> <em>for every K3 surface</em> <span><math><mi>X</mi></math></span><em>?</em> We show that the answer is negative by giving counterexamples over <span><math><mrow><mi>k</mi><mo>=</mo><mi>Q</mi></mrow></math></span>.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 4","pages":"Pages 610-621"},"PeriodicalIF":0.5,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0019357723000812/pdfft?md5=6dd74732aaf671c0aaa5195ba03e905f&pid=1-s2.0-S0019357723000812-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41575279","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Geometric progressions in the sets of values of rational functions","authors":"Maciej Ulas","doi":"10.1016/j.indag.2023.08.005","DOIUrl":"10.1016/j.indag.2023.08.005","url":null,"abstract":"<div><p>Let <span><math><mrow><mi>a</mi><mo>,</mo><mi>Q</mi><mo>∈</mo><mi>Q</mi></mrow></math></span> be given and consider the set <span><math><mrow><mi>G</mi><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>Q</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>{</mo><mi>a</mi><msup><mrow><mi>Q</mi></mrow><mrow><mi>i</mi></mrow></msup><mo>:</mo><mspace></mspace><mi>i</mi><mo>∈</mo><mi>N</mi><mo>}</mo></mrow></mrow></math></span> of terms of geometric progression with 0th term equal to <span><math><mi>a</mi></math></span> and the quotient <span><math><mi>Q</mi></math></span>. Let <span><math><mrow><mi>f</mi><mo>∈</mo><mi>Q</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>f</mi></mrow></msub></math></span> be the set of finite values of <span><math><mi>f</mi></math></span>. We consider the problem of existence of <span><math><mrow><mi>a</mi><mo>,</mo><mi>Q</mi><mo>∈</mo><mi>Q</mi></mrow></math></span> such that <span><math><mrow><mi>G</mi><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>Q</mi><mo>)</mo></mrow><mo>⊂</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></math></span>. In the first part of the paper we describe certain classes of rational functions for which our problem has a positive solution. In the second, experimental, part of the paper we study the stated problem for the rational function <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></mrow><mo>/</mo><mi>x</mi></mrow></math></span>. We relate the problem to the existence of rational points on certain elliptic curves and present interesting numerical observations which allow us to state several questions and conjectures.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 4","pages":"Pages 622-637"},"PeriodicalIF":0.5,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0019357723000824/pdfft?md5=6a0ec32c7eb19c5b691f6b150a52a65c&pid=1-s2.0-S0019357723000824-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46926096","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"p-linear schemes for sequences modulo pr","authors":"Frits Beukers","doi":"10.1016/j.indag.2023.12.003","DOIUrl":"10.1016/j.indag.2023.12.003","url":null,"abstract":"<div><p>Many interesting combinatorial sequences, such as Apéry numbers and Franel numbers, enjoy the so-called Lucas property modulo almost all primes <span><math><mi>p</mi></math></span>. Modulo prime powers <span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msup></math></span> such sequences have a more complicated behaviour which can be described by matrix versions of the Lucas property called <span><math><mi>p</mi></math></span>-linear schemes. They are generalizations of finite <span><math><mi>p</mi></math></span>-automata. In this paper we construct such <span><math><mi>p</mi></math></span>-linear schemes and give upper bounds for the number of states which, for fixed <span><math><mi>r</mi></math></span>, do not depend on <span><math><mi>p</mi></math></span>.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 4","pages":"Pages 698-707"},"PeriodicalIF":0.5,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0019357723001064/pdfft?md5=ea710133f3e4e343c282392434c744c9&pid=1-s2.0-S0019357723001064-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138629792","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Symmetry breaking operators for the reductive dual pair (Ul,Ul′)","authors":"M. McKee ,&nbsp;A. Pasquale ,&nbsp;T. Przebinda","doi":"10.1016/j.indag.2024.06.004","DOIUrl":"10.1016/j.indag.2024.06.004","url":null,"abstract":"&lt;div&gt;&lt;div&gt;We consider the dual pair &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;l&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;l&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; in the symplectic group &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Sp&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;l&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;l&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. Fix a Weil representation of the metaplectic group &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;Sp&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;˜&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;l&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;l&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. Let &lt;span&gt;&lt;math&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;˜&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;˜&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; be the preimages of &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; in &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;Sp&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;˜&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;l&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;l&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, and let &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;Π&lt;/mi&gt;&lt;mo&gt;⊗&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;Π&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; be a genuine irreducible representation of &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;˜&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;˜&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. We study the Weyl symbol &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;Π&lt;/mi&gt;&lt;mo&gt;⊗&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;Π&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt; of the (unique up to a possibly zero constant) symmetry breaking operator (SBO) intertwining the Weil representation with &lt;/span&gt;&lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;Π&lt;/mi&gt;&lt;mo&gt;⊗&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;Π&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. This SBO coincides with the orthogonal projection of the space of the Weil representation onto its &lt;span&gt;&lt;math&gt;&lt;mi&gt;Π&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-isotypic component and also with the orthogonal projection onto its &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;Π&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;-isotypic component. Hence &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;Π&lt;/mi&gt;&lt;mo&gt;⊗&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;Π&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; can be computed in two different ways, one using &lt;span&gt;&lt;math&gt;&lt;mi&gt;Π&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; and the o","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 2","pages":"Pages 413-449"},"PeriodicalIF":0.5,"publicationDate":"2024-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141569877","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Sharp estimates for distinguished random walks on affine buildings of type A˜r","authors":"Jean-Philippe Anker ,&nbsp;Bruno Schapira ,&nbsp;Bartosz Trojan","doi":"10.1016/j.indag.2024.06.002","DOIUrl":"10.1016/j.indag.2024.06.002","url":null,"abstract":"<div><div><span>We study a distinguished random walk on affine buildings of type </span><span><math><msub><mrow><mover><mrow><mi>A</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mspace></mspace><mi>r</mi></mrow></msub></math></span> , which was already considered by Cartwright, Saloff-Coste and Woess. In rank <span><math><mrow><mi>r</mi><mspace></mspace><mo>=</mo><mspace></mspace><mn>2</mn></mrow></math></span>, it is the simple random walk and we obtain optimal global bounds for its transition density (same upper and lower bound, up to multiplicative constants). In the higher rank case, we obtain sharp uniform bounds in fairly large space–time regions which are sufficient for most applications.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 2","pages":"Pages 383-412"},"PeriodicalIF":0.5,"publicationDate":"2024-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141504917","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}