MathematikaPub Date : 2026-03-19DOI: 10.1112/mtk.70085
Gergely Ambrus, Rainie Heck
{"title":"A note on the Steinitz lemma","authors":"Gergely Ambrus, Rainie Heck","doi":"10.1112/mtk.70085","DOIUrl":"10.1112/mtk.70085","url":null,"abstract":"<p>We establish the connection between the Steinitz problem for ordering vector families in arbitrary norms and its variant for not necessarily zero-sum families consisting of “nearly unit” vectors.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"72 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2026-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.70085","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147567060","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}
MathematikaPub Date : 2026-03-06DOI: 10.1112/mtk.70081
Tian Wang, Pengcheng Zhang
{"title":"Ordinary primes for -type abelian varieties and weight 2 modular forms","authors":"Tian Wang, Pengcheng Zhang","doi":"10.1112/mtk.70081","DOIUrl":"https://doi.org/10.1112/mtk.70081","url":null,"abstract":"<p>Let <span></span><math></math> be a <span></span><math></math>-dimensional abelian variety defined over a number field <span></span><math></math>. It is conjectured that the set of ordinary primes of <span></span><math></math> over <span></span><math></math> has positive density, and this is known to be true when <span></span><math></math>, or for certain abelian varieties with extra endomorphisms. In this paper, we extend the family of abelian varieties whose sets of ordinary primes have positive density. Specifically, we show that if the endomorphism algebra of <span></span><math></math> contains a number field <span></span><math></math> of degree <span></span><math></math>, then under certain conditions on the fields <span></span><math></math> and <span></span><math></math>, the set of ordinary primes of <span></span><math></math> over <span></span><math></math> has positive density. This includes <span></span><math></math>-type abelian varieties over <span></span><math></math> (resp., quadratic number fields) of dimension <span></span><math></math> or <span></span><math></math> (resp., <span></span><math></math>) for any rational prime <span></span><math></math>. The proof is carried out in the general setting of compatible systems of Galois representations, and as a consequence, it also implies a positive density result for the sets of ordinary primes of certain modular forms of weight 2.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"72 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2026-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.70081","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147563973","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}
MathematikaPub Date : 2026-03-06DOI: 10.1112/mtk.70081
Tian Wang, Pengcheng Zhang
{"title":"Ordinary primes for -type abelian varieties and weight 2 modular forms","authors":"Tian Wang, Pengcheng Zhang","doi":"10.1112/mtk.70081","DOIUrl":"10.1112/mtk.70081","url":null,"abstract":"<p>Let <span></span><math></math> be a <span></span><math></math>-dimensional abelian variety defined over a number field <span></span><math></math>. It is conjectured that the set of ordinary primes of <span></span><math></math> over <span></span><math></math> has positive density, and this is known to be true when <span></span><math></math>, or for certain abelian varieties with extra endomorphisms. In this paper, we extend the family of abelian varieties whose sets of ordinary primes have positive density. Specifically, we show that if the endomorphism algebra of <span></span><math></math> contains a number field <span></span><math></math> of degree <span></span><math></math>, then under certain conditions on the fields <span></span><math></math> and <span></span><math></math>, the set of ordinary primes of <span></span><math></math> over <span></span><math></math> has positive density. This includes <span></span><math></math>-type abelian varieties over <span></span><math></math> (resp., quadratic number fields) of dimension <span></span><math></math> or <span></span><math></math> (resp., <span></span><math></math>) for any rational prime <span></span><math></math>. The proof is carried out in the general setting of compatible systems of Galois representations, and as a consequence, it also implies a positive density result for the sets of ordinary primes of certain modular forms of weight 2.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"72 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2026-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.70081","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147563972","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}
MathematikaPub Date : 2026-03-05DOI: 10.1112/mtk.70080
Dung M. Ha, Anh D. Hoang, Hieu T. Ngo
{"title":"On the least almost-prime in an arithmetic progression","authors":"Dung M. Ha, Anh D. Hoang, Hieu T. Ngo","doi":"10.1112/mtk.70080","DOIUrl":"https://doi.org/10.1112/mtk.70080","url":null,"abstract":"<p>For all sufficiently large <span></span><math></math>, in any arithmetic progression <span></span><math></math> in which <span></span><math></math> and <span></span><math></math> are relatively prime there exists a positive integer with at most two prime factors (counted with multiplicity) which is asymptotically less than <span></span><math></math>. The proof uses the weighted sieve of Greaves–Halberstam–Richert with bilinear remainder terms and Selberg's sieve.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"72 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2026-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147563490","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}
MathematikaPub Date : 2026-03-05DOI: 10.1112/mtk.70080
Dung M. Ha, Anh D. Hoang, Hieu T. Ngo
{"title":"On the least almost-prime in an arithmetic progression","authors":"Dung M. Ha, Anh D. Hoang, Hieu T. Ngo","doi":"10.1112/mtk.70080","DOIUrl":"10.1112/mtk.70080","url":null,"abstract":"<p>For all sufficiently large <span></span><math></math>, in any arithmetic progression <span></span><math></math> in which <span></span><math></math> and <span></span><math></math> are relatively prime there exists a positive integer with at most two prime factors (counted with multiplicity) which is asymptotically less than <span></span><math></math>. The proof uses the weighted sieve of Greaves–Halberstam–Richert with bilinear remainder terms and Selberg's sieve.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"72 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2026-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147563292","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}