{"title":"A note on badly approximable systems of linear forms over a field of formal series","authors":"Mumtaz Hussain, N. Shulga","doi":"10.1007/s40993-023-00437-x","DOIUrl":"https://doi.org/10.1007/s40993-023-00437-x","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"27 1","pages":"1-13"},"PeriodicalIF":0.8,"publicationDate":"2023-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81483668","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":"On the Iwasawa invariants of prime cyclotomic fields","authors":"Sey Y. Kim","doi":"10.1007/s40993-023-00435-z","DOIUrl":"https://doi.org/10.1007/s40993-023-00435-z","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"15 1","pages":"1-20"},"PeriodicalIF":0.8,"publicationDate":"2023-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79099814","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":"On the adjoint of higher order Serre derivatives","authors":"M. Charan","doi":"10.1007/s40993-023-00438-w","DOIUrl":"https://doi.org/10.1007/s40993-023-00438-w","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"15 1","pages":"1-8"},"PeriodicalIF":0.8,"publicationDate":"2023-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85021942","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":"Identities associated to a generalized divisor function and modified Bessel function","authors":"D. Banerjee, B. Maji","doi":"10.1007/s40993-023-00431-3","DOIUrl":"https://doi.org/10.1007/s40993-023-00431-3","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"15 1","pages":"1-29"},"PeriodicalIF":0.8,"publicationDate":"2023-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89071851","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":"Modular forms on $${{,textrm{SU},}}(2,1)$$ with weight $$frac{1}{3}$$","authors":"E. Freitag, Richard M. Hill","doi":"10.1007/s40993-022-00361-6","DOIUrl":"https://doi.org/10.1007/s40993-022-00361-6","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"38 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87504722","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":"The probability of non-isomorphic group structures of isogenous elliptic curves in finite field extensions, I","authors":"J. Cullinan, N. Kaplan","doi":"10.1007/s40993-023-00456-8","DOIUrl":"https://doi.org/10.1007/s40993-023-00456-8","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"57 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72830781","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":"Unlikely intersections on the <i>p</i>-adic formal ball.","authors":"Vlad Serban","doi":"10.1007/s40993-023-00441-1","DOIUrl":"https://doi.org/10.1007/s40993-023-00441-1","url":null,"abstract":"<p><p>We investigate generalizations along the lines of the Mordell-Lang conjecture of the author's <i>p</i>-adic formal Manin-Mumford results for <i>n</i>-dimensional <i>p</i>-divisible formal groups <math><mi>F</mi></math> . In particular, given a finitely generated subgroup <math><mi>Γ</mi></math> of <math><mrow><mi>F</mi> <mo>(</mo> <msub><mover><mi>Q</mi> <mo>¯</mo></mover> <mi>p</mi></msub> <mo>)</mo></mrow> </math> and a closed subscheme <math><mrow><mi>X</mi> <mo>↪</mo> <mi>F</mi></mrow> </math> , we show under suitable assumptions that for any points <math><mrow><mi>P</mi> <mo>∈</mo> <mi>X</mi> <mo>(</mo> <msub><mi>C</mi> <mi>p</mi></msub> <mo>)</mo></mrow> </math> satisfying <math><mrow><mi>n</mi> <mi>P</mi> <mo>∈</mo> <mi>Γ</mi></mrow> </math> for some <math><mrow><mi>n</mi> <mo>∈</mo> <mi>N</mi></mrow> </math> , the minimal such orders <i>n</i> are uniformly bounded whenever <i>X</i> does not contain a formal subgroup translate of positive dimension. In contrast, we then provide counter-examples to a full <i>p</i>-adic formal Mordell-Lang result. Finally, we outline some consequences for the study of the Zariski-density of sets of automorphic objects in <i>p</i>-adic deformations. Specifically, we do so in the context of the nearly ordinary <i>p</i>-adic families of cuspidal cohomological automorphic forms for the general linear group constructed by Hida.</p>","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"9 2","pages":"34"},"PeriodicalIF":0.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10149481/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"9408242","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Sato-Tate distribution of <i>p</i>-adic hypergeometric functions.","authors":"Sudhir Pujahari, Neelam Saikia","doi":"10.1007/s40993-022-00414-w","DOIUrl":"10.1007/s40993-022-00414-w","url":null,"abstract":"<p><p>Recently Ono, Saad and the second author [21] initiated a study of value distribution of certain families of Gaussian hypergeometric functions over large finite fields. They investigated two families of Gaussian hypergeometric functions and showed that they satisfy semicircular and Batman distributions. Motivated by their results we aim to study distributions of certain families of hypergeometric functions in the <i>p</i>-adic setting over large finite fields. In particular, we consider two and six parameters families of hypergeometric functions in the <i>p</i>-adic setting and obtain that their limiting distributions are semicircular over large finite fields. In the process of doing this we also express the traces of <i>p</i>th Hecke operators acting on the spaces of cusp forms of even weight <math><mrow><mi>k</mi> <mo>≥</mo> <mn>4</mn></mrow> </math> and levels 4 and 8 in terms of <i>p</i>-adic hypergeometric function which is of independent interest. These results can be viewed as <i>p</i>-adic analogous of some trace formulas of [1, 2, 6].</p>","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":" ","pages":"4"},"PeriodicalIF":0.6,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9708806/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"35346191","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}