We study the resurgence structure of a formal normalization of a certain vector field to the normal form using “mould calculus” developed by J. Écalle. We also describe the resurgence structure of transseries solutions of a nonlinear ordinary differential equation.
{"title":"Resurgent transseries, mould calculus and Connes-Kreimer Hopf algebra","authors":"Shingo Kamimoto","doi":"10.3792/pjaa.99.013","DOIUrl":"https://doi.org/10.3792/pjaa.99.013","url":null,"abstract":"We study the resurgence structure of a formal normalization of a certain vector field to the normal form using “mould calculus” developed by J. Écalle. We also describe the resurgence structure of transseries solutions of a nonlinear ordinary differential equation.","PeriodicalId":49668,"journal":{"name":"Proceedings of the Japan Academy Series A-Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135614888","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}
In this paper, we construct a crepant resolution for the quotient singularity $mathbf{A}^{4}/A_{4}$ in characteristic 2, where $A_{4}$ is the alternating group of degree 4 with permutation action on $mathbf{A}^{4}$. By computing the Euler number of the crepant resolution, we obtain a new counterexample to an analogous statement of McKay correspondence in positive characteristic.
{"title":"Crepant resolution of $mathbf{A}^{4}/A_{4}$ in characteristic 2","authors":"Linghu Fan","doi":"10.3792/pjaa.99.014","DOIUrl":"https://doi.org/10.3792/pjaa.99.014","url":null,"abstract":"In this paper, we construct a crepant resolution for the quotient singularity $mathbf{A}^{4}/A_{4}$ in characteristic 2, where $A_{4}$ is the alternating group of degree 4 with permutation action on $mathbf{A}^{4}$. By computing the Euler number of the crepant resolution, we obtain a new counterexample to an analogous statement of McKay correspondence in positive characteristic.","PeriodicalId":49668,"journal":{"name":"Proceedings of the Japan Academy Series A-Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135615021","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}
Josep M. Miret, Daniel Sadornil, Juan Tena, Javier Valera
Let $E$ be an elliptic curve defined over a finite field $mathbf{F}_{q}$, $q = p^{d}$, $p > 3$, and a prime number $ell > 3$ such that $q equiv 1 pmod{ell}$ and $ell mid # E(mathbf{F}_{q})$. In this paper we study the possible factorisation patterns over $mathbf{F}_{q}[x]$ of the $ell^{k}$-division polynomials associated to $E$ with $k geq 2$, extending the work of Verdure [6] for $k=1$.
{"title":"A note on factorisation patterns of division polynomials of elliptic curves over finite fields","authors":"Josep M. Miret, Daniel Sadornil, Juan Tena, Javier Valera","doi":"10.3792/pjaa.99.011","DOIUrl":"https://doi.org/10.3792/pjaa.99.011","url":null,"abstract":"Let $E$ be an elliptic curve defined over a finite field $mathbf{F}_{q}$, $q = p^{d}$, $p > 3$, and a prime number $ell > 3$ such that $q equiv 1 pmod{ell}$ and $ell mid # E(mathbf{F}_{q})$. In this paper we study the possible factorisation patterns over $mathbf{F}_{q}[x]$ of the $ell^{k}$-division polynomials associated to $E$ with $k geq 2$, extending the work of Verdure [6] for $k=1$.","PeriodicalId":49668,"journal":{"name":"Proceedings of the Japan Academy Series A-Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134934890","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}
In 1977, Gosper introduced a conjectural evaluation for a hypergeometric series that has been described as strange by a number of authors. In 2013, Ebisu proved Gosper’s conjecture using contiguity operators. Subsequently, in 2017, Chu provided another proof of Gosper’s conjecture, using a telescoping argument together with Pfaff’s transformation. In this article, we present a new and simplified proof of Gosper’s conjecture that is inequivalent to the previous proofs due to Ebisu and Chu. Our proof relies on an evaluation technique that was previously given by Campbell and Cantarini and that involves the modified Abel lemma on summation by parts. We also show how this method may be applied to prove generalizations and variants of Gosper’s summation.
{"title":"Gosper’s strange series: A new, simplified proof and generalizations","authors":"John Campbell","doi":"10.3792/pjaa.99.012","DOIUrl":"https://doi.org/10.3792/pjaa.99.012","url":null,"abstract":"In 1977, Gosper introduced a conjectural evaluation for a hypergeometric series that has been described as strange by a number of authors. In 2013, Ebisu proved Gosper’s conjecture using contiguity operators. Subsequently, in 2017, Chu provided another proof of Gosper’s conjecture, using a telescoping argument together with Pfaff’s transformation. In this article, we present a new and simplified proof of Gosper’s conjecture that is inequivalent to the previous proofs due to Ebisu and Chu. Our proof relies on an evaluation technique that was previously given by Campbell and Cantarini and that involves the modified Abel lemma on summation by parts. We also show how this method may be applied to prove generalizations and variants of Gosper’s summation.","PeriodicalId":49668,"journal":{"name":"Proceedings of the Japan Academy Series A-Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134934891","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":"Non-purely non-symplectic automorphisms of odd order on $K3$ surfaces","authors":"Shingo Taki","doi":"10.3792/pjaa.99.009","DOIUrl":"https://doi.org/10.3792/pjaa.99.009","url":null,"abstract":"","PeriodicalId":49668,"journal":{"name":"Proceedings of the Japan Academy Series A-Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80447762","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":"Notes on a certain local time and excursions of simple symmetric random walks","authors":"T. Fujita, N. Yoshida","doi":"10.3792/pjaa.99.010","DOIUrl":"https://doi.org/10.3792/pjaa.99.010","url":null,"abstract":"","PeriodicalId":49668,"journal":{"name":"Proceedings of the Japan Academy Series A-Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81355391","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":"$q$-Log-concavity and $q$-unimodality of Gaussian polynomials and a problem of Andrews and Newman","authors":"Shane Chern","doi":"10.3792/pjaa.99.007","DOIUrl":"https://doi.org/10.3792/pjaa.99.007","url":null,"abstract":"","PeriodicalId":49668,"journal":{"name":"Proceedings of the Japan Academy Series A-Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86991181","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":"Proportion of modular forms with transcendental zeros for general levels","authors":"D. Choi, Youngmin Lee, Subong Lim, Jaegwang Ryu","doi":"10.3792/pjaa.99.004","DOIUrl":"https://doi.org/10.3792/pjaa.99.004","url":null,"abstract":"","PeriodicalId":49668,"journal":{"name":"Proceedings of the Japan Academy Series A-Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85949206","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":"Rational period functions for $Gamma_{0}^{+}(2)$ with poles only in $mathbf{Q}cup {infty}$","authors":"D. Y. Oh","doi":"10.3792/pjaa.99.002","DOIUrl":"https://doi.org/10.3792/pjaa.99.002","url":null,"abstract":"","PeriodicalId":49668,"journal":{"name":"Proceedings of the Japan Academy Series A-Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90886104","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":"On some arithmetic questions of reductive groups over algebraic extensions of local and global fields","authors":"N. Q. Thang","doi":"10.3792/pjaa.99.001","DOIUrl":"https://doi.org/10.3792/pjaa.99.001","url":null,"abstract":"","PeriodicalId":49668,"journal":{"name":"Proceedings of the Japan Academy Series A-Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85428188","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}
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