Pub Date : 2022-04-21DOI: 10.4153/S0008414X2200061X
V. Gould, Craig Miller, T. Quinn-Gregson, N. Ruškuc
Abstract A semigroup S is said to be right pseudo-finite if the universal right congruence can be generated by a finite set $Usubseteq Stimes S$ , and there is a bound on the length of derivations for an arbitrary pair $(s,t)in Stimes S$ as a consequence of those in U. This article explores the existence and nature of a minimal ideal in a right pseudo-finite semigroup. Continuing the theme started in an earlier work by Dandan et al., we show that in several natural classes of monoids, right pseudo-finiteness implies the existence of a completely simple minimal ideal. This is the case for orthodox monoids, completely regular monoids, and right reversible monoids, which include all commutative monoids. We also show that certain other conditions imply the existence of a minimal ideal, which need not be completely simple; notably, this is the case for semigroups in which one of the Green’s preorders ${leq _{mathcal {L}}}$ or ${leq _{mathcal {J}}}$ is left compatible with multiplication. Finally, we establish a number of examples of pseudo-finite monoids without a minimal ideal. We develop an explicit construction that yields such examples with additional desired properties, for instance, regularity or ${mathcal {J}}$ -triviality.
{"title":"On minimal ideals in pseudo-finite semigroups","authors":"V. Gould, Craig Miller, T. Quinn-Gregson, N. Ruškuc","doi":"10.4153/S0008414X2200061X","DOIUrl":"https://doi.org/10.4153/S0008414X2200061X","url":null,"abstract":"Abstract A semigroup S is said to be right pseudo-finite if the universal right congruence can be generated by a finite set \u0000$Usubseteq Stimes S$\u0000 , and there is a bound on the length of derivations for an arbitrary pair \u0000$(s,t)in Stimes S$\u0000 as a consequence of those in U. This article explores the existence and nature of a minimal ideal in a right pseudo-finite semigroup. Continuing the theme started in an earlier work by Dandan et al., we show that in several natural classes of monoids, right pseudo-finiteness implies the existence of a completely simple minimal ideal. This is the case for orthodox monoids, completely regular monoids, and right reversible monoids, which include all commutative monoids. We also show that certain other conditions imply the existence of a minimal ideal, which need not be completely simple; notably, this is the case for semigroups in which one of the Green’s preorders \u0000${leq _{mathcal {L}}}$\u0000 or \u0000${leq _{mathcal {J}}}$\u0000 is left compatible with multiplication. Finally, we establish a number of examples of pseudo-finite monoids without a minimal ideal. We develop an explicit construction that yields such examples with additional desired properties, for instance, regularity or \u0000${mathcal {J}}$\u0000 -triviality.","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"38 1","pages":"2007 - 2037"},"PeriodicalIF":0.7,"publicationDate":"2022-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85514702","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}
Pub Date : 2022-04-12DOI: 10.4153/s0008414x22000669
Jeffrey Im, G. Elliott
It is shown that the coloured isomorphism class of a unital, simple, Z -stable, separable amenable C ∗ -algebra satisfying the Universal Coefficient Theorem (UCT) is determined by its tracial simplex. This is a joint work with George A. Elliott.
证明了满足普适系数定理(Universal Coefficient Theorem, UCT)的一元、简单、Z稳定、可分可调C * -代数的彩色同构类是由它的迹单纯形决定的。这是和George a . Elliott合作的作品。
{"title":"Coloured Isomorphism of Classifiable C*-algebras","authors":"Jeffrey Im, G. Elliott","doi":"10.4153/s0008414x22000669","DOIUrl":"https://doi.org/10.4153/s0008414x22000669","url":null,"abstract":"It is shown that the coloured isomorphism class of a unital, simple, Z -stable, separable amenable C ∗ -algebra satisfying the Universal Coefficient Theorem (UCT) is determined by its tracial simplex. This is a joint work with George A. Elliott.","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"11 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84984689","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}
Pub Date : 2022-04-04DOI: 10.4153/S0008414X22000529
Maria Kourou, Oliver Roth
Abstract We prove several sharp distortion and monotonicity theorems for spherically convex functions defined on the unit disk involving geometric quantities such as spherical length, spherical area, and total spherical curvature. These results can be viewed as geometric variants of the classical Schwarz lemma for spherically convex functions.
{"title":"Geometric versions of Schwarz’s lemma for spherically convex functions","authors":"Maria Kourou, Oliver Roth","doi":"10.4153/S0008414X22000529","DOIUrl":"https://doi.org/10.4153/S0008414X22000529","url":null,"abstract":"Abstract We prove several sharp distortion and monotonicity theorems for spherically convex functions defined on the unit disk involving geometric quantities such as spherical length, spherical area, and total spherical curvature. These results can be viewed as geometric variants of the classical Schwarz lemma for spherically convex functions.","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"13 1","pages":"1780 - 1799"},"PeriodicalIF":0.7,"publicationDate":"2022-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81803059","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}
Pub Date : 2022-04-04DOI: 10.4153/s0008414x23000184
Thanasis Bouganis, Yubo Jin
In this paper we prove the algebraicity of some L-values attached to quaternionic modular forms. We follow the rather well established path of the doubling method. Our main contribution is that we include the case where the corresponding symmetric space is of non-tube type. We make various aspects very explicit such as, the doubling embedding, coset decomposition, and the definition of algebraicity of modular forms via CM points.
{"title":"Algebraicity of L-values attached to Quaternionic modular forms","authors":"Thanasis Bouganis, Yubo Jin","doi":"10.4153/s0008414x23000184","DOIUrl":"https://doi.org/10.4153/s0008414x23000184","url":null,"abstract":"In this paper we prove the algebraicity of some L-values attached to quaternionic modular forms. We follow the rather well established path of the doubling method. Our main contribution is that we include the case where the corresponding symmetric space is of non-tube type. We make various aspects very explicit such as, the doubling embedding, coset decomposition, and the definition of algebraicity of modular forms via CM points.","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"328 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77071218","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}
Pub Date : 2022-04-01DOI: 10.4153/s0008414x2200013x
{"title":"CJM volume 74 issue 2 Cover and Back matter","authors":"","doi":"10.4153/s0008414x2200013x","DOIUrl":"https://doi.org/10.4153/s0008414x2200013x","url":null,"abstract":"","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"2676 1","pages":"b1 - b2"},"PeriodicalIF":0.7,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81912926","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}
Pub Date : 2022-03-28DOI: 10.4153/s0008414x23000111
J. Rolin, Tamara Servi, P. Speissegger
We develop multisummability, in the positive real direction, for generalized power series with natural support, and we prove o-minimality of the expansion of the real field by all multisums of these series. This resulting structure expands both $mathbb{R}_{mathcal{G}}$ and the reduct of $mathbb{R}_{mathrm{an}^*}$ generated by all convergent generalized power series with natural support; in particular, its expansion by the exponential function defines both the Gamma function on $(0,infty)$ and the Zeta function on $(1,infty)$.
{"title":"MULTISUMMABILITY FOR GENERALIZED POWER SERIES","authors":"J. Rolin, Tamara Servi, P. Speissegger","doi":"10.4153/s0008414x23000111","DOIUrl":"https://doi.org/10.4153/s0008414x23000111","url":null,"abstract":"We develop multisummability, in the positive real direction, for generalized power series with natural support, and we prove o-minimality of the expansion of the real field by all multisums of these series. This resulting structure expands both $mathbb{R}_{mathcal{G}}$ and the reduct of $mathbb{R}_{mathrm{an}^*}$ generated by all convergent generalized power series with natural support; in particular, its expansion by the exponential function defines both the Gamma function on $(0,infty)$ and the Zeta function on $(1,infty)$.","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"6 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84424268","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}
Pub Date : 2022-03-26DOI: 10.4153/s0008414x23000068
Cipriana Anghel
A BSTRACT . We describe the small-time heat kernel asymptotics of real powers ∆ r , r ∈ (0 , 1) of a non-negative self-adjoint generalized Laplacian ∆ acting on the sections of a hermitian vector bundle E over a closed oriented manifold M . First we treat separately the asymptotic on the diagonal of M × M and in a compact set away from it. Logarithmic terms appear only if n is odd and r is rational with even denominator. We prove the non-triviality of the coefficients appearing in the diagonal asymptotics, and also the non-locality of some of the coefficients. In the special case r = 1 / 2 , we give a simultaneous formula by proving that the heat kernel of ∆ 1 / 2 is a polyhomogeneous conormal section in E ⊠ E ∗ on the standard blow-up space M heat of the diagonal at time t = 0 inside [0 , ∞ ) × M × M .
{"title":"Heat kernel asymptotics for real powers of Laplacians","authors":"Cipriana Anghel","doi":"10.4153/s0008414x23000068","DOIUrl":"https://doi.org/10.4153/s0008414x23000068","url":null,"abstract":"A BSTRACT . We describe the small-time heat kernel asymptotics of real powers ∆ r , r ∈ (0 , 1) of a non-negative self-adjoint generalized Laplacian ∆ acting on the sections of a hermitian vector bundle E over a closed oriented manifold M . First we treat separately the asymptotic on the diagonal of M × M and in a compact set away from it. Logarithmic terms appear only if n is odd and r is rational with even denominator. We prove the non-triviality of the coefficients appearing in the diagonal asymptotics, and also the non-locality of some of the coefficients. In the special case r = 1 / 2 , we give a simultaneous formula by proving that the heat kernel of ∆ 1 / 2 is a polyhomogeneous conormal section in E ⊠ E ∗ on the standard blow-up space M heat of the diagonal at time t = 0 inside [0 , ∞ ) × M × M .","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"15 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83387467","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}
Pub Date : 2022-03-07DOI: 10.4153/S0008414X22000244
Philippe Michaud-Jacobs
Abstract Let K be a number field. For which primes p does there exist an elliptic curve $E / K$ admitting a K-rational p-isogeny? Although we have an answer to this question over the rationals, extending this to other number fields is a fundamental open problem in number theory. In this paper, we study this question in the case that K is a quadratic field, subject to the assumption that E is semistable at the primes of K above p. We prove results both for families of quadratic fields and for specific quadratic fields.
{"title":"On elliptic curves with p-isogenies over quadratic fields","authors":"Philippe Michaud-Jacobs","doi":"10.4153/S0008414X22000244","DOIUrl":"https://doi.org/10.4153/S0008414X22000244","url":null,"abstract":"Abstract Let K be a number field. For which primes p does there exist an elliptic curve \u0000$E / K$\u0000 admitting a K-rational p-isogeny? Although we have an answer to this question over the rationals, extending this to other number fields is a fundamental open problem in number theory. In this paper, we study this question in the case that K is a quadratic field, subject to the assumption that E is semistable at the primes of K above p. We prove results both for families of quadratic fields and for specific quadratic fields.","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"64 1","pages":"945 - 964"},"PeriodicalIF":0.7,"publicationDate":"2022-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78640508","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}
Pub Date : 2022-03-07DOI: 10.4153/S0008414X22000086
A. Cojocaru, T. Wang
Abstract Let $g geq 1$ be an integer and let $A/mathbb Q$ be an abelian variety that is isogenous over $mathbb Q$ to a product of g elliptic curves defined over $mathbb Q$ , pairwise non-isogenous over $overline {mathbb Q}$ and each without complex multiplication. For an integer t and a positive real number x, denote by $pi _A(x, t)$ the number of primes $p leq x$ , of good reduction for A, for which the Frobenius trace $a_{1, p}(A)$ associated to the reduction of A modulo p equals t. Assuming the Generalized Riemann Hypothesis for Dedekind zeta functions, we prove that $pi _A(x, 0) ll _A x^{1 - frac {1}{3 g+1 }}/(operatorname {log} x)^{1 - frac {2}{3 g+1}}$ and $pi _A(x, t) ll _A x^{1 - frac {1}{3 g + 2}}/(operatorname {log} x)^{1 - frac {2}{3 g + 2}}$ if $t neq 0$ . These bounds largely improve upon recent ones obtained for $g = 2$ by Chen, Jones, and Serban, and may be viewed as generalizations to arbitrary g of the bounds obtained for $g=1$ by Murty, Murty, and Saradha, combined with a refinement in the power of $operatorname {log} x$ by Zywina. Under the assumptions stated above, we also prove the existence of a density one set of primes p satisfying $|a_{1, p}(A)|>p^{frac {1}{3 g + 1} - varepsilon }$ for any fixed $varepsilon>0$ .
{"title":"Bounds for the distribution of the Frobenius traces associated to products of non-CM elliptic curves","authors":"A. Cojocaru, T. Wang","doi":"10.4153/S0008414X22000086","DOIUrl":"https://doi.org/10.4153/S0008414X22000086","url":null,"abstract":"Abstract Let \u0000$g geq 1$\u0000 be an integer and let \u0000$A/mathbb Q$\u0000 be an abelian variety that is isogenous over \u0000$mathbb Q$\u0000 to a product of g elliptic curves defined over \u0000$mathbb Q$\u0000 , pairwise non-isogenous over \u0000$overline {mathbb Q}$\u0000 and each without complex multiplication. For an integer t and a positive real number x, denote by \u0000$pi _A(x, t)$\u0000 the number of primes \u0000$p leq x$\u0000 , of good reduction for A, for which the Frobenius trace \u0000$a_{1, p}(A)$\u0000 associated to the reduction of A modulo p equals t. Assuming the Generalized Riemann Hypothesis for Dedekind zeta functions, we prove that \u0000$pi _A(x, 0) ll _A x^{1 - frac {1}{3 g+1 }}/(operatorname {log} x)^{1 - frac {2}{3 g+1}}$\u0000 and \u0000$pi _A(x, t) ll _A x^{1 - frac {1}{3 g + 2}}/(operatorname {log} x)^{1 - frac {2}{3 g + 2}}$\u0000 if \u0000$t neq 0$\u0000 . These bounds largely improve upon recent ones obtained for \u0000$g = 2$\u0000 by Chen, Jones, and Serban, and may be viewed as generalizations to arbitrary g of the bounds obtained for \u0000$g=1$\u0000 by Murty, Murty, and Saradha, combined with a refinement in the power of \u0000$operatorname {log} x$\u0000 by Zywina. Under the assumptions stated above, we also prove the existence of a density one set of primes p satisfying \u0000$|a_{1, p}(A)|>p^{frac {1}{3 g + 1} - varepsilon }$\u0000 for any fixed \u0000$varepsilon>0$\u0000 .","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"4 1","pages":"687 - 712"},"PeriodicalIF":0.7,"publicationDate":"2022-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79966815","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}
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