We prove a finiteness theorem for the first flat cohomology group of finite flat group schemes over integral normal proper varieties over finite fields. As a consequence, we can prove the invariance of the finiteness of the Tate-Shafarevich group of Abelian schemes over higher dimensional bases under isogenies and alterations over/of such bases for the p-part. Along the way, we generalize previous results on the Tate-Shafarevich group in this situation.
{"title":"On the p-torsion of the Tate–Shafarevich group of abelian varieties over higher dimensional bases over finite fields","authors":"Timo Keller","doi":"10.5802/jtnb.1211","DOIUrl":"https://doi.org/10.5802/jtnb.1211","url":null,"abstract":"We prove a finiteness theorem for the first flat cohomology group of finite flat group schemes over integral normal proper varieties over finite fields. As a consequence, we can prove the invariance of the finiteness of the Tate-Shafarevich group of Abelian schemes over higher dimensional bases under isogenies and alterations over/of such bases for the p-part. Along the way, we generalize previous results on the Tate-Shafarevich group in this situation.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47360413","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}
The article revisits a result of [4] concerning the structure of the image by the Iwasawa regulator map of the Iwasawa module associated with a semi-stable p-adic representation on an unramified finite extension of Qp and gives a direct proof based on the results of [7] in the crystalline case and [8] in the semi-stable case.
{"title":"Note sur les diviseurs élémentaires du régulateur d’Iwasawa","authors":"B. Perrin-Riou","doi":"10.5802/jtnb.1188","DOIUrl":"https://doi.org/10.5802/jtnb.1188","url":null,"abstract":"The article revisits a result of [4] concerning the structure of the image by the Iwasawa regulator map of the Iwasawa module associated with a semi-stable p-adic representation on an unramified finite extension of Qp and gives a direct proof based on the results of [7] in the crystalline case and [8] in the semi-stable case.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45043033","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}
Chromatic Selmer groups are modified Selmer groups with local information for supersingular primes p. We sketch their role in establishing the p-primary part of the Birch–Swinnerton-Dyer formula in Sections 2–5, and then study the growth of the Mordell–Weil rank along the Zp-extension of a quadratic imaginary number field in which p splits in Section 6.
{"title":"Chromatic Selmer groups and arithmetic invariants of elliptic curves","authors":"Florian Ito Sprung","doi":"10.5802/jtnb.1190","DOIUrl":"https://doi.org/10.5802/jtnb.1190","url":null,"abstract":"Chromatic Selmer groups are modified Selmer groups with local information for supersingular primes p. We sketch their role in establishing the p-primary part of the Birch–Swinnerton-Dyer formula in Sections 2–5, and then study the growth of the Mordell–Weil rank along the Zp-extension of a quadratic imaginary number field in which p splits in Section 6.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47053852","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 the late 1990s, Vatsal showed that a congruence modulo p between two modular forms implied a congruence between their respective p-adic L-functions. We prove an analogous statement for both the double product and triple product p-adic L-functions, Lp(f ⊗ g) and Lp(f ⊗ g ⊗ h): the former is cyclotomic in its nature, while the latter is over the weight-space. As a corollary, we derive transition formulae relating analytic λ-invariants of congruent Galois representations for Vf⊗Vg, and for Vf⊗Vg⊗Vh, respectively.
{"title":"Controlling λ-invariants for the double and triple product p-adic L-functions","authors":"D. Delbourgo, H. Gilmore","doi":"10.5802/jtnb.1177","DOIUrl":"https://doi.org/10.5802/jtnb.1177","url":null,"abstract":"In the late 1990s, Vatsal showed that a congruence modulo p between two modular forms implied a congruence between their respective p-adic L-functions. We prove an analogous statement for both the double product and triple product p-adic L-functions, Lp(f ⊗ g) and Lp(f ⊗ g ⊗ h): the former is cyclotomic in its nature, while the latter is over the weight-space. As a corollary, we derive transition formulae relating analytic λ-invariants of congruent Galois representations for Vf⊗Vg, and for Vf⊗Vg⊗Vh, respectively.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44728888","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}
We use our previous work [4] on the Galois module structure of `–adic realizations of Picard 1–motives to construct explicit representatives in the `–adified Tate class (i.e. explicit `–adic Tate sequences, as defined in [8]) for general Galois extensions of characteristic p > 0 global fields. If combined with the Equivariant Main Conjecture proved in [4], these results lead to a very direct proof of the Equivariant Tamagawa Number Conjecture for characteristic p > 0 Artin motives with abelian coefficients.
{"title":"Picard 1-motives and Tate sequences for function fields","authors":"C. Greither, C. Popescu","doi":"10.5802/jtnb.1180","DOIUrl":"https://doi.org/10.5802/jtnb.1180","url":null,"abstract":"We use our previous work [4] on the Galois module structure of `–adic realizations of Picard 1–motives to construct explicit representatives in the `–adified Tate class (i.e. explicit `–adic Tate sequences, as defined in [8]) for general Galois extensions of characteristic p > 0 global fields. If combined with the Equivariant Main Conjecture proved in [4], these results lead to a very direct proof of the Equivariant Tamagawa Number Conjecture for characteristic p > 0 Artin motives with abelian coefficients.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49365219","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}
The use of overconvergent cohomology in constructing p -adic L -functions, initiated by Stevens and Pollack–Stevens in the setting of classical modular forms, has now been estab-lished in a number of settings. The method is compatible with constructions of eigenvarieties by Ash–Stevens, Urban and Hansen
{"title":"Overconvergent cohomology, p-adic L-functions and families for GL(2) over CM fields","authors":"Daniel Barrera Salazar, Chris Williams","doi":"10.5802/jtnb.1175","DOIUrl":"https://doi.org/10.5802/jtnb.1175","url":null,"abstract":"The use of overconvergent cohomology in constructing p -adic L -functions, initiated by Stevens and Pollack–Stevens in the setting of classical modular forms, has now been estab-lished in a number of settings. The method is compatible with constructions of eigenvarieties by Ash–Stevens, Urban and Hansen","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42372240","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 define a Kolyvagin system of rank 0 and develop the theory of Kolyvagin systems of rank 0. In particular, we prove that the module of Kolyvagin systems of rank 0 is free of rank one under standard assumptions.
{"title":"On the theory of Kolyvagin systems of rank 0","authors":"Ryotaro Sakamoto","doi":"10.5802/jtnb.1189","DOIUrl":"https://doi.org/10.5802/jtnb.1189","url":null,"abstract":"In this paper, we define a Kolyvagin system of rank 0 and develop the theory of Kolyvagin systems of rank 0. In particular, we prove that the module of Kolyvagin systems of rank 0 is free of rank one under standard assumptions.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44460928","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}
We prove inequalities that compare the regulator of a number field with its absolute discriminant. We refine some ideas in Silverman's work in 1984 where such general inequalities are first proven. In order to prove our main theorems, we combine these refinements with the authors' recent results on bounding the product of heights of relative units in a number field extension.
{"title":"Lower bounds for regulators of number fields in terms of their discriminants","authors":"S. Akhtari, J. Vaaler","doi":"10.5802/jtnb.1245","DOIUrl":"https://doi.org/10.5802/jtnb.1245","url":null,"abstract":"We prove inequalities that compare the regulator of a number field with its absolute discriminant. We refine some ideas in Silverman's work in 1984 where such general inequalities are first proven. In order to prove our main theorems, we combine these refinements with the authors' recent results on bounding the product of heights of relative units in a number field extension.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46005890","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}
We prove that there are finitely many perfect powers in elliptic divisibility sequences generated by a non-integral point on elliptic curves of the from $y^2=x(x^2+b)$, where $b$ is any positive integer. We achieve this by using the modularity of elliptic curves over real quadratic number fields.
{"title":"On the Finiteness of Perfect Powers in Elliptic Divisibility Sequences","authors":"Abdulmuhsin Alfaraj","doi":"10.5802/jtnb.1244","DOIUrl":"https://doi.org/10.5802/jtnb.1244","url":null,"abstract":"We prove that there are finitely many perfect powers in elliptic divisibility sequences generated by a non-integral point on elliptic curves of the from $y^2=x(x^2+b)$, where $b$ is any positive integer. We achieve this by using the modularity of elliptic curves over real quadratic number fields.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43447081","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}
We define spherical Heron triangles (spherical triangles with"rational"side-lengths and angles) and parametrize them via rational points of certain families of elliptic curves. We show that the congruent number problem has infinitely many solutions for most areas in the spherical setting and we find a spherical Heron triangle with rational medians. We also explore the question of spherical triangles with a single rational median or a single a rational area bisector (median splitting the triangle in half), and discuss various problems involving isosceles spherical triangles.
{"title":"Spherical Heron triangles and elliptic curves","authors":"T. Huang, Matilde Lal'in, Olivier Mila","doi":"10.5802/jtnb.1243","DOIUrl":"https://doi.org/10.5802/jtnb.1243","url":null,"abstract":"We define spherical Heron triangles (spherical triangles with\"rational\"side-lengths and angles) and parametrize them via rational points of certain families of elliptic curves. We show that the congruent number problem has infinitely many solutions for most areas in the spherical setting and we find a spherical Heron triangle with rational medians. We also explore the question of spherical triangles with a single rational median or a single a rational area bisector (median splitting the triangle in half), and discuss various problems involving isosceles spherical triangles.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43615095","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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