It is well-known that the Hasse principle holds for quadric hypersurfaces. The Hasse principle fails for integral points on smooth quadric hypersurfaces of dimension 2 but the failure can be completely explained by the Brauer-Manin obstruction. We investigate how often the family of quadric hypersurfaces $ax^2 + by^2 +cz^2 = n$ has a Brauer-Manin obstruction. We improve previous bounds of Mitankin.
{"title":"Integral points on affine quadric surfaces","authors":"Tim Santens","doi":"10.5802/jtnb.1196","DOIUrl":"https://doi.org/10.5802/jtnb.1196","url":null,"abstract":"It is well-known that the Hasse principle holds for quadric hypersurfaces. The Hasse principle fails for integral points on smooth quadric hypersurfaces of dimension 2 but the failure can be completely explained by the Brauer-Manin obstruction. We investigate how often the family of quadric hypersurfaces $ax^2 + by^2 +cz^2 = n$ has a Brauer-Manin obstruction. We improve previous bounds of Mitankin.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41747103","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}
Given a number field $k$ and a finite $k$-group $G$, the Tame Approximation Problem for $G$ asks whether the restriction map $H^1(k,G)toprod_{vinSigma}H^1(k_v,G)$ is surjective for every finite set of places $SigmasubseteqOmega_k$ disjoint from $text{Bad}_G$, where $text{Bad}_G$ is the finite set of places that either divides the order of $G$ or ramifies in the minimal extension splitting $G$. In this paper we prove that the set $text{Bad}_G$ is "sharp". To achieve this we prove that there are finite abelian $k$-groups $A$ where the map $H^1(k,A)toprod_{vinSigma_0}H^1(k_v,A)$ is not surjective in a set $Sigma_0subseteqtext{Bad}_A$ with particular properties, namely $Sigma_0$ is the set of places that do not divide the order of $A$ and ramify in the minimal extension splitting $A$.
{"title":"Bad places for the approximation property for finite groups","authors":"Felipe Rivera-Mesas","doi":"10.5802/jtnb.1199","DOIUrl":"https://doi.org/10.5802/jtnb.1199","url":null,"abstract":"Given a number field $k$ and a finite $k$-group $G$, the Tame Approximation Problem for $G$ asks whether the restriction map $H^1(k,G)toprod_{vinSigma}H^1(k_v,G)$ is surjective for every finite set of places $SigmasubseteqOmega_k$ disjoint from $text{Bad}_G$, where $text{Bad}_G$ is the finite set of places that either divides the order of $G$ or ramifies in the minimal extension splitting $G$. In this paper we prove that the set $text{Bad}_G$ is \"sharp\". To achieve this we prove that there are finite abelian $k$-groups $A$ where the map $H^1(k,A)toprod_{vinSigma_0}H^1(k_v,A)$ is not surjective in a set $Sigma_0subseteqtext{Bad}_A$ with particular properties, namely $Sigma_0$ is the set of places that do not divide the order of $A$ and ramify in the minimal extension splitting $A$.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45045453","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 Tate’s conjecture for the elliptic modular surface of level $N$ over a prime field of characteristic $1 protect mathrm{mod} N$","authors":"R. Lodh","doi":"10.5802/jtnb.1117","DOIUrl":"https://doi.org/10.5802/jtnb.1117","url":null,"abstract":"","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88598294","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}
Kontsevich and Manin gave a formula for the number Ne of rational plane curves of degree e through 3e−1 points in general position in the plane. When these 3e−1 points have coordinates in the rational numbers, the corresponding set of Ne rational curves has a natural Galois-module structure. We make some extremely preliminary investigations into this Galois module structure, and relate this to the deck transformations of the generic fibre of the product of the evaluation maps on the moduli space of maps. We then study the asymptotics of the number of rational points on hypersurfaces of low degree, and use this to generalise our results by replacing the projective plane by such a hypersurface.
{"title":"Fields of definition of rational curves of a given degree","authors":"D. Holmes, Nick Rome","doi":"10.5802/jtnb.1123","DOIUrl":"https://doi.org/10.5802/jtnb.1123","url":null,"abstract":"Kontsevich and Manin gave a formula for the number Ne of rational plane curves of degree e through 3e−1 points in general position in the plane. When these 3e−1 points have coordinates in the rational numbers, the corresponding set of Ne rational curves has a natural Galois-module structure. We make some extremely preliminary investigations into this Galois module structure, and relate this to the deck transformations of the generic fibre of the product of the evaluation maps on the moduli space of maps. We then study the asymptotics of the number of rational points on hypersurfaces of low degree, and use this to generalise our results by replacing the projective plane by such a hypersurface.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82531836","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 an earlier paper (joint with Min Ru), we proved a result on diophantine approximation to Cartier divisors, extending a 2011 result of P. Autissier. This was recently extended to certain closed subschemes (in place of divisors) by Ru and Wang. In this paper we extend this result to a broader class of closed subschemes. We also show that some notions of $beta(mathscr L,D)$ coincide, and that they can all be evaluated as limits.
{"title":"Birational Nevanlinna Constants, Beta Constants, and Diophantine Approximation to Closed Subschemes","authors":"Paul Vojta","doi":"10.5802/jtnb.1237","DOIUrl":"https://doi.org/10.5802/jtnb.1237","url":null,"abstract":"In an earlier paper (joint with Min Ru), we proved a result on diophantine approximation to Cartier divisors, extending a 2011 result of P. Autissier. This was recently extended to certain closed subschemes (in place of divisors) by Ru and Wang. In this paper we extend this result to a broader class of closed subschemes. We also show that some notions of $beta(mathscr L,D)$ coincide, and that they can all be evaluated as limits.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44528938","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 give an optimal version of the classical ``three-gap theorem'' on the fractional parts of $n theta$, in the case where $theta$ is an irrational number that is badly approximable. As a consequence, we deduce a version of Kronecker's inhomogeneous approximation theorem in one dimension for badly approximable numbers. We apply these results to obtain an improved measure of sequence diversity for characteristic Sturmian sequences, where the slope is badly approximable.
{"title":"Badly approximable numbers, Kronecker’s theorem, and diversity of Sturmian characteristic sequences","authors":"Dmitry Badziahin, J. Shallit","doi":"10.5802/jtnb.1236","DOIUrl":"https://doi.org/10.5802/jtnb.1236","url":null,"abstract":"We give an optimal version of the classical ``three-gap theorem'' on the fractional parts of $n theta$, in the case where $theta$ is an irrational number that is badly approximable. As a consequence, we deduce a version of Kronecker's inhomogeneous approximation theorem in one dimension for badly approximable numbers. We apply these results to obtain an improved measure of sequence diversity for characteristic Sturmian sequences, where the slope is badly approximable.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42533367","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 show, under some natural conditions, that the set of abelian points on the non-anomalous dense subset of a closed irreducible subvariety $X$ intersected with the union of connected algebraic subgroups of codimension at least $dim X$ in a torus is finite, generalising results of Ostafe, Sha, Shparlinski and Zannier (2017). We also generalise their structure theorem for such sets when the algebraic subgroups are not necessarily connected, and obtain a related result in the context of curves and arithmetic dynamics.
{"title":"On abelian points of varieties intersecting subgroups in a torus","authors":"J. Mello","doi":"10.5802/jtnb.1203","DOIUrl":"https://doi.org/10.5802/jtnb.1203","url":null,"abstract":"We show, under some natural conditions, that the set of abelian points on the non-anomalous dense subset of a closed irreducible subvariety $X$ intersected with the union of connected algebraic subgroups of codimension at least $dim X$ in a torus is finite, generalising results of Ostafe, Sha, Shparlinski and Zannier (2017). We also generalise their structure theorem for such sets when the algebraic subgroups are not necessarily connected, and obtain a related result in the context of curves and arithmetic dynamics.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41534332","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, for a CM abelian extension $K/k$ of number fields, we propose a conjecture which describes completely the Fitting ideal of the minus part of the Pontryagin dual of the $T$-ray class group of $K$ for a set $T$ of primes as a ${rm Gal}(K/k)$-module. Here, we emphasize that we consider the full class group, and do not throw away the ramifying primes (namely, the object we study is not the quotient of the class group by the subgroup generated by the classes of ramifying primes). We prove that our conjecture is a consequence of the equivariant Tamagawa number conjecture, and also prove that the Iwasawa theoretic version of our conjecture holds true under the assumption $mu=0$ without assuming eTNC.
{"title":"Notes on the dual of the ideal class groups of CM-fields","authors":"M. Kurihara","doi":"10.5802/jtnb.1184","DOIUrl":"https://doi.org/10.5802/jtnb.1184","url":null,"abstract":"In this paper, for a CM abelian extension $K/k$ of number fields, we propose a conjecture which describes completely the Fitting ideal of the minus part of the Pontryagin dual of the $T$-ray class group of $K$ for a set $T$ of primes as a ${rm Gal}(K/k)$-module. Here, we emphasize that we consider the full class group, and do not throw away the ramifying primes (namely, the object we study is not the quotient of the class group by the subgroup generated by the classes of ramifying primes). We prove that our conjecture is a consequence of the equivariant Tamagawa number conjecture, and also prove that the Iwasawa theoretic version of our conjecture holds true under the assumption $mu=0$ without assuming eTNC.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47244764","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}
Let $p$ be an odd prime. What are the possible Newton polygons for a curve in characteristic $p$? Equivalently, which Newton strata intersect the Torelli locus in $mathcal{A}_g$? In this note, we study the Newton polygons of certain curves with $mathbb{Z}/pmathbb{Z}$-actions. Many of these curves exhibit unlikely intersections between the Torelli locus and the Newton stratification in $mathcal{A}_g$. Here is one example of particular interest: fix a genus $g$. We show that for any $k$ with $frac{2g}{3}-frac{2p(p-1)}{3}geq 2k(p-1)$, there exists a curve of genus $g$ whose Newton polygon has slopes ${0,1}^{g-k(p-1)} sqcup {frac{1}{2}}^{2k(p-1)}$. This provides evidence for Oort's conjecture that the amalgamation of the Newton polygons of two curves is again the Newton polygon of a curve. We also construct families of curves ${C_g}_{g geq 1}$, where $C_g$ is a curve of genus $g$, whose Newton polygons have interesting asymptotic properties. For example, we construct a family of curves whose Newton polygons are asymptotically bounded below by the graph $y=frac{x^2}{4g}$. The proof uses a Newton-over-Hodge result for $mathbb{Z}/pmathbb{Z}$-covers of curves due to the author, in addition to recent work of Booher-Pries on the realization of this Hodge bound.
{"title":"Some unlikely intersections between the Torelli locus and Newton strata in 𝒜 g","authors":"Joe Kramer-Miller","doi":"10.5802/JTNB.1159","DOIUrl":"https://doi.org/10.5802/JTNB.1159","url":null,"abstract":"Let $p$ be an odd prime. What are the possible Newton polygons for a curve in characteristic $p$? Equivalently, which Newton strata intersect the Torelli locus in $mathcal{A}_g$? In this note, we study the Newton polygons of certain curves with $mathbb{Z}/pmathbb{Z}$-actions. Many of these curves exhibit unlikely intersections between the Torelli locus and the Newton stratification in $mathcal{A}_g$. Here is one example of particular interest: fix a genus $g$. We show that for any $k$ with $frac{2g}{3}-frac{2p(p-1)}{3}geq 2k(p-1)$, there exists a curve of genus $g$ whose Newton polygon has slopes ${0,1}^{g-k(p-1)} sqcup {frac{1}{2}}^{2k(p-1)}$. This provides evidence for Oort's conjecture that the amalgamation of the Newton polygons of two curves is again the Newton polygon of a curve. We also construct families of curves ${C_g}_{g geq 1}$, where $C_g$ is a curve of genus $g$, whose Newton polygons have interesting asymptotic properties. For example, we construct a family of curves whose Newton polygons are asymptotically bounded below by the graph $y=frac{x^2}{4g}$. The proof uses a Newton-over-Hodge result for $mathbb{Z}/pmathbb{Z}$-covers of curves due to the author, in addition to recent work of Booher-Pries on the realization of this Hodge bound.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45799642","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}
Inspired by the classical setting, Goss defined the $L$-series of Drinfeld $A$-modules corresponding to representations of the absolute Galois group of a rational function field. In this paper, for a given Drinfeld $A$-module $phi$ of rank 2 defined over the finite field $mathbb{F}_q$, we give explicit formulas for the values of Goss $L$-series at positive integers $n$ such that $2n+1leq q$ in terms of polylogarithms and coefficients of the logarithm series of $phi$.
{"title":"Special values of Goss L-series attached to Drinfeld modules of rank 2","authors":"Oğuz Gezmiş","doi":"10.5802/jtnb.1168","DOIUrl":"https://doi.org/10.5802/jtnb.1168","url":null,"abstract":"Inspired by the classical setting, Goss defined the $L$-series of Drinfeld $A$-modules corresponding to representations of the absolute Galois group of a rational function field. In this paper, for a given Drinfeld $A$-module $phi$ of rank 2 defined over the finite field $mathbb{F}_q$, we give explicit formulas for the values of Goss $L$-series at positive integers $n$ such that $2n+1leq q$ in terms of polylogarithms and coefficients of the logarithm series of $phi$.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41765968","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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