We investigate the question of whether the existence of a family of local zero-cycles of degree $d$ orthogonal to the Brauer group implies the non-emptiness of the Brauer-Manin set for certain varieties. We provide various examples of Brauer-Manin obstruction to the existence of zero-cycles of appropriate degrees.
{"title":"Brauer–Manin obstruction for zero-cycles on certain varieties","authors":"Evis Ieronymou","doi":"10.5802/jtnb.1241","DOIUrl":"https://doi.org/10.5802/jtnb.1241","url":null,"abstract":"We investigate the question of whether the existence of a family of local zero-cycles of degree $d$ orthogonal to the Brauer group implies the non-emptiness of the Brauer-Manin set for certain varieties. We provide various examples of Brauer-Manin obstruction to the existence of zero-cycles of appropriate degrees.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44764447","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 develop a new strategy for studying low weight specializations of $p$-adic families of ordinary modular forms. In the elliptic case, we give a new proof of a result of Ghate--Vatsal which states that a Hida family contains infinitely many classical eigenforms of weight one if and only if it has complex multiplication. Our strategy is designed to explicitly avoid use of the related facts that the Galois representation attached to a classical weight one eigenform has finite image, and that classical weight one eigenforms satisfy the Ramanujan conjecture. We indicate how this strategy might be used to prove similar statement in the case of partial weight one Hilbert modular forms, given a suitable development of Hida theory in that setting.
{"title":"Classical forms of weight one in ordinary families","authors":"Eric Stubley","doi":"10.5802/jtnb.1242","DOIUrl":"https://doi.org/10.5802/jtnb.1242","url":null,"abstract":"We develop a new strategy for studying low weight specializations of $p$-adic families of ordinary modular forms. In the elliptic case, we give a new proof of a result of Ghate--Vatsal which states that a Hida family contains infinitely many classical eigenforms of weight one if and only if it has complex multiplication. Our strategy is designed to explicitly avoid use of the related facts that the Galois representation attached to a classical weight one eigenform has finite image, and that classical weight one eigenforms satisfy the Ramanujan conjecture. We indicate how this strategy might be used to prove similar statement in the case of partial weight one Hilbert modular forms, given a suitable development of Hida theory in that setting.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48022238","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}
Generalising results of ErdH{o}s-Freud and Lindstr"om, we prove that the largest Sidon subset of a bounded interval of integers is equidistributed in Bohr neighbourhoods. We establish this by showing that extremal Sidon sets are Fourier-pseudorandom, in that they have no large non-trivial Fourier coefficients. As a further application we deduce that, for any partition regular equation in five or more variables, every finite colouring of an extremal Sidon set has a monochromatic solution.
{"title":"Extremal Sidon Sets are Fourier Uniform, with Applications to Partition Regularity","authors":"Miquel Ortega, Sean M. Prendiville","doi":"10.5802/jtnb.1239","DOIUrl":"https://doi.org/10.5802/jtnb.1239","url":null,"abstract":"Generalising results of ErdH{o}s-Freud and Lindstr\"om, we prove that the largest Sidon subset of a bounded interval of integers is equidistributed in Bohr neighbourhoods. We establish this by showing that extremal Sidon sets are Fourier-pseudorandom, in that they have no large non-trivial Fourier coefficients. As a further application we deduce that, for any partition regular equation in five or more variables, every finite colouring of an extremal Sidon set has a monochromatic solution.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48887107","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 $ mathbb{Q}mathcal{E}_{mathbb{Z}} $ be the set of power sums whose characteristic roots belong to $ mathbb{Z} $ and whose coefficients belong to $ mathbb{Q} $, i.e. $ G : mathbb{N} rightarrow mathbb{Q} $ satisfies begin{equation*} G(n) = G_n = b_1 c_1^n + cdots + b_h c_h^n end{equation*} with $ c_1,ldots,c_h in mathbb{Z} $ and $ b_1,ldots,b_h in mathbb{Q} $. Furthermore, let $ f in mathbb{Q}[x,y] $ be absolutely irreducible and $ alpha : mathbb{N} rightarrow overline{mathbb{Q}} $ be a solution $ y $ of $ f(G_n,y) = 0 $, i.e. $ f(G_n,alpha(n)) = 0 $ identically in $ n $. Then we will prove under suitable assumptions a lower bound, valid for all but finitely many positive integers $ n $, for the approximation error if $ alpha(n) $ is approximated by rational numbers with bounded denominator. After that we will also consider the case that $ alpha $ is a solution of begin{equation*} f(G_n^{(0)}, ldots, G_n^{(d)},y) = 0, end{equation*} i.e. defined by using more than one power sum and a polynomial $ f $ satisfying some suitable conditions. This extends results of Bugeaud, Corvaja, Luca, Scremin and Zannier.
设$ mathbb{Q}mathcal{E}_{mathbb{Z}} $为特征根为$ mathbb{Z} $且系数为$ mathbb{Q} $的幂和集合，即$ G : mathbb{N} rightarrow mathbb{Q} $满足begin{equation*} G(n) = G_n = b_1 c_1^n + cdots + b_h c_h^n end{equation*}的$ c_1,ldots,c_h in mathbb{Z} $和$ b_1,ldots,b_h in mathbb{Q} $。进一步，设$ f in mathbb{Q}[x,y] $为绝对不可约，$ alpha : mathbb{N} rightarrow overline{mathbb{Q}} $为$ f(G_n,y) = 0 $的解$ y $，即$ f(G_n,alpha(n)) = 0 $与$ n $相同。然后，我们将在适当的假设下证明一个下界，适用于除有限多个正整数$ n $以外的所有整数，对于$ alpha(n) $由有界分母的有理数近似时的近似误差。之后，我们还将考虑$ alpha $是begin{equation*} f(G_n^{(0)}, ldots, G_n^{(d)},y) = 0, end{equation*}的解的情况，即通过使用多个幂和和满足某些适当条件的多项式$ f $来定义。这扩展了Bugeaud、Corvaja、Luca、Scremin和Zannier的研究结果。
{"title":"Approximation of values of algebraic elements over the ring of power sums","authors":"C. Fuchs, Sebastian Heintze","doi":"10.5802/jtnb.1247","DOIUrl":"https://doi.org/10.5802/jtnb.1247","url":null,"abstract":"Let $ mathbb{Q}mathcal{E}_{mathbb{Z}} $ be the set of power sums whose characteristic roots belong to $ mathbb{Z} $ and whose coefficients belong to $ mathbb{Q} $, i.e. $ G : mathbb{N} rightarrow mathbb{Q} $ satisfies begin{equation*} G(n) = G_n = b_1 c_1^n + cdots + b_h c_h^n end{equation*} with $ c_1,ldots,c_h in mathbb{Z} $ and $ b_1,ldots,b_h in mathbb{Q} $. Furthermore, let $ f in mathbb{Q}[x,y] $ be absolutely irreducible and $ alpha : mathbb{N} rightarrow overline{mathbb{Q}} $ be a solution $ y $ of $ f(G_n,y) = 0 $, i.e. $ f(G_n,alpha(n)) = 0 $ identically in $ n $. Then we will prove under suitable assumptions a lower bound, valid for all but finitely many positive integers $ n $, for the approximation error if $ alpha(n) $ is approximated by rational numbers with bounded denominator. After that we will also consider the case that $ alpha $ is a solution of begin{equation*} f(G_n^{(0)}, ldots, G_n^{(d)},y) = 0, end{equation*} i.e. defined by using more than one power sum and a polynomial $ f $ satisfying some suitable conditions. This extends results of Bugeaud, Corvaja, Luca, Scremin and Zannier.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41246434","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 will first investigate the linear relations of a one parameter family of Siegel Poincaré series. Then we give the applications to the non-vanishing of Fourier coefficients of Siegel cusp eigenforms and the central values.
{"title":"Linear Relations of Siegel Poincaré Series and Non-vanishing of the Central Values of Spinor L-functions","authors":"Zhining Wei","doi":"10.5802/jtnb.1226","DOIUrl":"https://doi.org/10.5802/jtnb.1226","url":null,"abstract":"In this paper, we will first investigate the linear relations of a one parameter family of Siegel Poincaré series. Then we give the applications to the non-vanishing of Fourier coefficients of Siegel cusp eigenforms and the central values.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41706869","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":"Correction to the paper “Kolyvagin’s result on the vanishing of Ш(E/K)[p ∞ ] and its consequences for anticyclotomic Iwasawa theory”","authors":"Ahmed Matar, J. Nekovář","doi":"10.5802/jtnb.1172","DOIUrl":"https://doi.org/10.5802/jtnb.1172","url":null,"abstract":"","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48835980","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}
L’accès aux articles de la revue « Journal de Théorie des Nombres de Bordeaux » (http://jtnb.centre-mersenne.org/), implique l’accord avec les conditions générales d’utilisation (http://jtnb. centre-mersenne.org/legal/). Toute reproduction en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation à fin strictement personnelle du copiste est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
{"title":"The distribution of numbers with many ordered factorizations","authors":"Noah Lebowitz-Lockard","doi":"10.5802/jtnb.1170","DOIUrl":"https://doi.org/10.5802/jtnb.1170","url":null,"abstract":"L’accès aux articles de la revue « Journal de Théorie des Nombres de Bordeaux » (http://jtnb.centre-mersenne.org/), implique l’accord avec les conditions générales d’utilisation (http://jtnb. centre-mersenne.org/legal/). Toute reproduction en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation à fin strictement personnelle du copiste est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46423084","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 generalize Jacobi’s derivative formula for odd m by writing an m × m determinant of higher order derivatives at 0 of theta functions in 1 variable with characteristic vectors with entries in 1 2mZ as an explicit constant times a power of Dedekind’s η-function. We do so by deriving it from an algebraic geometric version that holds in characteristic not dividing 6m. Introduction In the vast pantheon of theta function identities, a central position is held by Jacobi’s derivative formula. Recall that for τ ∈ h = {x+ iy | y > 0}, and a, b ∈ R, we define the theta function in one variable z ∈ C with characteristic vector [ a b ] by (1) θ [ a b ] (z, τ) = ∑ n∈Z eπi(n+a) τ+2πi(n+a)(z+b). A characteristic vector [ a b ] with a, b ∈ 1 2Z is called a theta characteristic, which is called odd or even depending on whether θ [ a b ] (z, τ) is an odd or even function of z. Modulo 1 there is a unique odd theta characteristic δ := [ 1/2 1/2 ] , and three even ones, 1 := [ 0 0 ] , 2 := [ 1/2 0 ] , 3 := [ 0 1/2 ] . Manuscrit reçu le 6 février 2020, révisé le 2 février 2021, accepté le 18 mai 2021. 2010 Mathematics Subject Classification. 14K25, 14H42. Mots-clefs. Theta functions, elliptic curves.
我们推广了奇m的Jacobi导数公式，通过在1个变量中写θ函数的0处的高阶导数的m×m行列式，其中特征向量的项为1 2mZ，作为显式常数乘以Dedekindη-函数的幂。我们通过从代数几何版本中导出它来实现这一点，该版本具有不划分6m的特性。引言在θ函数恒等式的万神殿中，Jacobi的导数公式占据了中心位置。回想一下，对于τ∈h={x+iy|y>0}，a，b∈R，我们定义了特征向量为[ab]的一个变量z∈C中的θ函数：（1）θ[ab]（z，τ）=∑n∈z eπi（n+a）τ+2πi（n+a）（z+b）。具有A，b∈12Z的特征向量[ab]称为θ特征，根据θ[ab]（z，τ）是z的奇函数还是偶函数，称为奇函数或偶函数。Manuscrit reçu le 6 février 2020，réviséle 2 février2021，acceptéle 18 maié2021。2010年数学学科分类。14K25、14H42。Mots clefs。Theta函数，椭圆曲线。
{"title":"A higher-order generalization of Jacobi’s derivative formula and its algebraic geometric analogue","authors":"David Grant","doi":"10.5802/jtnb.1164","DOIUrl":"https://doi.org/10.5802/jtnb.1164","url":null,"abstract":"We generalize Jacobi’s derivative formula for odd m by writing an m × m determinant of higher order derivatives at 0 of theta functions in 1 variable with characteristic vectors with entries in 1 2mZ as an explicit constant times a power of Dedekind’s η-function. We do so by deriving it from an algebraic geometric version that holds in characteristic not dividing 6m. Introduction In the vast pantheon of theta function identities, a central position is held by Jacobi’s derivative formula. Recall that for τ ∈ h = {x+ iy | y > 0}, and a, b ∈ R, we define the theta function in one variable z ∈ C with characteristic vector [ a b ] by (1) θ [ a b ] (z, τ) = ∑ n∈Z eπi(n+a) τ+2πi(n+a)(z+b). A characteristic vector [ a b ] with a, b ∈ 1 2Z is called a theta characteristic, which is called odd or even depending on whether θ [ a b ] (z, τ) is an odd or even function of z. Modulo 1 there is a unique odd theta characteristic δ := [ 1/2 1/2 ] , and three even ones, 1 := [ 0 0 ] , 2 := [ 1/2 0 ] , 3 := [ 0 1/2 ] . Manuscrit reçu le 6 février 2020, révisé le 2 février 2021, accepté le 18 mai 2021. 2010 Mathematics Subject Classification. 14K25, 14H42. Mots-clefs. Theta functions, elliptic curves.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48757562","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}
Jacobi sums are ubiquitous in number theory, and congruences often provide a helpful way to study them. A p-adic congruence for Jacobi sums comes from Stickelberger’s congruence, and various `-adic congruences have been studied in [Eva98], [Mik87], [Iwa75], [Iha86], and [Ueh87]. We establish a new `-adic congruence for certain Jacobi sums.
{"title":"On the ℓ-adic valuation of certain Jacobi sums","authors":"V. Arul","doi":"10.5802/jtnb.1171","DOIUrl":"https://doi.org/10.5802/jtnb.1171","url":null,"abstract":"Jacobi sums are ubiquitous in number theory, and congruences often provide a helpful way to study them. A p-adic congruence for Jacobi sums comes from Stickelberger’s congruence, and various `-adic congruences have been studied in [Eva98], [Mik87], [Iwa75], [Iha86], and [Ueh87]. We establish a new `-adic congruence for certain Jacobi sums.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44322211","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 generalize a classical result of Andrianov on the decomposition of Hecke polynomials. Let F be a non-archimedean local fied. For every connected reductive group G, we give a criterion for when a polynomial with coefficients in the spherical parahoric Hecke algebra of G(F) decomposes over a parabolic Hecke algebra associated with a non-obtuse parabolic subgroup of G. We classify the non-obtuse parabolics. This then shows that our decomposition theorem covers all the classical cases due to Andrianov and Gritsenko. We also obtain new cases when the relative root system of G contains factors of types E6 or E7.
{"title":"On the Decomposition of Hecke Polynomials over Parabolic Hecke Algebras","authors":"Claudius Heyer","doi":"10.5802/jtnb.1235","DOIUrl":"https://doi.org/10.5802/jtnb.1235","url":null,"abstract":"We generalize a classical result of Andrianov on the decomposition of Hecke polynomials. Let F be a non-archimedean local fied. For every connected reductive group G, we give a criterion for when a polynomial with coefficients in the spherical parahoric Hecke algebra of G(F) decomposes over a parabolic Hecke algebra associated with a non-obtuse parabolic subgroup of G. We classify the non-obtuse parabolics. This then shows that our decomposition theorem covers all the classical cases due to Andrianov and Gritsenko. We also obtain new cases when the relative root system of G contains factors of types E6 or E7.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49527766","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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