Pub Date : 2024-06-25DOI: 10.1016/j.jnt.2024.05.004
Haowen Zhang
Over function fields of p-adic curves, we construct stably rational varieties in the form of homogeneous spaces of with semisimple simply connected stabilizers and we show that strong approximation away from a non-empty set of places fails for such varieties. The construction combines the Lichtenbaum duality and the degree 3 cohomological invariants of the stabilizers. We then establish a reciprocity obstruction which accounts for this failure of strong approximation. We show that this reciprocity obstruction to strong approximation is the only one for counterexamples we constructed, and also for classifying varieties of tori. We also show that this reciprocity obstruction to strong approximation is compatible with known results for tori. At the end, we explain how a similar point of view shows that the reciprocity obstruction to weak approximation is the only one for classifying varieties of tori over p-adic function fields.
在-adic 曲线的函数域上,我们以半简单简单连接稳定器的均质空间的形式构造了稳定有理变种,并证明了对于这类变种,从非空位集出发的强逼近是失败的。这一构造结合了利希滕鲍姆对偶性和稳定子的 3 级同调不变式。然后,我们建立了一个互易障碍来解释强近似的失败。我们证明,这个强近似的互易性障碍是我们构造的反例以及环的分类变体的唯一障碍。我们还证明,强近似的互易性障碍与已知的环状结果是一致的。最后,我们将解释如何从类似的角度说明,弱逼近的互易性障碍是对-二次函数域上的 tori varieties 进行分类的唯一障碍。
{"title":"Reciprocity obstruction to strong approximation over p-adic function fields","authors":"Haowen Zhang","doi":"10.1016/j.jnt.2024.05.004","DOIUrl":"10.1016/j.jnt.2024.05.004","url":null,"abstract":"<div><p>Over function fields of <em>p</em>-adic curves, we construct stably rational varieties in the form of homogeneous spaces of <span><math><msub><mrow><mi>SL</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> with semisimple simply connected stabilizers and we show that strong approximation away from a non-empty set of places fails for such varieties. The construction combines the Lichtenbaum duality and the degree 3 cohomological invariants of the stabilizers. We then establish a reciprocity obstruction which accounts for this failure of strong approximation. We show that this reciprocity obstruction to strong approximation is the only one for counterexamples we constructed, and also for classifying varieties of tori. We also show that this reciprocity obstruction to strong approximation is compatible with known results for tori. At the end, we explain how a similar point of view shows that the reciprocity obstruction to weak approximation is the only one for classifying varieties of tori over <em>p</em>-adic function fields.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"264 ","pages":"Pages 99-134"},"PeriodicalIF":0.6,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022314X24001331/pdfft?md5=cac87e78166d928ce995f053684adbf1&pid=1-s2.0-S0022314X24001331-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141529654","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-25DOI: 10.1016/j.jnt.2024.06.001
Ananth N. Shankar, Yunqing Tang
{"title":"Reductions of abelian varieties and K3 surfaces","authors":"Ananth N. Shankar, Yunqing Tang","doi":"10.1016/j.jnt.2024.06.001","DOIUrl":"https://doi.org/10.1016/j.jnt.2024.06.001","url":null,"abstract":"","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"29 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141507462","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 : 2024-06-25DOI: 10.1016/j.jnt.2024.05.009
Yanqiu Guo, Michael Ilyin
This paper is inspired by Richards' work on large gaps between sums of two squares [10]. It is shown in [10] that there exist arbitrarily large values of λ and μ, where , such that intervals do not contain any sums of two squares. Geometrically, these gaps between sums of two squares correspond to annuli in that do not contain any integer lattice points. A major objective of this paper is to investigate the sparse distribution of integer lattice points within annular regions in . Specifically, we establish the existence of annuli with arbitrarily large λ and for , satisfying that any two integer lattice points within any one of these annuli must be sufficiently far apart. This result is sharp, as such a property ceases to hold at and beyond the threshold . Furthermore, we extend our analysis to include the sparse distribution of lattice points in spherical shells in .
{"title":"Sparse distribution of lattice points in annular regions","authors":"Yanqiu Guo, Michael Ilyin","doi":"10.1016/j.jnt.2024.05.009","DOIUrl":"10.1016/j.jnt.2024.05.009","url":null,"abstract":"<div><p>This paper is inspired by Richards' work on large gaps between sums of two squares <span>[10]</span>. It is shown in <span>[10]</span> that there exist arbitrarily large values of <em>λ</em> and <em>μ</em>, where <span><math><mi>μ</mi><mo>≥</mo><mi>C</mi><mi>log</mi><mo></mo><mi>λ</mi></math></span>, such that intervals <span><math><mo>[</mo><mi>λ</mi><mo>,</mo><mspace></mspace><mi>λ</mi><mo>+</mo><mi>μ</mi><mo>]</mo></math></span> do not contain any sums of two squares. Geometrically, these gaps between sums of two squares correspond to annuli in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> that do not contain any integer lattice points. A major objective of this paper is to investigate the sparse distribution of integer lattice points within annular regions in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>. Specifically, we establish the existence of annuli <span><math><mo>{</mo><mi>x</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>:</mo><mi>λ</mi><mo>≤</mo><mo>|</mo><mi>x</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>≤</mo><mi>λ</mi><mo>+</mo><mi>κ</mi><mo>}</mo></math></span> with arbitrarily large <em>λ</em> and <span><math><mi>κ</mi><mo>≥</mo><mi>C</mi><msup><mrow><mi>λ</mi></mrow><mrow><mi>s</mi></mrow></msup></math></span> for <span><math><mn>0</mn><mo><</mo><mi>s</mi><mo><</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></math></span>, satisfying that any two integer lattice points within any one of these annuli must be sufficiently far apart. This result is sharp, as such a property ceases to hold at and beyond the threshold <span><math><mi>s</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></math></span>. Furthermore, we extend our analysis to include the sparse distribution of lattice points in spherical shells in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"264 ","pages":"Pages 277-294"},"PeriodicalIF":0.6,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141507454","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 : 2024-06-25DOI: 10.1016/j.jnt.2024.05.011
Timothy Gillespie
Let be integers with D square free and k even. Let N be a positive integer so that when D has residue one modulo four and when D has residue two or three modulo four. In this paper the asymptotic behavior of a resonance sum attached to the quadratic base change lift of a holomorphic cusp form f of level N and weight k over the quadratic extension generated by is computed. First a Voronoi summation formula is derived that expresses in terms of the Meier-G function. Then, using the known asymptotics of the Meier-G function the asymptotic behavior of as X approaches infinity is determined. It is then shown that using only finitely many Fourier coefficients of the form, one can recover the weight k and the level N, which is a special case of the multiplicity one theorem.
设为无平方和偶数的整数。设为正整数,则 when 的余数为 1,且 when 的余数为 2 或 3,且 when 的余数为 4。在本文中,我们计算了一个全形尖顶形式的级数和权重在由其生成的二次扩展上的二次基变提升所附共振和的渐近行为。首先推导出一个用 Meier-G 函数表示的 Voronoi 求和公式。然后,利用已知的 Meier-G 函数渐近线,确定接近无穷大时的渐近行为。然后证明,只需使用有限个傅里叶系数的形式,就能恢复权重和水平,这是乘数一定理的一个特例。
{"title":"Quadratic base change and resonance sums for holomorphic cusp forms on Γ0(N)","authors":"Timothy Gillespie","doi":"10.1016/j.jnt.2024.05.011","DOIUrl":"10.1016/j.jnt.2024.05.011","url":null,"abstract":"<div><p>Let <span><math><mi>D</mi><mo>,</mo><mi>k</mi></math></span> be integers with <em>D</em> square free and <em>k</em> even. Let <em>N</em> be a positive integer so that <span><math><mo>(</mo><mi>N</mi><mo>,</mo><mi>D</mi><mo>)</mo><mo>=</mo><mn>1</mn></math></span> when <em>D</em> has residue one modulo four and <span><math><mo>(</mo><mi>N</mi><mo>,</mo><mn>4</mn><mi>D</mi><mo>)</mo><mo>=</mo><mn>1</mn></math></span> when <em>D</em> has residue two or three modulo four. In this paper the asymptotic behavior of a resonance sum <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>;</mo><mi>π</mi><mo>)</mo></math></span> attached to the quadratic base change lift of a holomorphic cusp form <em>f</em> of level <em>N</em> and weight <em>k</em> over the quadratic extension generated by <span><math><msqrt><mrow><mi>D</mi></mrow></msqrt></math></span> is computed. First a Voronoi summation formula is derived that expresses <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>;</mo><mi>π</mi><mo>)</mo></math></span> in terms of the Meier-G function. Then, using the known asymptotics of the Meier-G function the asymptotic behavior of <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>;</mo><mi>π</mi><mo>)</mo></math></span> as <em>X</em> approaches infinity is determined. It is then shown that using only finitely many Fourier coefficients of the form, one can recover the weight <em>k</em> and the level <em>N</em>, which is a special case of the multiplicity one theorem.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"264 ","pages":"Pages 184-210"},"PeriodicalIF":0.6,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141507457","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 : 2024-06-25DOI: 10.1016/j.jnt.2024.05.002
Paul M. Voutier
We investigate the number of squares in a very broad family of binary recurrence sequences with . We show that there are at most two distinct squares in such sequences (the best possible result), except under very special conditions where we prove there are at most three such squares.
{"title":"Bounds on the number of squares in recurrence sequences","authors":"Paul M. Voutier","doi":"10.1016/j.jnt.2024.05.002","DOIUrl":"10.1016/j.jnt.2024.05.002","url":null,"abstract":"<div><p>We investigate the number of squares in a very broad family of binary recurrence sequences with <span><math><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mn>1</mn></math></span>. We show that there are at most two distinct squares in such sequences (the best possible result), except under very special conditions where we prove there are at most three such squares.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"265 ","pages":"Pages 291-343"},"PeriodicalIF":0.6,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141930230","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 : 2024-06-25DOI: 10.1016/j.jnt.2024.05.007
Naoto Dainobu
Let E be an elliptic curve over , p an odd prime number and n a positive integer. In this article, we investigate the ideal class group of the -division field of E. We introduce a certain subgroup of and study the p-adic valuation of the class number .
In addition, when , we further study as a -module. More precisely, we study the semi-simplification of as a -module. We obtain a lower bound of the multiplicity of the -component in the semi-simplification when is an irreducible -module.
{"title":"Ideal class groups of division fields of elliptic curves and everywhere unramified rational points","authors":"Naoto Dainobu","doi":"10.1016/j.jnt.2024.05.007","DOIUrl":"10.1016/j.jnt.2024.05.007","url":null,"abstract":"<div><p>Let <em>E</em> be an elliptic curve over <span><math><mi>Q</mi></math></span>, <em>p</em> an odd prime number and <em>n</em> a positive integer. In this article, we investigate the ideal class group <span><math><mrow><mi>Cl</mi></mrow><mo>(</mo><mi>Q</mi><mo>(</mo><mi>E</mi><mo>[</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>]</mo><mo>)</mo><mo>)</mo></math></span> of the <span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>-division field <span><math><mi>Q</mi><mo>(</mo><mi>E</mi><mo>[</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>]</mo><mo>)</mo></math></span> of <em>E</em>. We introduce a certain subgroup <span><math><mi>E</mi><msub><mrow><mo>(</mo><mi>Q</mi><mo>)</mo></mrow><mrow><mrow><mi>ur</mi></mrow><mo>,</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span> of <span><math><mi>E</mi><mo>(</mo><mi>Q</mi><mo>)</mo></math></span> and study the <em>p</em>-adic valuation of the class number <span><math><mi>#</mi><mrow><mi>Cl</mi></mrow><mo>(</mo><mi>Q</mi><mo>(</mo><mi>E</mi><mo>[</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>]</mo><mo>)</mo><mo>)</mo></math></span>.</p><p>In addition, when <span><math><mi>n</mi><mo>=</mo><mn>1</mn></math></span>, we further study <span><math><mrow><mi>Cl</mi></mrow><mo>(</mo><mi>Q</mi><mo>(</mo><mi>E</mi><mo>[</mo><mi>p</mi><mo>]</mo><mo>)</mo><mo>)</mo></math></span> as a <span><math><mi>Gal</mi><mo>(</mo><mi>Q</mi><mo>(</mo><mi>E</mi><mo>[</mo><mi>p</mi><mo>]</mo><mo>)</mo><mo>/</mo><mi>Q</mi><mo>)</mo></math></span>-module. More precisely, we study the semi-simplification <span><math><msup><mrow><mo>(</mo><mrow><mi>Cl</mi></mrow><mo>(</mo><mi>Q</mi><mo>(</mo><mi>E</mi><mo>[</mo><mi>p</mi><mo>]</mo><mo>)</mo><mo>)</mo><mo>⊗</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>ss</mi></mrow></msup></math></span> of <span><math><mrow><mi>Cl</mi></mrow><mo>(</mo><mi>Q</mi><mo>(</mo><mi>E</mi><mo>[</mo><mi>p</mi><mo>]</mo><mo>)</mo><mo>)</mo><mo>⊗</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> as a <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>[</mo><mi>Gal</mi><mo>(</mo><mi>Q</mi><mo>(</mo><mi>E</mi><mo>[</mo><mi>p</mi><mo>]</mo><mo>)</mo><mo>/</mo><mi>Q</mi><mo>)</mo><mo>]</mo></math></span>-module. We obtain a lower bound of the multiplicity of the <span><math><mi>E</mi><mo>[</mo><mi>p</mi><mo>]</mo></math></span>-component in the semi-simplification when <span><math><mi>E</mi><mo>[</mo><mi>p</mi><mo>]</mo></math></span> is an irreducible <span><math><mi>Gal</mi><mo>(</mo><mi>Q</mi><mo>(</mo><mi>E</mi><mo>[</mo><mi>p</mi><mo>]</mo><mo>)</mo><mo>/</mo><mi>Q</mi><mo>)</mo></math></span>-module.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"264 ","pages":"Pages 211-232"},"PeriodicalIF":0.6,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141507458","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 : 2024-06-25DOI: 10.1016/j.jnt.2024.05.008
Pierre J. Clavier , Dorian Perrot
We show that any convergent (shuffle) arborified zeta value admits a series representation. This justifies the introduction of a new generalisation to rooted forests of multiple zeta values, and we study its algebraic properties. As a consequence of the series representation, we derive elementary proofs of some results of Bradley and Zhou for Mordell-Tornheim zeta values and give explicit formulas. The series representation for shuffle arborified zeta values also implies that they are conical zeta values. We characterise which conical zeta values are arborified zeta values and evaluate them as sums of multiple zeta values with rational coefficients.
{"title":"Generalisations of multiple zeta values to rooted forests","authors":"Pierre J. Clavier , Dorian Perrot","doi":"10.1016/j.jnt.2024.05.008","DOIUrl":"10.1016/j.jnt.2024.05.008","url":null,"abstract":"<div><p>We show that any convergent (shuffle) arborified zeta value admits a series representation. This justifies the introduction of a new generalisation to rooted forests of multiple zeta values, and we study its algebraic properties. As a consequence of the series representation, we derive elementary proofs of some results of Bradley and Zhou for Mordell-Tornheim zeta values and give explicit formulas. The series representation for shuffle arborified zeta values also implies that they are conical zeta values. We characterise which conical zeta values are arborified zeta values and evaluate them as sums of multiple zeta values with rational coefficients.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"264 ","pages":"Pages 233-276"},"PeriodicalIF":0.6,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141507459","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 : 2024-06-25DOI: 10.1016/j.jnt.2024.05.014
In this paper we prove that if A and B are infinite subsets of positive integers such that every positive integer n can be written as , , , then . We present some tight density bounds in connection with multiplicative complements.
{"title":"Multiplicative complements, II.","authors":"","doi":"10.1016/j.jnt.2024.05.014","DOIUrl":"10.1016/j.jnt.2024.05.014","url":null,"abstract":"<div><p>In this paper we prove that if <em>A</em> and <em>B</em> are infinite subsets of positive integers such that every positive integer <em>n</em> can be written as <span><math><mi>n</mi><mo>=</mo><mi>a</mi><mi>b</mi></math></span>, <span><math><mi>a</mi><mo>∈</mo><mi>A</mi></math></span>, <span><math><mi>b</mi><mo>∈</mo><mi>B</mi></math></span>, then <span><math><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mo>∞</mo></mrow></munder><mo></mo><mfrac><mrow><mi>A</mi><mo>(</mo><mi>x</mi><mo>)</mo><mi>B</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mrow><mi>x</mi></mrow></mfrac><mo>=</mo><mo>∞</mo></math></span>. We present some tight density bounds in connection with multiplicative complements.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"265 ","pages":"Pages 1-19"},"PeriodicalIF":0.6,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141507460","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 : 2024-06-25DOI: 10.1016/j.jnt.2024.05.006
Byeong-Kweon Oh , Jongheun Yoon
For a positive integer m, a (positive definite integral) quadratic form is called primitively m-universal if it primitively represents all quadratic forms of rank m. It was proved in [9] that there are exactly 107 equivalence classes of primitively 1-universal quaternary quadratic forms. In this article, we prove that the minimal rank of primitively 2-universal quadratic forms is six, and there are exactly 201 equivalence classes of primitively 2-universal senary quadratic forms.
对于一个正整数 ,如果一个(正定积分)二次型原始地代表了所有秩的二次型,那么这个二次型就叫做原始通用二次型。有学者证明,基元 1-Universal 四元二次函数形式的等价类恰好有 107 个。在本文中,我们将证明 primitively 2-universal quadratic forms 的最小秩是 6,并且正好有 201 个 primitively 2-universal senary quadratic forms 的等价类。
{"title":"Primitively 2-universal senary integral quadratic forms","authors":"Byeong-Kweon Oh , Jongheun Yoon","doi":"10.1016/j.jnt.2024.05.006","DOIUrl":"10.1016/j.jnt.2024.05.006","url":null,"abstract":"<div><p>For a positive integer <em>m</em>, a (positive definite integral) quadratic form is called primitively <em>m</em>-universal if it primitively represents all quadratic forms of rank <em>m</em>. It was proved in <span>[9]</span> that there are exactly 107 equivalence classes of primitively 1-universal quaternary quadratic forms. In this article, we prove that the minimal rank of primitively 2-universal quadratic forms is six, and there are exactly 201 equivalence classes of primitively 2-universal senary quadratic forms.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"264 ","pages":"Pages 148-183"},"PeriodicalIF":0.6,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141507461","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 : 2024-06-25DOI: 10.1016/j.jnt.2024.05.003
Noam Kimmel , Vivian Kuperberg
We study the distribution of consecutive sums of two squares in arithmetic progressions. If is the sequence of sums of two squares in increasing order, we show that for any modulus q and any congruence classes which are admissible in the sense that there are solutions to , there exist infinitely many n with , for . We also show that for any , there exist infinitely many n with for and for .
{"title":"Consecutive runs of sums of two squares","authors":"Noam Kimmel , Vivian Kuperberg","doi":"10.1016/j.jnt.2024.05.003","DOIUrl":"10.1016/j.jnt.2024.05.003","url":null,"abstract":"<div><p>We study the distribution of consecutive sums of two squares in arithmetic progressions. If <span><math><msub><mrow><mo>{</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span> is the sequence of sums of two squares in increasing order, we show that for any modulus <em>q</em> and any congruence classes <span><math><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>3</mn></mrow></msub><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><mi>q</mi></math></span> which are admissible in the sense that there are solutions to <span><math><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>≡</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><mi>q</mi></math></span>, there exist infinitely many <em>n</em> with <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>n</mi><mo>+</mo><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>≡</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><mi>q</mi></math></span>, for <span><math><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn></math></span>. We also show that for any <span><math><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>≥</mo><mn>1</mn></math></span>, there exist infinitely many <em>n</em> with <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>n</mi><mo>+</mo><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>≡</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><mi>q</mi></math></span> for <span><math><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>n</mi><mo>+</mo><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>≡</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><mi>q</mi></math></span> for <span><math><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"264 ","pages":"Pages 135-147"},"PeriodicalIF":0.6,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141529652","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}