Journal of Number Theory最新文献

{"title":"Asymptotic and non-asymptotic results for a binary additive problem involving Piatetski-Shapiro numbers","authors":"Yuuya Yoshida","doi":"10.1016/j.jnt.2024.06.012","DOIUrl":"10.1016/j.jnt.2024.06.012","url":null,"abstract":"<div><p>For all <span><math><msub><mrow><mi>α</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span> with <span><math><mn>1</mn><mo>/</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mn>1</mn><mo>/</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>&gt;</mo><mn>5</mn><mo>/</mo><mn>3</mn></math></span>, we show that the number of pairs <span><math><mo>(</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> of positive integers with <span><math><mi>N</mi><mo>=</mo><mo>⌊</mo><msubsup><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>α</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>⌋</mo><mo>+</mo><mo>⌊</mo><msubsup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>α</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mo>⌋</mo></math></span> is equal to <span><math><mi>Γ</mi><mo>(</mo><mn>1</mn><mo>+</mo><mn>1</mn><mo>/</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>Γ</mi><mo>(</mo><mn>1</mn><mo>+</mo><mn>1</mn><mo>/</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>Γ</mi><msup><mrow><mo>(</mo><mn>1</mn><mo>/</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mn>1</mn><mo>/</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><msup><mrow><mi>N</mi></mrow><mrow><mn>1</mn><mo>/</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mn>1</mn><mo>/</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><mi>o</mi><mo>(</mo><msup><mrow><mi>N</mi></mrow><mrow><mn>1</mn><mo>/</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mn>1</mn><mo>/</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span> as <span><math><mi>N</mi><mo>→</mo><mo>∞</mo></math></span>, where Γ denotes the gamma function. Moreover, we show a non-asymptotic result for the same counting problem when <span><math><msub><mrow><mi>α</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span> lie in a larger range than the above. Finally, we give some asymptotic formulas for similar counting problems in a heuristic way.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022314X24001562/pdfft?md5=f340a4f5d9777bfe3886facce83ff86f&pid=1-s2.0-S0022314X24001562-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141780785","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}

{"title":"Nonvanishing of L-function of some Hecke characters on cyclotomic fields","authors":"Keunyoung Jeong ,&nbsp;Yeong-Wook Kwon ,&nbsp;Junyeong Park","doi":"10.1016/j.jnt.2024.06.002","DOIUrl":"10.1016/j.jnt.2024.06.002","url":null,"abstract":"<div><p>In this paper, we show the nonvanishing of some Hecke characters on cyclotomic fields. The main ingredient of this paper is a computation of eigenfunctions and the action of Weil representation at some primes including the primes above 2. As an application, we show that for each isogeny factor of the Jacobian of the <em>p</em>-th Fermat curve where 2 is a quadratic residue modulo <em>p</em>, there are infinitely many twists whose analytic rank is zero. Also, for a certain hyperelliptic curve over the 11-th cyclotomic field whose Jacobian has complex multiplication, there are infinitely many twists whose analytic rank is zero.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141780788","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}

{"title":"Orbits in lattices","authors":"Matthew Dawes","doi":"10.1016/j.jnt.2024.06.013","DOIUrl":"10.1016/j.jnt.2024.06.013","url":null,"abstract":"<div><p>Let <em>L</em> be a lattice. We exhibit algorithms for calculating Tits buildings and orbits of vectors in <em>L</em> for certain subgroups of the orthogonal group <span><math><mi>O</mi><mo>(</mo><mi>L</mi><mo>)</mo></math></span>. We discuss how these algorithms can be applied to determine the configuration of boundary components in the Baily-Borel compactification of orthogonal modular varieties and to improve the performance of computer arithmetic of orthogonal modular forms.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141786072","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}

{"title":"Representations of the p-adic GSpin4 and GSpin6 and the adjoint L-function","authors":"Mahdi Asgari ,&nbsp;Kwangho Choiy","doi":"10.1016/j.jnt.2024.06.004","DOIUrl":"10.1016/j.jnt.2024.06.004","url":null,"abstract":"<div><p>We prove a conjecture of B. Gross and D. Prasad about determination of generic <em>L</em>-packets in terms of the analytic properties of the adjoint <em>L</em>-function for <em>p</em>-adic general even spin groups of semi-simple ranks 2 and 3. We also explicitly write the adjoint <em>L</em>-function for each <em>L</em>-packet in terms of the local Langlands <em>L</em>-functions for the general linear groups.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141780543","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}

{"title":"Degeneracy loci in the universal family of Abelian varieties","authors":"Ziyang Gao, Philipp Habegger","doi":"10.1016/j.jnt.2024.05.015","DOIUrl":"https://doi.org/10.1016/j.jnt.2024.05.015","url":null,"abstract":"Recent developments on the uniformity of the number of rational points on curves and subvarieties in a moving abelian variety rely on the geometric concept of the degeneracy locus. The first-named author investigated the degeneracy locus in certain mixed Shimura varieties. In this expository note we revisit some of these results while minimizing the use of mixed Shimura varieties while working in a family of principally polarized abelian varieties. We also explain their relevance for applications in diophantine geometry.","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141507452","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}

{"title":"Benford's law and random integer decomposition with congruence stopping condition","authors":"Xinyu Fang ,&nbsp;Steven J. Miller ,&nbsp;Maxwell Sun ,&nbsp;Amanda Verga","doi":"10.1016/j.jnt.2024.05.005","DOIUrl":"10.1016/j.jnt.2024.05.005","url":null,"abstract":"<div><p>Benford's law is a statement about the frequency that each digit arises as the leading digit of numbers in a dataset. It is satisfied by various common integer sequences, such as the Fibonacci numbers, the factorials, and the powers of most integers. In this paper, we prove that integer sequences resulting from a random integral decomposition process (which we model as discrete “stick breaking”) subject to a certain congruence stopping condition approach Benford distribution asymptotically. We also show that our requirement on the number of congruence classes defining the congruence stopping condition is necessary for Benford behavior to occur and is a critical point; deviation from that would result in drastically different behavior.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141507453","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}

{"title":"Hecke eigenspaces for the projective line","authors":"Roberto Alvarenga ,&nbsp;Nans Bonnel","doi":"10.1016/j.jnt.2024.05.010","DOIUrl":"10.1016/j.jnt.2024.05.010","url":null,"abstract":"<div><p>In this article we investigate the action of (ramified and unramified) Hecke operators on automorphic forms for the function field of the projective line defined over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> and for the group <span><math><msub><mrow><mi>GL</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>. We first compute the dimension of the Hecke eigenspaces for every generator of the unramified Hecke algebra. Thus, we consider the ramification in a point of degree one and explicitly describe the action of certain ramified Hecke operators on automorphic forms. Moreover, we also compute the dimensions of its eigenspaces for those ramified Hecke operators. We finish the article considering more general ramifications, namely those one attached to a closed point of higher degree.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141507455","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}

{"title":"A conjecture of Flach and Morin","authors":"Bruno Chiarellotto ,&nbsp;Nicola Mazzari ,&nbsp;Yukihide Nakada","doi":"10.1016/j.jnt.2024.05.013","DOIUrl":"https://doi.org/10.1016/j.jnt.2024.05.013","url":null,"abstract":"<div><p>A conjecture recently stated by Flach and Morin relates the action of the monodromy on the Galois invariant part of the <em>p</em>-adic Beilinson–Hyodo–Kato cohomology of the generic fiber of a scheme defined over a DVR of mixed characteristic to (the cohomology of) its special fiber. We prove the conjecture in the case that the special fiber of the given arithmetic scheme is also a fiber of a geometric family over a curve in positive characteristic.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141484692","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}

{"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":null,"pages":null},"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}

{"title":"On some local properties of sequences of big Galois representations","authors":"Jyoti Prakash Saha,&nbsp;Aniruddha Sudarshan","doi":"10.1016/j.jnt.2024.05.012","DOIUrl":"10.1016/j.jnt.2024.05.012","url":null,"abstract":"<div><p>In this article, we prove that for a convergent sequence of residually absolutely irreducible representations of the absolute Galois group of a number field <em>F</em> with coefficients in a domain, which admits a finite monomorphism from a power series ring over a <em>p</em>-adic integer ring, the set of places of <em>F</em> where some of the representations ramifies has density zero. Using this, we extend a result of Das–Rajan to such convergent sequences. We also establish a strong multiplicity one theorem for big Galois representations.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141507456","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}