Pub Date : 2024-07-12DOI: 10.1142/s1793042124501276
Rupam Barman, Sulakashna
{"title":"p-Adic hypergeometric functions and the trace of Frobenius of elliptic curves","authors":"Rupam Barman, Sulakashna","doi":"10.1142/s1793042124501276","DOIUrl":"https://doi.org/10.1142/s1793042124501276","url":null,"abstract":"","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141652554","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 : 2024-07-12DOI: 10.1142/s1793042125500071
Li Zhu
{"title":"Almost prime triples and Chen's theorem","authors":"Li Zhu","doi":"10.1142/s1793042125500071","DOIUrl":"https://doi.org/10.1142/s1793042125500071","url":null,"abstract":"","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141653739","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 : 2024-07-12DOI: 10.1142/s1793042125500010
Di Zhang
{"title":"On the Shintani lifting of integral weight modular forms to half-integral weight modular forms","authors":"Di Zhang","doi":"10.1142/s1793042125500010","DOIUrl":"https://doi.org/10.1142/s1793042125500010","url":null,"abstract":"","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141654839","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 : 2024-07-12DOI: 10.1142/s1793042125500125
Sandro Mattarei, R. Tauraso
{"title":"Congruences for partial sums of the generating series for 3kk","authors":"Sandro Mattarei, R. Tauraso","doi":"10.1142/s1793042125500125","DOIUrl":"https://doi.org/10.1142/s1793042125500125","url":null,"abstract":"","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141652494","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 : 2024-07-12DOI: 10.1142/s1793042125500083
Ji-Zhen Xu, Yong-Gao Chen
{"title":"On integers of the form p + 2a2 + 2b2","authors":"Ji-Zhen Xu, Yong-Gao Chen","doi":"10.1142/s1793042125500083","DOIUrl":"https://doi.org/10.1142/s1793042125500083","url":null,"abstract":"","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141653534","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 : 2024-07-12DOI: 10.1142/s1793042125500034
Toru Komatsu
{"title":"Pairs of cyclic cubic units with rational difference","authors":"Toru Komatsu","doi":"10.1142/s1793042125500034","DOIUrl":"https://doi.org/10.1142/s1793042125500034","url":null,"abstract":"","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141655067","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 : 2024-05-30DOI: 10.1142/s1793042124500982
SoYoung Choi
We prove that for even integer , almost all of zeros of the period polynomial associated to a cusp form of weight on are on the circle under some conditions.
我们证明，对于偶数整数 k，在某些条件下，Γ0(N) 上与权重为 k 的尖顶形式相关的周期多项式的几乎所有零点都位于圆 |z|=1/N 上。
{"title":"Riemann hypothesis for period polynomials for cusp forms on Γ0(N)","authors":"SoYoung Choi","doi":"10.1142/s1793042124500982","DOIUrl":"https://doi.org/10.1142/s1793042124500982","url":null,"abstract":"<p>We prove that for even integer <span><math altimg=\"eq-00003.gif\" display=\"inline\"><mi>k</mi></math></span><span></span>, almost all of zeros of the period polynomial associated to a cusp form of weight <span><math altimg=\"eq-00004.gif\" display=\"inline\"><mi>k</mi></math></span><span></span> on <span><math altimg=\"eq-00005.gif\" display=\"inline\"><msub><mrow><mi mathvariant=\"normal\">Γ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo stretchy=\"false\">(</mo><mi>N</mi><mo stretchy=\"false\">)</mo></math></span><span></span> are on the circle <span><math altimg=\"eq-00006.gif\" display=\"inline\"><mo>|</mo><mi>z</mi><mo>|</mo><mo>=</mo><mn>1</mn><mo stretchy=\"false\">/</mo><msqrt><mrow><mi>N</mi></mrow></msqrt></math></span><span></span> under some conditions.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141253809","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 : 2024-05-29DOI: 10.1142/s1793042124501045
Narasimha Kumar, Satyabrat Sahoo
In this paper, we study Lehmer-type bounds for the Néron–Tate height of -points on abelian varieties over number fields . Then, we estimate the number of -rational points on with Néron–Tate height for . This estimate involves a constant , which is not explicit. However, for elliptic curves and the product of elliptic curves over , we make the constant explicitly computable.
在本文中，我们研究了数域 K 上的无性变项 A 上 K̄ 点的奈伦-塔特高度的雷默型边界。然后，我们估计了 B≫0 时 A 上奈伦-塔特高度≤logB 的 K 有理点的数量。这个估计涉及一个常数 C，它并不明确。然而，对于椭圆曲线和 K 上的椭圆曲线乘积，我们可以明确地计算这个常数。
{"title":"Lehmer-type bounds and counting rational points of bounded heights on Abelian varieties","authors":"Narasimha Kumar, Satyabrat Sahoo","doi":"10.1142/s1793042124501045","DOIUrl":"https://doi.org/10.1142/s1793042124501045","url":null,"abstract":"<p>In this paper, we study Lehmer-type bounds for the Néron–Tate height of <span><math altimg=\"eq-00001.gif\" display=\"inline\"><mover accent=\"true\"><mrow><mi>K</mi></mrow><mo>̄</mo></mover></math></span><span></span>-points on abelian varieties <span><math altimg=\"eq-00002.gif\" display=\"inline\"><mi>A</mi></math></span><span></span> over number fields <span><math altimg=\"eq-00003.gif\" display=\"inline\"><mi>K</mi></math></span><span></span>. Then, we estimate the number of <span><math altimg=\"eq-00004.gif\" display=\"inline\"><mi>K</mi></math></span><span></span>-rational points on <span><math altimg=\"eq-00005.gif\" display=\"inline\"><mi>A</mi></math></span><span></span> with Néron–Tate height <span><math altimg=\"eq-00006.gif\" display=\"inline\"><mo>≤</mo><mo>log</mo><mi>B</mi></math></span><span></span> for <span><math altimg=\"eq-00007.gif\" display=\"inline\"><mi>B</mi><mo>≫</mo><mn>0</mn></math></span><span></span>. This estimate involves a constant <span><math altimg=\"eq-00008.gif\" display=\"inline\"><mi>C</mi></math></span><span></span>, which is not explicit. However, for elliptic curves and the product of elliptic curves over <span><math altimg=\"eq-00009.gif\" display=\"inline\"><mi>K</mi></math></span><span></span>, we make the constant explicitly computable.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141253929","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 : 2024-05-29DOI: 10.1142/s1793042124500970
Mohammad Sadek, Mohamed Wafik, Tuğba Yesin
Let be a polynomial with integer coefficients whose degree is at least 2. We consider the problem of covering the orbit , where is an integer, using arithmetic progressions each of which contains . Fixing an integer , we prove that it is impossible to cover using such arithmetic progressions unless is contained in one of these progressions. In fact, we show that the relative density of terms covered by such arithmetic progressions in is uniformly bounded from above by a bound that depends solely on . In addition, the latter relative density can be made as close as desired to by an appropriate choice of arithmetic progressions containing if is allowed to be large enough.
我们考虑的问题是用算术级数覆盖轨道 Orbf(t)={t,f(t),f(f(t)),...}，其中 t 是整数，而每个算术级数都包含 t。固定整数 k≥2，我们证明除非 Orbf(t) 包含在其中一个算术级数中，否则不可能用 k 个这样的算术级数覆盖 Orbf(t)。事实上，我们证明了在 Orbf(t) 中，由 k 个这样的算术级数所覆盖的项的相对密度是由一个完全取决于 k 的约束从上均匀限定的。
{"title":"Arithmetic progressions in polynomial orbits","authors":"Mohammad Sadek, Mohamed Wafik, Tuğba Yesin","doi":"10.1142/s1793042124500970","DOIUrl":"https://doi.org/10.1142/s1793042124500970","url":null,"abstract":"<p>Let <span><math altimg=\"eq-00001.gif\" display=\"inline\"><mi>f</mi></math></span><span></span> be a polynomial with integer coefficients whose degree is at least 2. We consider the problem of covering the orbit <span><math altimg=\"eq-00002.gif\" display=\"inline\"><msub><mrow><mo>Orb</mo></mrow><mrow><mi>f</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>t</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mo stretchy=\"false\">{</mo><mi>t</mi><mo>,</mo><mi>f</mi><mo stretchy=\"false\">(</mo><mi>t</mi><mo stretchy=\"false\">)</mo><mo>,</mo><mi>f</mi><mo stretchy=\"false\">(</mo><mi>f</mi><mo stretchy=\"false\">(</mo><mi>t</mi><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">)</mo><mo>,</mo><mo>…</mo><mo stretchy=\"false\">}</mo></math></span><span></span>, where <span><math altimg=\"eq-00003.gif\" display=\"inline\"><mi>t</mi></math></span><span></span> is an integer, using arithmetic progressions each of which contains <span><math altimg=\"eq-00004.gif\" display=\"inline\"><mi>t</mi></math></span><span></span>. Fixing an integer <span><math altimg=\"eq-00005.gif\" display=\"inline\"><mi>k</mi><mo>≥</mo><mn>2</mn></math></span><span></span>, we prove that it is impossible to cover <span><math altimg=\"eq-00006.gif\" display=\"inline\"><msub><mrow><mo>Orb</mo></mrow><mrow><mi>f</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>t</mi><mo stretchy=\"false\">)</mo></math></span><span></span> using <span><math altimg=\"eq-00007.gif\" display=\"inline\"><mi>k</mi></math></span><span></span> such arithmetic progressions unless <span><math altimg=\"eq-00008.gif\" display=\"inline\"><msub><mrow><mo>Orb</mo></mrow><mrow><mi>f</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>t</mi><mo stretchy=\"false\">)</mo></math></span><span></span> is contained in one of these progressions. In fact, we show that the relative density of terms covered by <span><math altimg=\"eq-00009.gif\" display=\"inline\"><mi>k</mi></math></span><span></span> such arithmetic progressions in <span><math altimg=\"eq-00010.gif\" display=\"inline\"><msub><mrow><mo>Orb</mo></mrow><mrow><mi>f</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>t</mi><mo stretchy=\"false\">)</mo></math></span><span></span> is uniformly bounded from above by a bound that depends solely on <span><math altimg=\"eq-00011.gif\" display=\"inline\"><mi>k</mi></math></span><span></span>. In addition, the latter relative density can be made as close as desired to <span><math altimg=\"eq-00012.gif\" display=\"inline\"><mn>1</mn></math></span><span></span> by an appropriate choice of <span><math altimg=\"eq-00013.gif\" display=\"inline\"><mi>k</mi></math></span><span></span> arithmetic progressions containing <span><math altimg=\"eq-00014.gif\" display=\"inline\"><mi>t</mi></math></span><span></span> if <span><math altimg=\"eq-00015.gif\" display=\"inline\"><mi>k</mi></math></span><span></span> is allowed to be large enough.</p>","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141253805","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}
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