Pub Date : 2023-11-07DOI: 10.1007/s40993-022-00415-9
Noam D. Elkies, Gaurav Goel
{"title":"On powerful integers expressible as sums of two coprime fourth powers","authors":"Noam D. Elkies, Gaurav Goel","doi":"10.1007/s40993-022-00415-9","DOIUrl":"https://doi.org/10.1007/s40993-022-00415-9","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135474994","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-01DOI: 10.1007/s40993-023-00479-1
Masatake Hirao, Hiroshi Nozaki, Koji Tasaka
{"title":"Spherical designs and modular forms of the $$D_4$$ lattice","authors":"Masatake Hirao, Hiroshi Nozaki, Koji Tasaka","doi":"10.1007/s40993-023-00479-1","DOIUrl":"https://doi.org/10.1007/s40993-023-00479-1","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135325739","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-31DOI: 10.1007/s40993-023-00482-6
Grant Molnar, John Voight
{"title":"Counting elliptic curves over the rationals with a 7-isogeny","authors":"Grant Molnar, John Voight","doi":"10.1007/s40993-023-00482-6","DOIUrl":"https://doi.org/10.1007/s40993-023-00482-6","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135869816","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-31DOI: 10.1007/s40993-023-00483-5
Qing Liu
{"title":"Computing minimal Weierstrass equations of hyperelliptic curves","authors":"Qing Liu","doi":"10.1007/s40993-023-00483-5","DOIUrl":"https://doi.org/10.1007/s40993-023-00483-5","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135808249","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-21DOI: 10.1007/s40993-023-00478-2
Jaitra Chattopadhyay, H. Laxmi, Anupam Saikia
{"title":"Structure of 2-class groups in the $${mathbb {Z}}_{2}$$-extensions of certain real quadratic fields","authors":"Jaitra Chattopadhyay, H. Laxmi, Anupam Saikia","doi":"10.1007/s40993-023-00478-2","DOIUrl":"https://doi.org/10.1007/s40993-023-00478-2","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135511505","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-21DOI: 10.1007/s40993-023-00477-3
Olivier Bordellès, László Tóth
{"title":"Additive arithmetic functions meet the inclusion-exclusion principle, II","authors":"Olivier Bordellès, László Tóth","doi":"10.1007/s40993-023-00477-3","DOIUrl":"https://doi.org/10.1007/s40993-023-00477-3","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135512512","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-03DOI: 10.1007/s40993-023-00476-4
D. R. Johnston, O. Ramaré, T. Trudgian
Abstract We consider Dirichlet L -functions $$L(s, chi )$$ L(s,χ) where $$chi $$ χ is a non-principal quadratic character to the modulus q . We make explicit a result due to Pintz and Stephens by showing that $$|L(1, chi )|leqslant frac{1}{2}log q$$ |L(1,χ)|⩽12logq for all $$qgeqslant 2cdot 10^{23}$$ q⩾2·1023 and $$|L(1, chi )|leqslant frac{9}{20}log q$$ |L(1,χ)|⩽920logq for all $$qgeqslant 5cdot 10^{50}$$ q⩾5·1050 .
我们考虑Dirichlet L -函数$$L(s, chi )$$ L (s， χ)，其中$$chi $$ χ是模q的非主二次特征。我们通过显示所有$$qgeqslant 2cdot 10^{23}$$ q大于或等于2·10 23的$$|L(1, chi )|leqslant frac{1}{2}log q$$ | L (1， χ) |≤1 2 log q和所有$$qgeqslant 5cdot 10^{50}$$ q大于或等于5·10 50的$$|L(1, chi )|leqslant frac{9}{20}log q$$ | L (1， χ) |≤9 20 log q来明确pinz和Stephens的结果。
{"title":"An explicit upper bound for $$L(1,chi )$$ when $$chi $$ is quadratic","authors":"D. R. Johnston, O. Ramaré, T. Trudgian","doi":"10.1007/s40993-023-00476-4","DOIUrl":"https://doi.org/10.1007/s40993-023-00476-4","url":null,"abstract":"Abstract We consider Dirichlet L -functions $$L(s, chi )$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo>(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>,</mml:mo> <mml:mi>χ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> where $$chi $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>χ</mml:mi> </mml:math> is a non-principal quadratic character to the modulus q . We make explicit a result due to Pintz and Stephens by showing that $$|L(1, chi )|leqslant frac{1}{2}log q$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>L</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>χ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> <mml:mo>⩽</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mo>log</mml:mo> <mml:mi>q</mml:mi> </mml:mrow> </mml:math> for all $$qgeqslant 2cdot 10^{23}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>⩾</mml:mo> <mml:mn>2</mml:mn> <mml:mo>·</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mn>23</mml:mn> </mml:msup> </mml:mrow> </mml:math> and $$|L(1, chi )|leqslant frac{9}{20}log q$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>L</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>χ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> <mml:mo>⩽</mml:mo> <mml:mfrac> <mml:mn>9</mml:mn> <mml:mn>20</mml:mn> </mml:mfrac> <mml:mo>log</mml:mo> <mml:mi>q</mml:mi> </mml:mrow> </mml:math> for all $$qgeqslant 5cdot 10^{50}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>⩾</mml:mo> <mml:mn>5</mml:mn> <mml:mo>·</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mn>50</mml:mn> </mml:msup> </mml:mrow> </mml:math> .","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135739322","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-29DOI: 10.1007/s40993-023-00471-9
Yuqi Deng, Riku Kurimaru, Toshiki Matsusaka
{"title":"Arithmetic Dijkgraaf–Witten invariants for real quadratic fields, quadratic residue graphs, and density formulas","authors":"Yuqi Deng, Riku Kurimaru, Toshiki Matsusaka","doi":"10.1007/s40993-023-00471-9","DOIUrl":"https://doi.org/10.1007/s40993-023-00471-9","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135194021","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-29DOI: 10.1007/s40993-023-00474-6
Diana Mocanu
Abstract Let K be a totally real field, and $$rge 5$$ r≥5 a fixed rational prime. In this paper, we use the modular method as presented in the work of Freitas and Siksek to study non-trivial, primitive solutions $$(x,y,z) in mathcal {O}_K^3$$ (x,y,z)∈OK3 of the signature ( r , r , p ) equation $$x^r+y^r=z^p$$ xr+yr=zp (where p is a prime that varies). An adaptation of the modular method is needed, and we follow the work of Freitas which constructs Frey curves over totally real subfields of $$K(zeta _r)$$ K(ζr) . When $$K=mathbb {Q}$$ K=Q we get that for most of the primes $$r<150$$ r<150 with $$r not equiv 1 mod 8$$ r≢1mod8 there are no non-trivial, primitive integer solutions ( x , y , z ) with 2| z for signatures ( r , r , p ) when p is sufficiently large. Similar results hold for quadratic fields, for example when $$K=mathbb {Q}(sqrt{2})$$ K=Q(2) there are no non-trivial, primitive solutions $$(x,y,z)in mathcal {O}_K^3$$ (x,y,z)∈OK3
设K为全实域，且 $$rge 5$$ R≥5是一个定有理数。在本文中，我们使用Freitas和Siksek的工作中提出的模块化方法来研究非平凡的原始解 $$(x,y,z) in mathcal {O}_K^3$$ (x, y, z)∈ok3的签名(r, r, p)方程 $$x^r+y^r=z^p$$ xr + yr = zp (p是变化的质数)需要对模方法进行改进，我们遵循Freitas的工作，在全实数子域上构造Frey曲线 $$K(zeta _r)$$ K (ζ r)什么时候 $$K=mathbb {Q}$$ K = Q对于大多数质数都是这样的 $$r<150$$ R ＆lt;150 with $$r not equiv 1 mod 8$$ 当p足够大时，对于特征(R, R, p)，不存在具有2| z的非平凡原始整数解(x, y, z)。类似的结果适用于二次域，例如当 $$K=mathbb {Q}(sqrt{2})$$ K = Q(2)没有非平凡的原始解 $$(x,y,z)in mathcal {O}_K^3$$ (x, y, z)∈O k3 with $$sqrt{2}|z$$ 2 | z用于签名(5,5,p)， (11,11, p)， (13,13, p)和足够大的p。
{"title":"Asymptotic Fermat for signatures (r, r, p) using the modular approach","authors":"Diana Mocanu","doi":"10.1007/s40993-023-00474-6","DOIUrl":"https://doi.org/10.1007/s40993-023-00474-6","url":null,"abstract":"Abstract Let K be a totally real field, and $$rge 5$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>5</mml:mn> </mml:mrow> </mml:math> a fixed rational prime. In this paper, we use the modular method as presented in the work of Freitas and Siksek to study non-trivial, primitive solutions $$(x,y,z) in mathcal {O}_K^3$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo>,</mml:mo> <mml:mi>z</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>∈</mml:mo> <mml:msubsup> <mml:mi>O</mml:mi> <mml:mi>K</mml:mi> <mml:mn>3</mml:mn> </mml:msubsup> </mml:mrow> </mml:math> of the signature ( r , r , p ) equation $$x^r+y^r=z^p$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msup> <mml:mi>x</mml:mi> <mml:mi>r</mml:mi> </mml:msup> <mml:mo>+</mml:mo> <mml:msup> <mml:mi>y</mml:mi> <mml:mi>r</mml:mi> </mml:msup> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>z</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:mrow> </mml:math> (where p is a prime that varies). An adaptation of the modular method is needed, and we follow the work of Freitas which constructs Frey curves over totally real subfields of $$K(zeta _r)$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>ζ</mml:mi> <mml:mi>r</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . When $$K=mathbb {Q}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo>=</mml:mo> <mml:mi>Q</mml:mi> </mml:mrow> </mml:math> we get that for most of the primes $$r<150$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo><</mml:mo> <mml:mn>150</mml:mn> </mml:mrow> </mml:math> with $$r not equiv 1 mod 8$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>≢</mml:mo> <mml:mn>1</mml:mn> <mml:mspace /> <mml:mo>mod</mml:mo> <mml:mspace /> <mml:mn>8</mml:mn> </mml:mrow> </mml:math> there are no non-trivial, primitive integer solutions ( x , y , z ) with 2| z for signatures ( r , r , p ) when p is sufficiently large. Similar results hold for quadratic fields, for example when $$K=mathbb {Q}(sqrt{2})$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo>=</mml:mo> <mml:mi>Q</mml:mi> <mml:mo>(</mml:mo> <mml:msqrt> <mml:mn>2</mml:mn> </mml:msqrt> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> there are no non-trivial, primitive solutions $$(x,y,z)in mathcal {O}_K^3$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo>,</mml:mo> <mml:mi>z</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>∈</mml:mo> <mml:msubsup> <mml:mi>O</mml:mi> <mml:mi>K</mml:mi> <mml:mn>3</mml:mn> </mml:msubsup> </mml","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135193836","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-28DOI: 10.1007/s40993-023-00472-8
Neelam Kandhil, Rashi Lunia, Jyothsnaa Sivaraman
{"title":"Explicit upper bounds on the average of Euler–Kronecker constants of narrow ray class fields","authors":"Neelam Kandhil, Rashi Lunia, Jyothsnaa Sivaraman","doi":"10.1007/s40993-023-00472-8","DOIUrl":"https://doi.org/10.1007/s40993-023-00472-8","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135387553","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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