Research in Number Theory最新文献_第4页

On powerful integers expressible as sums of two coprime fourth powers 在可表示为两个素数四次方和的强整数上

Research in Number Theory Pub Date : 2023-11-07 DOI: 10.1007/s40993-022-00415-9

Noam D. Elkies, Gaurav Goel

引用次数: 0

Spherical designs and modular forms of the $$D_4$$ lattice 球形设计和$$D_4$$晶格的模块化形式

Research in Number Theory Pub Date : 2023-11-01 DOI: 10.1007/s40993-023-00479-1

Masatake Hirao, Hiroshi Nozaki, Koji Tasaka

引用次数: 1

Counting elliptic curves over the rationals with a 7-isogeny 计算具有7等构的有理数上的椭圆曲线

Research in Number Theory Pub Date : 2023-10-31 DOI: 10.1007/s40993-023-00482-6

Grant Molnar, John Voight

引用次数: 0

Computing minimal Weierstrass equations of hyperelliptic curves 超椭圆曲线的最小Weierstrass方程的计算

Research in Number Theory Pub Date : 2023-10-31 DOI: 10.1007/s40993-023-00483-5

Qing Liu

引用次数: 2

Structure of 2-class groups in the $${mathbb {Z}}_{2}$$-extensions of certain real quadratic fields 某些实数二次域$${mathbb {Z}}_{2}$$ -扩展中2类群的结构

Research in Number Theory Pub Date : 2023-10-21 DOI: 10.1007/s40993-023-00478-2

Jaitra Chattopadhyay, H. Laxmi, Anupam Saikia

引用次数: 0

Additive arithmetic functions meet the inclusion-exclusion principle, II 加性算术函数满足含不相容原则

Research in Number Theory Pub Date : 2023-10-21 DOI: 10.1007/s40993-023-00477-3

Olivier Bordellès, László Tóth

引用次数: 3

An explicit upper bound for $$L(1,chi )$$ when $$chi $$ is quadratic 当$$chi $$是二次元时，$$L(1,chi )$$的显式上界

Research in Number Theory Pub Date : 2023-10-03 DOI: 10.1007/s40993-023-00476-4

D. R. Johnston, O. Ramaré, T. Trudgian

{"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}

引用次数: 0

Arithmetic Dijkgraaf–Witten invariants for real quadratic fields, quadratic residue graphs, and density formulas 算术Dijkgraaf-Witten不变量的实二次域，二次剩余图，和密度公式

Research in Number Theory Pub Date : 2023-09-29 DOI: 10.1007/s40993-023-00471-9

Yuqi Deng, Riku Kurimaru, Toshiki Matsusaka

引用次数: 0

Asymptotic Fermat for signatures (r, r, p) using the modular approach 用模方法求签名(r, r, p)的渐近费马

Research in Number Theory Pub Date : 2023-09-29 DOI: 10.1007/s40993-023-00474-6

Diana Mocanu

{"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}

引用次数: 3

Explicit upper bounds on the average of Euler–Kronecker constants of narrow ray class fields 窄射线类场的Euler-Kronecker常数平均值的显式上界

Research in Number Theory Pub Date : 2023-09-28 DOI: 10.1007/s40993-023-00472-8

Neelam Kandhil, Rashi Lunia, Jyothsnaa Sivaraman

引用次数: 0