Journal of Number Theory最新文献

{"title":"Galois trace forms of type An,Dn,En for odd n","authors":"Riku Higa ,&nbsp;Yoshinosuke Hirakawa","doi":"10.1016/j.jnt.2024.12.007","DOIUrl":"10.1016/j.jnt.2024.12.007","url":null,"abstract":"<div><div>Let <em>p</em> be an odd prime number and <span><math><msub><mrow><mi>ζ</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>:</mo><mo>=</mo><mi>exp</mi><mo>⁡</mo><mo>(</mo><mn>2</mn><mi>π</mi><mi>i</mi><mo>/</mo><mi>p</mi><mo>)</mo></math></span>. Then, it is well-known that the <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span>-root lattice can be realized as the (Hermitian) trace form of the <em>p</em>-th cyclotomic extension <span><math><mi>Q</mi><mo>(</mo><msub><mrow><mi>ζ</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo><mo>/</mo><mi>Q</mi></math></span> restricted to the fractional ideal generated by <span><math><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><msub><mrow><mi>ζ</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></mrow><mrow><mo>−</mo><mo>(</mo><mi>p</mi><mo>−</mo><mn>3</mn><mo>)</mo><mo>/</mo><mn>2</mn></mrow></msup></math></span>. In this paper, in contrast with the case of the <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span>-root lattice, we prove the following theorem: Let <em>n</em> be an odd positive integer and <span><math><mi>F</mi><mo>/</mo><mi>Q</mi></math></span> be a Galois extension of degree <em>n</em>. Then, the number field <em>F</em> does not contain a fractional ideal Λ such that the restricted trace form <span><math><mo>(</mo><mi>Λ</mi><mo>,</mo><mi>Tr</mi><mspace></mspace><msub><mrow><mo>|</mo></mrow><mrow><mi>Λ</mi><mo>×</mo><mi>Λ</mi></mrow></msub><mo>)</mo></math></span> is of type <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. In the proof, we use the prime ideal factorization in <em>F</em> with care of certain 2-adic obstruction for Λ being of type <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. Additionally, we prove that every cyclic cubic field contains infinitely many lattices of type <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> (i.e., normalized face centered cubic lattices) having normal <span><math><mi>Z</mi></math></span>-bases. The latter fact is in contrast with another fact that among quadratic fields only <span><math><mi>Q</mi><mo>(</mo><msqrt><mrow><mo>±</mo><mn>3</mn></mrow></msqrt><mo>)</mo></math></span> contain lattices of type <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"275 ","pages":"Pages 196-213"},"PeriodicalIF":0.6,"publicationDate":"2025-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143609381","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":"Class fields and form class groups for solving certain quadratic Diophantine equations","authors":"Ho Yun Jung ,&nbsp;Ja Kyung Koo ,&nbsp;Dong Hwa Shin ,&nbsp;Dong Sung Yoon","doi":"10.1016/j.jnt.2025.01.018","DOIUrl":"10.1016/j.jnt.2025.01.018","url":null,"abstract":"<div><div>Let <em>K</em> be an imaginary quadratic field and <span><math><mi>O</mi></math></span> be an order in <em>K</em>. We construct class fields associated with form class groups which are isomorphic to certain <span><math><mi>O</mi></math></span>-ideal class groups in terms of the theory of canonical models due to Shimura. As its applications, by using such class fields, for a positive integer <em>n</em> we first find primes of the form <span><math><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>n</mi><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> with additional conditions on <em>x</em> and <em>y</em>. Second, by utilizing these form class groups, we derive a congruence relation on special values of a modular function of higher level as an analogue of Kronecker's congruence relation.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"275 ","pages":"Pages 1-34"},"PeriodicalIF":0.6,"publicationDate":"2025-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143552299","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":"Surjectivity of the adelic Galois representation associated to a Drinfeld module of prime rank","authors":"Chien-Hua Chen","doi":"10.1016/j.jnt.2024.12.012","DOIUrl":"10.1016/j.jnt.2024.12.012","url":null,"abstract":"<div><div>In this paper, let <em>ϕ</em> be the Drinfeld module over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>T</mi><mo>)</mo></math></span> of prime rank <em>r</em> defined by<span><span><span><math><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>=</mo><mi>T</mi><mo>+</mo><msup><mrow><mi>τ</mi></mrow><mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><msup><mrow><mi>T</mi></mrow><mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow></msup><msup><mrow><mi>τ</mi></mrow><mrow><mi>r</mi></mrow></msup><mo>.</mo></math></span></span></span> We prove that under certain conditions on <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>, the adelic Galois representation<span><span><span><math><msub><mrow><mi>ρ</mi></mrow><mrow><mi>ϕ</mi></mrow></msub><mo>:</mo><mrow><mi>Gal</mi></mrow><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><msup><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow><mrow><mi>sep</mi></mrow></msup><mo>/</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>T</mi><mo>)</mo><mo>)</mo><mo>⟶</mo><munder><munder><mi>lim</mi><mo>←</mo></munder><mrow><mi>a</mi></mrow></munder><mrow><mi>Aut</mi></mrow><mo>(</mo><mi>ϕ</mi><mo>[</mo><mi>a</mi><mo>]</mo><mo>)</mo><mo>≅</mo><mrow><mi>G</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow><mo>(</mo><mover><mrow><mi>A</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>)</mo></math></span></span></span> is surjective.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"274 ","pages":"Pages 180-218"},"PeriodicalIF":0.6,"publicationDate":"2025-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143548217","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":"Two properties of symmetric cube transfers of modular forms","authors":"Debargha Banerjee ,&nbsp;Tathagata Mandal ,&nbsp;Sudipa Mondal","doi":"10.1016/j.jnt.2024.12.013","DOIUrl":"10.1016/j.jnt.2024.12.013","url":null,"abstract":"<div><div>In this article, we study two important properties of the symmetric cube transfer of the automorphic representation <em>π</em> associated to a modular form. We first show how the local epsilon factor at each prime changes by twisting in terms of the local Weil-Deligne representation. From this variation number, for each prime <em>p</em>, we classify the types of <span><math><msup><mrow><mi>sym</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> transfers of the local representations <span><math><msub><mrow><mi>π</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>. We also compute the conductor of <span><math><msup><mrow><mi>sym</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>(</mo><mi>π</mi><mo>)</mo></math></span> as it is involved in the variation number. For <span><math><msup><mrow><mi>sym</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> transfer, the most difficult prime is <span><math><mi>p</mi><mo>=</mo><mn>3</mn></math></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"275 ","pages":"Pages 160-195"},"PeriodicalIF":0.6,"publicationDate":"2025-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143601075","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":"An effective open image theorem for products of principally polarized abelian varieties","authors":"Jacob Mayle ,&nbsp;Tian Wang","doi":"10.1016/j.jnt.2024.12.011","DOIUrl":"10.1016/j.jnt.2024.12.011","url":null,"abstract":"<div><div>Let <span><math><mi>A</mi><mo>=</mo><msub><mrow><mo>∏</mo></mrow><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>n</mi></mrow></msub><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> be the product of principally polarized abelian varieties <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> of dimensions <span><math><msub><mrow><mi>g</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, respectively, each defined over a number field <em>K</em>, and pairwise nonisogenous over <span><math><mover><mrow><mi>K</mi></mrow><mo>‾</mo></mover></math></span>. We make effective an open image theorem for <em>A</em> due to Hindry and Ratazzi. More specifically, we give an explicit bound of the constant <span><math><mi>c</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> under GRH, in terms of standard invariants of <em>K</em> and each <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>, where <span><math><mi>c</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> is defined to be the smallest positive integer such that for any prime <span><math><mi>ℓ</mi><mo>&gt;</mo><mi>c</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span>, the image of the <em>ℓ</em>-adic Galois representation of <em>A</em> is “as large as possible” in a suitable sense.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"274 ","pages":"Pages 140-179"},"PeriodicalIF":0.6,"publicationDate":"2025-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143548168","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":"Arboreal Galois groups for cubic polynomials with colliding critical points","authors":"Robert L. Benedetto ,&nbsp;William DeGroot ,&nbsp;Xinyu Ni ,&nbsp;Jesse Seid ,&nbsp;Annie Wei ,&nbsp;Samantha Winton","doi":"10.1016/j.jnt.2025.01.021","DOIUrl":"10.1016/j.jnt.2025.01.021","url":null,"abstract":"<div><div>Let <em>K</em> be a field, and let <span><math><mi>f</mi><mo>∈</mo><mi>K</mi><mo>(</mo><mi>z</mi><mo>)</mo></math></span> be a rational function of degree <span><math><mi>d</mi><mo>≥</mo><mn>2</mn></math></span>. The Galois group of the field extension generated by the preimages of <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∈</mo><mi>K</mi></math></span> under all iterates of <em>f</em> naturally embeds in the automorphism group of an infinite <em>d</em>-ary rooted tree. In some cases the Galois group can be the full automorphism group of the tree, but in other cases it is known to have infinite index. In this paper, we consider a previously unstudied such case: that <em>f</em> is a polynomial of degree <span><math><mi>d</mi><mo>=</mo><mn>3</mn></math></span>, and the two finite critical points of <em>f</em> collide at the <em>ℓ</em>-th iteration, for some <span><math><mi>ℓ</mi><mo>≥</mo><mn>2</mn></math></span>. We describe an explicit subgroup <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>ℓ</mi><mo>,</mo><mo>∞</mo></mrow></msub></math></span> of automorphisms of the 3-ary tree in which the resulting Galois group must always embed, and we present sufficient conditions for this embedding to be an isomorphism.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"274 ","pages":"Pages 72-103"},"PeriodicalIF":0.6,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143548346","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":"Locally induced Galois representations with exceptional residual images","authors":"Chengyang Bao","doi":"10.1016/j.jnt.2024.12.015","DOIUrl":"10.1016/j.jnt.2024.12.015","url":null,"abstract":"<div><div>In this paper, we classify all continuous Galois representations <span><math><mi>ρ</mi><mo>:</mo><mrow><mi>Gal</mi></mrow><mo>(</mo><mover><mrow><mi>Q</mi></mrow><mo>‾</mo></mover><mo>/</mo><mi>Q</mi><mo>)</mo><mo>→</mo><msub><mrow><mi>GL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><msub><mrow><mover><mrow><mi>Q</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></math></span> which are unramified outside <span><math><mo>{</mo><mi>p</mi><mo>,</mo><mo>∞</mo><mo>}</mo></math></span> and locally induced at <em>p</em>, under the assumption that <span><math><mover><mrow><mi>ρ</mi></mrow><mo>‾</mo></mover></math></span> is exceptional, that is, has image of order prime to <em>p</em>. We prove two results. If <em>f</em> is a level one cuspidal eigenform and one of the <em>p</em>-adic Galois representations <span><math><msub><mrow><mi>ρ</mi></mrow><mrow><mi>f</mi></mrow></msub></math></span> associated to <em>f</em> has exceptional residual image, then <span><math><msub><mrow><mi>ρ</mi></mrow><mrow><mi>f</mi></mrow></msub></math></span> is not locally induced and <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>f</mi><mo>)</mo><mo>≠</mo><mn>0</mn></math></span>. If <em>ρ</em> is locally induced at <em>p</em> and with exceptional residual image, and furthermore certain subfields of the fixed field of the kernel of <span><math><mover><mrow><mi>ρ</mi></mrow><mo>‾</mo></mover></math></span> are assumed to have class numbers prime to <em>p</em>, then <em>ρ</em> has finite image up to a twist.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"275 ","pages":"Pages 49-66"},"PeriodicalIF":0.6,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143552300","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":"Asymptotic solutions of the generalized Fermat-type equation of signature (p,p,3) over totally real number fields","authors":"Satyabrat Sahoo ,&nbsp;Narasimha Kumar","doi":"10.1016/j.jnt.2025.01.020","DOIUrl":"10.1016/j.jnt.2025.01.020","url":null,"abstract":"<div><div>In this article, we study the asymptotic solutions of the generalized Fermat-type equation of signature <span><math><mo>(</mo><mi>p</mi><mo>,</mo><mi>p</mi><mo>,</mo><mn>3</mn><mo>)</mo></math></span> over totally real number fields <em>K</em>, i.e., <span><math><mi>A</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>+</mo><mi>B</mi><msup><mrow><mi>y</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>=</mo><mi>C</mi><msup><mrow><mi>z</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> with prime exponent <em>p</em> and <span><math><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo>∈</mo><msub><mrow><mi>O</mi></mrow><mrow><mi>K</mi></mrow></msub><mo>∖</mo><mo>{</mo><mn>0</mn><mo>}</mo></math></span>. For certain class of fields <em>K</em>, we prove that <span><math><mi>A</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>+</mo><mi>B</mi><msup><mrow><mi>y</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>=</mo><mi>C</mi><msup><mrow><mi>z</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> has no asymptotic solutions over <em>K</em> (resp., solutions of certain type over <em>K</em>) with restrictions on <span><math><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi></math></span> (resp., for all <span><math><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo>∈</mo><msub><mrow><mi>O</mi></mrow><mrow><mi>K</mi></mrow></msub><mo>∖</mo><mo>{</mo><mn>0</mn><mo>}</mo></math></span>). Finally, we present several local criteria over <em>K</em>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"274 ","pages":"Pages 56-71"},"PeriodicalIF":0.6,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143529155","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":"On the factor rings of Z[n3]","authors":"Tomasz Jędrzejak","doi":"10.1016/j.jnt.2025.01.022","DOIUrl":"10.1016/j.jnt.2025.01.022","url":null,"abstract":"<div><div>We give a description of the structure of factor rings of <span><math><mi>Z</mi><mrow><mo>[</mo><mroot><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></mroot><mo>]</mo></mrow></math></span> where (without loss of generality) <em>n</em> is a positive integer which is not a cube. For example, we prove that <span><math><mi>Z</mi><mrow><mo>[</mo><mroot><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></mroot><mo>]</mo></mrow><mo>/</mo><mrow><mo>(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mroot><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></mroot><mo>+</mo><mi>c</mi><mroot><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>3</mn></mrow></mroot><mo>)</mo></mrow></math></span> is isomorphic to the ring of integers modulo <span><math><mo>|</mo><mi>N</mi><mo>|</mo></math></span>, if <span><math><mi>gcd</mi><mo>⁡</mo><mrow><mo>(</mo><msup><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mi>a</mi><mi>c</mi><mo>,</mo><mi>N</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></math></span> where <span><math><mi>N</mi><mo>=</mo><msup><mrow><mi>a</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><mi>n</mi><msup><mrow><mi>b</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mi>c</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>−</mo><mn>3</mn><mi>n</mi><mi>a</mi><mi>b</mi><mi>c</mi></math></span> is the norm of the generator. We also characterize the structure of these factor rings for others integers <span><math><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi></math></span>. Finally, we describe <span><math><mi>Z</mi><mrow><mo>[</mo><mroot><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></mroot><mo>]</mo></mrow><mo>/</mo><mi>I</mi></math></span> for certain non-principal ideals <em>I</em>. We also present many corollaries regarding irreducible and prime elements in <span><math><mi>Z</mi><mrow><mo>[</mo><mroot><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></mroot><mo>]</mo></mrow></math></span> and give numerous examples.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"275 ","pages":"Pages 104-118"},"PeriodicalIF":0.6,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143563378","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":"On torsion subgroups of elliptic curves over quartic, quintic and sextic number fields","authors":"Mustafa Umut Kazancıoğlu,&nbsp;Mohammad Sadek","doi":"10.1016/j.jnt.2025.01.017","DOIUrl":"10.1016/j.jnt.2025.01.017","url":null,"abstract":"<div><div>The list of all groups that can appear as torsion subgroups of elliptic curves over number fields of degree <em>d</em>, <span><math><mi>d</mi><mo>=</mo><mn>4</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>6</mn></math></span>, is not completely determined. However, the list of groups <span><math><msup><mrow><mi>Φ</mi></mrow><mrow><mo>∞</mo></mrow></msup><mo>(</mo><mi>d</mi><mo>)</mo></math></span>, <span><math><mi>d</mi><mo>=</mo><mn>4</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>6</mn></math></span>, that can be realized as torsion subgroups for infinitely many non-isomorphic elliptic curves over these fields is known. We address the question of which torsion subgroups can arise over a given number field of degree <em>d</em>. In fact, given <span><math><mi>G</mi><mo>∈</mo><msup><mrow><mi>Φ</mi></mrow><mrow><mo>∞</mo></mrow></msup><mo>(</mo><mi>d</mi><mo>)</mo></math></span> and a number field <em>K</em> of degree <em>d</em>, we give explicit criteria telling whether <em>G</em> is realized finitely or infinitely often over <em>K</em>. We also give results on the field with the smallest absolute value of its discriminant such that there exists an elliptic curve with torsion <em>G</em>. Finally, we give examples of number fields <em>K</em> of degree <em>d</em>, <span><math><mi>d</mi><mo>=</mo><mn>4</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>6</mn></math></span>, over which the Mordell-Weil rank of elliptic curves with prescribed torsion is bounded from above.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"274 ","pages":"Pages 37-55"},"PeriodicalIF":0.6,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143529156","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}