Journal of Number Theory最新文献

{"title":"3-Selmer groups, ideal class groups and the cube sum problem","authors":"Somnath Jha ,&nbsp;Dipramit Majumdar ,&nbsp;Pratiksha Shingavekar","doi":"10.1016/j.jnt.2025.02.004","DOIUrl":"10.1016/j.jnt.2025.02.004","url":null,"abstract":"<div><div>Consider a Mordell curve <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>a</mi></mrow></msub><mo>:</mo><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><mi>a</mi></math></span> with <span><math><mi>a</mi><mo>∈</mo><mi>Z</mi></math></span>. These curves have a rational 3-isogeny, say <em>φ</em>. We give an upper and a lower bound on the rank of the <em>φ</em>-Selmer group of <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>a</mi></mrow></msub></math></span> over <span><math><mi>Q</mi><mo>(</mo><msub><mrow><mi>ζ</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>)</mo></math></span> in terms of the 3-part of the ideal class group of certain quadratic extension of <span><math><mi>Q</mi><mo>(</mo><msub><mrow><mi>ζ</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>)</mo></math></span>. Using our bounds on the Selmer groups, we prove some cases of the rational cube sum problem. Further, using these bounds, we prove that a positive proportion of Mordell curves, explicitly determined by some congruence conditions, have vanishing 3-Selmer group over <span><math><mi>Q</mi></math></span> (respectively have <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>-rank of the 3-Selmer group over <span><math><mi>Q</mi></math></span> equal to 1).</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"277 ","pages":"Pages 165-200"},"PeriodicalIF":0.6,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143946686","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":"Eventual tightness of projective dimension growth bounds: Quadratic in the degree","authors":"Raf Cluckers ,&nbsp;Itay Glazer","doi":"10.1016/j.jnt.2025.03.010","DOIUrl":"10.1016/j.jnt.2025.03.010","url":null,"abstract":"<div><div>In projective dimension growth results, one bounds the number of rational points of height at most <em>H</em> on an irreducible hypersurface in <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> of degree <span><math><mi>d</mi><mo>&gt;</mo><mn>3</mn></math></span> by <span><math><mi>C</mi><mo>(</mo><mi>n</mi><mo>)</mo><msup><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mi>H</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><msup><mrow><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>H</mi><mo>)</mo></mrow><mrow><mi>M</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup></math></span>, where the quadratic dependence in <em>d</em> has been recently obtained by Binyamini, Cluckers and Kato in 2024 <span><span>[1]</span></span>. For these bounds, it was already shown by Castryck, Cluckers, Dittmann and Nguyen in 2020 <span><span>[3]</span></span> that one cannot do better than a linear dependence in <em>d</em>. In this paper we show that, for the mentioned projective dimension growth bounds, the quadratic dependence in <em>d</em> is eventually tight when <em>n</em> grows. More precisely the upper bounds cannot be better than <span><math><mi>c</mi><mo>(</mo><mi>n</mi><mo>)</mo><msup><mrow><mi>d</mi></mrow><mrow><mn>2</mn><mo>−</mo><mn>2</mn><mo>/</mo><mi>n</mi></mrow></msup><msup><mrow><mi>H</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> in general. Note that for affine dimension growth (for affine hypersurfaces of degree <em>d</em>, satisfying some extra conditions), the dependence on <em>d</em> is also quadratic by <span><span>[1]</span></span>, which is already known to be optimal by <span><span>[3]</span></span>. Our projective case thus complements the picture of tightness for dimension growth bounds for hypersurfaces.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"276 ","pages":"Pages 72-80"},"PeriodicalIF":0.6,"publicationDate":"2025-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143935354","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":"Models of CM elliptic curves with a prescribed ℓ-adic Galois image","authors":"Enrique González–Jiménez ,&nbsp;Álvaro Lozano-Robledo ,&nbsp;Benjamin York","doi":"10.1016/j.jnt.2025.03.001","DOIUrl":"10.1016/j.jnt.2025.03.001","url":null,"abstract":"<div><div>For each prime number <em>ℓ</em> and for each imaginary quadratic order of class number one or two, we determine all the possible <em>ℓ</em>-adic Galois representations that occur for any elliptic curve with complex multiplication by such an order over its minimal field of definition, and then we determine all the isomorphism classes of elliptic curves that have a prescribed <em>ℓ</em>-adic Galois representation.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"277 ","pages":"Pages 19-62"},"PeriodicalIF":0.6,"publicationDate":"2025-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143942093","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":"Quadratic twists of genus one curves","authors":"Lukas Novak","doi":"10.1016/j.jnt.2025.02.001","DOIUrl":"10.1016/j.jnt.2025.02.001","url":null,"abstract":"<div><div>For a given irreducible and monic polynomial <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>∈</mo><mi>Z</mi><mo>[</mo><mi>x</mi><mo>]</mo></math></span> of degree 4, we consider the quadratic twists by square-free integers <em>q</em> of the genus one quartic <span><math><mi>H</mi><mspace></mspace><mo>:</mo><mspace></mspace><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span><span><span><span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>q</mi></mrow></msub><mspace></mspace><mo>:</mo><mspace></mspace><mi>q</mi><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>.</mo></math></span></span></span></div><div>Let <em>L</em> denote the set of positive square-free integers <em>q</em> for which <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> is everywhere locally solvable. For a real number <em>x</em>, let <span><math><mi>L</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>#</mi><mo>{</mo><mi>q</mi><mo>∈</mo><mi>L</mi><mo>:</mo><mspace></mspace><mi>q</mi><mo>≤</mo><mi>x</mi><mo>}</mo></math></span> be the number of elements in <em>L</em> that are less than or equal to <em>x</em>.</div><div>In this paper, we obtain that<span><span><span><math><mi>L</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>f</mi></mrow></msub><mfrac><mrow><mi>x</mi></mrow><mrow><msup><mrow><mo>(</mo><mi>ln</mi><mo>⁡</mo><mi>x</mi><mo>)</mo></mrow><mrow><mi>m</mi></mrow></msup></mrow></mfrac><mo>+</mo><mi>O</mi><mrow><mo>(</mo><mfrac><mrow><mi>x</mi></mrow><mrow><msup><mrow><mo>(</mo><mi>ln</mi><mo>⁡</mo><mi>x</mi><mo>)</mo></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfrac><mo>)</mo></mrow></math></span></span></span> for some constants <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>&gt;</mo><mn>0</mn></math></span>, <em>m</em> and <em>α</em> only depending on <em>f</em> such that <span><math><mi>m</mi><mo>&lt;</mo><mi>α</mi><mo>≤</mo><mn>1</mn><mo>+</mo><mi>m</mi></math></span>. We also express the Dirichlet series <span><math><mi>F</mi><mo>(</mo><mi>s</mi><mo>)</mo><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>n</mi><mo>∈</mo><mi>L</mi></mrow></msub><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mi>s</mi></mrow></msup></math></span> associated to the set <em>L</em> in terms of Dedekind zeta functions of certain number fields.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"276 ","pages":"Pages 1-22"},"PeriodicalIF":0.6,"publicationDate":"2025-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143917288","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":"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}