Journal of Number Theory最新文献

{"title":"Reduction of hyperelliptic curves in residue characteristic 2","authors":"Tim Gehrunger,&nbsp;Richard Pink","doi":"10.1016/j.jnt.2025.10.004","DOIUrl":"10.1016/j.jnt.2025.10.004","url":null,"abstract":"<div><div>Consider a hyperelliptic curve of genus <em>g</em> over a field <em>K</em> of characteristic zero. After extending <em>K</em> we can view it as a marked curve with its <span><math><mn>2</mn><mi>g</mi><mo>+</mo><mn>2</mn></math></span> Weierstrass points. We present an explicit algorithm to compute the stable reduction of this marked curve for a valuation of residue characteristic 2 over a finite extension of <em>K</em>. In the cases <span><math><mi>g</mi><mo>⩽</mo><mn>2</mn></math></span> we work out relatively simple conditions for the structure of this reduction.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"281 ","pages":"Pages 429-491"},"PeriodicalIF":0.7,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145467142","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":"A uniform formula on the number of integer matrices with given determinant and height","authors":"Muhammad Afifurrahman","doi":"10.1016/j.jnt.2025.10.011","DOIUrl":"10.1016/j.jnt.2025.10.011","url":null,"abstract":"<div><div>We obtain an asymptotic formula for the number of integer <span><math><mn>2</mn><mo>×</mo><mn>2</mn></math></span> matrices that have determinant Δ and whose absolute values of the entries are at most <em>H</em>. The result holds uniformly for a large range of Δ with respect to <em>H</em>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"281 ","pages":"Pages 741-770"},"PeriodicalIF":0.7,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145617773","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":"Parameters of solvable automorphic forms","authors":"Peter Vang Uttenthal","doi":"10.1016/j.jnt.2025.10.016","DOIUrl":"10.1016/j.jnt.2025.10.016","url":null,"abstract":"<div><div>In a letter from Tate to Serre dated March 26, 1974, Tate suggested a classification of weight one modular forms of prime level in terms of their associated odd Artin representations. This paper carries out an analogous classification of Maass wave forms of prime power level in terms of complex even representations. The parameters are identified with techniques from class field theory and Galois representations. The classification reveals that there exist distinct Maass cusp forms of tetrahedral type on <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>ℓ</mi><mo>)</mo></math></span> that remain inequivalent modulo 3 for <span><math><mi>ℓ</mi><mo>=</mo><mn>7687</mn><mo>,</mo><mn>16363</mn></math></span> and 20887, and that these <em>ℓ</em> are the three smallest such primes.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"281 ","pages":"Pages 771-794"},"PeriodicalIF":0.7,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145618275","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":"New truncated sums on the Jacobi triple product identity and the quintuple product identity","authors":"Ernest X.W. Xia","doi":"10.1016/j.jnt.2025.10.006","DOIUrl":"10.1016/j.jnt.2025.10.006","url":null,"abstract":"<div><div>In their study of the truncated sums of the classical theta functions, Andrews-Merca and Guo-Zeng posed a conjecture on truncated sums of a special case of the Jacobi triple product identity which was confirmed independently by Mao and Yee. In 2016, Chan, Ho and Mao examined the truncated series arising from two consequences of the quintuple product identity. In this paper, we establish an explicit series form with nonnegative coefficients on a new truncated sum of the Jacobi triple product identity which is stronger than the conjecture due to Andrews-Merca and Guo-Zeng by taking different truncated series and using Bailey pairs. As a corollary of our results, we obtain a new truncated sum of Jacobi's identity which implies another conjecture given by Guo and Zeng. In addition, we determine the signs of coefficients of two well-known identities derived from the quintuple product identity which can be considered as the companion results of a theorem proved by Chan, Ho and Mao.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"281 ","pages":"Pages 562-587"},"PeriodicalIF":0.7,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145467130","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":"Artin's conjecture for Abelian varieties with Frobenius condition","authors":"Florian Hess ,&nbsp;Leonard Tomczak","doi":"10.1016/j.jnt.2025.09.017","DOIUrl":"10.1016/j.jnt.2025.09.017","url":null,"abstract":"<div><div>Let <em>A</em> be an abelian variety over a number field <em>K</em> of dimension <em>r</em>, <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>g</mi></mrow></msub><mo>∈</mo><mi>A</mi><mo>(</mo><mi>K</mi><mo>)</mo></math></span> and <span><math><mi>F</mi><mo>/</mo><mi>K</mi></math></span> a finite Galois extension. We consider the density of primes <span><math><mi>p</mi></math></span> of <em>K</em> such that the quotient <span><math><mover><mrow><mi>A</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>(</mo><mi>k</mi><mo>(</mo><mi>p</mi><mo>)</mo><mo>)</mo><mo>/</mo><mo>〈</mo><msub><mrow><mover><mrow><mi>a</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mover><mrow><mi>a</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mi>g</mi></mrow></msub><mo>〉</mo></math></span> has at most <span><math><mn>2</mn><mi>r</mi><mo>−</mo><mn>1</mn></math></span> cyclic components and <span><math><mi>p</mi></math></span> satisfies a Frobenius condition with respect to <span><math><mi>F</mi><mo>/</mo><mi>K</mi></math></span>, where <span><math><mover><mrow><mi>A</mi></mrow><mrow><mo>¯</mo></mrow></mover></math></span> is the reduction of <em>A</em> modulo <span><math><mi>p</mi></math></span>, <span><math><mi>k</mi><mo>(</mo><mi>p</mi><mo>)</mo></math></span> is the residue field of <span><math><mi>p</mi></math></span> and <span><math><mo>〈</mo><msub><mrow><mover><mrow><mi>a</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mover><mrow><mi>a</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mi>g</mi></mrow></msub><mo>〉</mo></math></span> is the subgroup generated by the reductions <span><math><msub><mrow><mover><mrow><mi>a</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mover><mrow><mi>a</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mi>g</mi></mrow></msub></math></span>. We develop a general framework to prove the existence of this density under the Generalized Riemann Hypothesis.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"281 ","pages":"Pages 43-58"},"PeriodicalIF":0.7,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145365628","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":"Unconditional lower bounds for the sixth and eighth moments of the Riemann zeta function","authors":"Timothy Page","doi":"10.1016/j.jnt.2025.08.017","DOIUrl":"10.1016/j.jnt.2025.08.017","url":null,"abstract":"<div><div>Unconditional bounds on the sixth and eighth moments of the Riemann zeta function are improved by bounding twisted second and fourth moments that arise upon application of the Cauchy-Schwarz inequality and Hölder's inequality. An unconditional bound on the sixth moment of the derivative of the Riemann zeta function is also deduced.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 318-369"},"PeriodicalIF":0.7,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145220651","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":"Higher moments for non-normal fields with Galois group Ad and Sd","authors":"Jiong Yang","doi":"10.1016/j.jnt.2025.08.018","DOIUrl":"10.1016/j.jnt.2025.08.018","url":null,"abstract":"<div><div>Let <em>K</em> be a non-normal number field of degree <em>d</em> with Galois group <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>d</mi></mrow></msub></math></span> or <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>d</mi></mrow></msub></math></span>. Let <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>K</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> be the number of integral ideals of norm <em>n</em> in <em>K</em>. We obtain an asymptotic formula for the summation <span><math><munder><mo>∑</mo><mrow><mi>n</mi><mo>≤</mo><mi>x</mi></mrow></munder><msubsup><mrow><mi>a</mi></mrow><mrow><mi>K</mi></mrow><mrow><mi>l</mi></mrow></msubsup><mo>(</mo><mi>n</mi><mo>)</mo></math></span> for any <span><math><mi>l</mi><mo>≥</mo><mn>2</mn></math></span>. As a consequence, we obtain such an asymptotic formula for any number field <em>K</em> of degree less or equal to 8 unconditionally.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 370-389"},"PeriodicalIF":0.7,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145220652","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":"Abelian varieties with real multiplication: Classification and isogeny classes over finite fields","authors":"Tejasi Bhatnagar ,&nbsp;Yu Fu","doi":"10.1016/j.jnt.2025.08.006","DOIUrl":"10.1016/j.jnt.2025.08.006","url":null,"abstract":"<div><div>In this paper, we provide a classification of certain points on Hilbert modular varieties over finite fields under a mild assumption on Newton polygon. Furthermore, we use this characterization to prove estimates for the size of isogeny classes.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 171-190"},"PeriodicalIF":0.7,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145096123","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":"M-functions and screw functions: Applications to Goldbach's problem and zeros of the Riemann zeta-function","authors":"Kohji Matsumoto ,&nbsp;Masatoshi Suzuki","doi":"10.1016/j.jnt.2025.09.013","DOIUrl":"10.1016/j.jnt.2025.09.013","url":null,"abstract":"<div><div>We study the <em>M</em>-functions, which describe the limit theorem for the value-distributions of the secondary main terms in the asymptotic formulas for the summatory functions of the Goldbach counting function. One of the new aspects is a sufficient condition for the Riemann hypothesis provided by some formulas of the <em>M</em>-functions, which was a necessary condition in previous work. The other new aspect is the relation between the secondary main terms and the screw functions, which provides another necessary and sufficient condition for the Riemann hypothesis. We study such <em>M</em>-functions and screw functions in generalized settings by axiomatizing them.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 918-946"},"PeriodicalIF":0.7,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145262238","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":"Torsion of rational elliptic curves over the Zp-extensions of quadratic fields","authors":"Ömer Avcı","doi":"10.1016/j.jnt.2025.08.009","DOIUrl":"10.1016/j.jnt.2025.08.009","url":null,"abstract":"<div><div>Let <em>E</em> be an elliptic curve defined over <span><math><mi>Q</mi></math></span>. For a quadratic number field <em>K</em> and an odd prime number <em>p</em>, let <em>L</em> be a <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-extension of <em>K</em>. We prove that <span><math><mi>E</mi><msub><mrow><mo>(</mo><mi>L</mi><mo>)</mo></mrow><mrow><mtext>tors</mtext></mrow></msub><mo>=</mo><mi>E</mi><msub><mrow><mo>(</mo><mi>K</mi><mo>)</mo></mrow><mrow><mtext>tors</mtext></mrow></msub></math></span> when <span><math><mi>p</mi><mo>&gt;</mo><mn>5</mn></math></span>. It enables us to classify the groups that can be realized as the torsion subgroup <span><math><mi>E</mi><msub><mrow><mo>(</mo><mi>L</mi><mo>)</mo></mrow><mrow><mtext>tors</mtext></mrow></msub></math></span>, by using the classification of torsion subgroups over the quadratic fields.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 153-170"},"PeriodicalIF":0.7,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145096122","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}