Journal of Number Theory最新文献

{"title":"Common values of linear recurrences related to Shank's simplest cubics","authors":"Attila Pethő ,&nbsp;Szabolcs Tengely","doi":"10.1016/j.jnt.2024.09.001","DOIUrl":"10.1016/j.jnt.2024.09.001","url":null,"abstract":"<div><div>Let <span><math><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo>∈</mo><mi>Z</mi></math></span> not all zeroes and let <span><math><mi>F</mi><mo>(</mo><mi>u</mi><mo>,</mo><mi>n</mi><mo>)</mo><mo>=</mo><mi>F</mi><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo>,</mo><mi>u</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span> be the linear recursive sequence, which is defined by the initial terms <span><math><mi>F</mi><mo>(</mo><mi>u</mi><mo>,</mo><mn>0</mn><mo>)</mo><mo>=</mo><mi>A</mi><mo>,</mo><mi>F</mi><mo>(</mo><mi>u</mi><mo>,</mo><mn>1</mn><mo>)</mo><mo>=</mo><mi>B</mi><mo>,</mo><mi>F</mi><mo>(</mo><mi>u</mi><mo>,</mo><mn>2</mn><mo>)</mo><mo>=</mo><mi>C</mi></math></span> and whose characteristic polynomial is Daniel Shanks simplest cubic <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>u</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>X</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>−</mo><mo>(</mo><mi>u</mi><mo>−</mo><mn>1</mn><mo>)</mo><msup><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mo>(</mo><mi>u</mi><mo>+</mo><mn>2</mn><mo>)</mo><mi>X</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>u</mi><mo>∈</mo><mi>Z</mi></math></span>. We prove that there exists an effectively computable constant <em>c</em> depending only on <span><math><mi>L</mi><mo>=</mo><mi>max</mi><mo>⁡</mo><mo>{</mo><mo>|</mo><mi>A</mi><mo>|</mo><mo>,</mo><mo>|</mo><mi>B</mi><mo>|</mo><mo>,</mo><mo>|</mo><mi>C</mi><mo>|</mo><mo>}</mo></math></span> such that if <span><math><mo>|</mo><mi>F</mi><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo>,</mo><mi>u</mi><mo>,</mo><mi>n</mi><mo>)</mo><mo>|</mo><mo>=</mo><mo>|</mo><mi>F</mi><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo>,</mo><mi>u</mi><mo>,</mo><mi>m</mi><mo>)</mo><mo>|</mo></math></span> holds for some integers <span><math><mi>u</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>m</mi></math></span> with <span><math><mi>n</mi><mo>≠</mo><mi>m</mi></math></span> then <span><math><mo>|</mo><mi>n</mi><mo>|</mo><mo>,</mo><mo>|</mo><mi>m</mi><mo>|</mo><mo>&lt;</mo><mi>c</mi></math></span>. For the choices <span><math><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo>)</mo><mo>∈</mo><mo>{</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>,</mo><mo>(</mo><mn>1</mn><mo>,</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>}</mo></math></span> we solve the above equations completely. At the end we give an outlook to the equation <span><math><mi>F</mi><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mi>u</mi><mo>,</mo><mi>n</mi><mo>)</mo><mo>=</mo><mi>F</mi><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mi>v</mi><mo>,</mo><mi>m</mi><mo>)</mo></math></span> for some fixed integers <span><math><mi>n</mi><mo>,</mo><mi>m</mi></math></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"267 ","pages":"Pages 34-79"},"PeriodicalIF":0.6,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142328260","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":"On the number of prime factors with a given multiplicity over h-free and h-full numbers","authors":"Sourabhashis Das,&nbsp;Wentang Kuo,&nbsp;Yu-Ru Liu","doi":"10.1016/j.jnt.2024.08.007","DOIUrl":"10.1016/j.jnt.2024.08.007","url":null,"abstract":"<div><div>Let <em>k</em> and <em>n</em> be natural numbers. Let <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> denote the number of distinct prime factors of <em>n</em> with multiplicity <em>k</em> as studied by Elma and the third author <span><span>[5]</span></span>. We obtain asymptotic estimates for the first and the second moments of <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> when restricted to the set of <em>h</em>-free and <em>h</em>-full numbers. We prove that <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> has normal order <span><math><mi>log</mi><mo>⁡</mo><mi>log</mi><mo>⁡</mo><mi>n</mi></math></span> over <em>h</em>-free numbers, <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mi>h</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> has normal order <span><math><mi>log</mi><mo>⁡</mo><mi>log</mi><mo>⁡</mo><mi>n</mi></math></span> over <em>h</em>-full numbers, and both of them satisfy the Erdős-Kac Theorem. Finally, we prove that the functions <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> with <span><math><mn>1</mn><mo>&lt;</mo><mi>k</mi><mo>&lt;</mo><mi>h</mi></math></span> do not have normal order over <em>h</em>-free numbers and <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> with <span><math><mi>k</mi><mo>&gt;</mo><mi>h</mi></math></span> do not have normal order over <em>h</em>-full numbers.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"267 ","pages":"Pages 176-201"},"PeriodicalIF":0.6,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142328287","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":"Maximally elastic quadratic fields","authors":"Paul Pollack","doi":"10.1016/j.jnt.2024.08.003","DOIUrl":"10.1016/j.jnt.2024.08.003","url":null,"abstract":"<div><div>Recall that for a domain <em>R</em> where every nonzero nonunit factors into irreducibles, the <span>elasticity</span> of <em>R</em> is defined as<span><span><span><math><mi>sup</mi><mo>⁡</mo><mrow><mo>{</mo><mfrac><mrow><mi>s</mi></mrow><mrow><mi>r</mi></mrow></mfrac><mo>:</mo><msub><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>⋯</mo><msub><mrow><mi>π</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>=</mo><msub><mrow><mi>ρ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>⋯</mo><msub><mrow><mi>ρ</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>,</mo><mrow><mtext> with all </mtext><msub><mrow><mi>π</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>ρ</mi></mrow><mrow><mi>j</mi></mrow></msub><mtext> irreducible</mtext></mrow><mo>}</mo></mrow><mo>.</mo></math></span></span></span> We call a quadratic field <em>K</em> <span>maximally elastic</span> if the ring of integers of <em>K</em> is a UFD and each element of <span><math><mo>{</mo><mn>1</mn><mo>,</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo><mn>2</mn><mo>,</mo><mfrac><mrow><mn>5</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo><mn>3</mn><mo>,</mo><mo>…</mo><mo>}</mo><mo>∪</mo><mo>{</mo><mo>∞</mo><mo>}</mo></math></span> appears as an elasticity of infinitely many orders inside <em>K</em>. This corresponds to the orders in <em>K</em> exhibiting, to the extent possible for a quadratic field, maximal variation in terms of the failure of unique factorization. Assuming the Generalized Riemann Hypothesis, we prove that <span><math><mi>K</mi><mo>=</mo><mi>Q</mi><mo>(</mo><msqrt><mrow><mn>2</mn></mrow></msqrt><mo>)</mo></math></span> is universally elastic, and we provide evidence for a conjectured characterization of maximally elastic quadratic fields.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"267 ","pages":"Pages 80-100"},"PeriodicalIF":0.6,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142328259","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 dihedral Pólya fields","authors":"Charles Wend-Waoga Tougma","doi":"10.1016/j.jnt.2024.08.005","DOIUrl":"10.1016/j.jnt.2024.08.005","url":null,"abstract":"<div><div>A number field is a Pólya field when the module of integer-valued polynomials over its ring of integers has a regular basis. A quartic field is a <span><math><msub><mrow><mi>D</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-field when the Galois group of its splitting field is the dihedral group <span><math><msub><mrow><mi>D</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> of 8 elements. In this paper, we prove that there are infinitely many <span><math><msub><mrow><mi>D</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-Pólya fields with <em>ℓ</em> ramified prime numbers for each <span><math><mi>ℓ</mi><mo>∈</mo><mo>{</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>5</mn><mo>}</mo></math></span> and a <span><math><msub><mrow><mi>D</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> Pólya field with <span><math><mi>ℓ</mi><mo>=</mo><mn>1</mn></math></span> ramified prime number, is, up to <span><math><mi>Q</mi></math></span>-isomorphism, <span><math><mi>Q</mi><mrow><mo>(</mo><msqrt><mrow><mn>1</mn><mo>+</mo><msqrt><mrow><mn>2</mn></mrow></msqrt></mrow></msqrt><mo>)</mo></mrow></math></span>, <span><math><mi>Q</mi><mrow><mo>(</mo><msqrt><mrow><mo>−</mo><mn>1</mn><mo>+</mo><msqrt><mrow><mn>2</mn></mrow></msqrt></mrow></msqrt><mo>)</mo></mrow></math></span> or a pure field. Consequently, we answer a question raised in <span><span>[29]</span></span> on <span><math><msub><mrow><mi>D</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-fields. The same question arises on pure fields. We find an upper bound for such fields. And for any integer <em>ℓ</em> less that this bound, we show that there are infinitely many pure Pólya fields with <em>ℓ</em> ramified prime numbers except when <span><math><mi>ℓ</mi><mo>=</mo><mn>1</mn></math></span> where we proved that there are only 2 fields (and their two conjugate fields)</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"267 ","pages":"Pages 221-250"},"PeriodicalIF":0.6,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142425074","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 Selmer group and rank of a family of elliptic curves and curves of genus one violating the Hasse principle","authors":"Eleni Agathocleous","doi":"10.1016/j.jnt.2024.08.001","DOIUrl":"10.1016/j.jnt.2024.08.001","url":null,"abstract":"<div><div>We study an infinite family of <em>j</em>-invariant zero elliptic curves <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>D</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><mn>16</mn><mi>D</mi></math></span> and their <em>λ</em>-isogenous curves <span><math><msub><mrow><mi>E</mi></mrow><mrow><msup><mrow><mi>D</mi></mrow><mrow><mo>′</mo></mrow></msup></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><mn>27</mn><mo>⋅</mo><mn>16</mn><mi>D</mi></math></span>, where <em>D</em> and <span><math><msup><mrow><mi>D</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>=</mo><mo>−</mo><mn>3</mn><mi>D</mi></math></span> are fundamental discriminants of a specific form, and <em>λ</em> is an isogeny of degree 3. A result of Honda guarantees that for our discriminants <em>D</em>, the quadratic number field <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>D</mi></mrow></msub><mo>=</mo><mi>Q</mi><mo>(</mo><msqrt><mrow><mi>D</mi></mrow></msqrt><mo>)</mo></math></span> always has non-trivial 3-class group. We prove a series of results related to the set of rational points <span><math><msub><mrow><mi>E</mi></mrow><mrow><msup><mrow><mi>D</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></msub><mo>(</mo><mi>Q</mi><mo>)</mo><mo>∖</mo><mi>λ</mi><mo>(</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>D</mi></mrow></msub><mo>(</mo><mi>Q</mi><mo>)</mo><mo>)</mo></math></span>, and the <span><math><mi>S</mi><mi>L</mi><mo>(</mo><mn>2</mn><mo>,</mo><mi>Z</mi><mo>)</mo></math></span>-equivalence classes of irreducible integral binary cubic forms of discriminant <em>D</em>. By assuming finiteness of the Tate-Shafarevich group, we derive a parity result between the rank of <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>D</mi></mrow></msub></math></span> and the rank of its 3-Selmer group, and we establish lower and upper bounds for the rank of our elliptic curves. Finally, we give explicit classes of genus-1 curves that correspond to irreducible integral binary cubic forms of discriminant <span><math><mi>D</mi><mo>=</mo><mn>48035713</mn></math></span>, and we show that every curve in these classes violates the Hasse Principle.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"267 ","pages":"Pages 101-133"},"PeriodicalIF":0.6,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142328257","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":"The characteristic cycle of a non-confluent ℓ-adic GKZ hypergeometric sheaf","authors":"Peijiang Liu","doi":"10.1016/j.jnt.2024.07.014","DOIUrl":"10.1016/j.jnt.2024.07.014","url":null,"abstract":"<div><div>An <em>ℓ</em>-adic GKZ hypergeometric sheaf is defined analogously to a GKZ hypergeometric <span><math><mi>D</mi></math></span>-module. We introduce an algorithm of computing the characteristic cycle of an <em>ℓ</em>-adic GKZ hypergeometric sheaf of certain type. Our strategy is to apply a formula of the characteristic cycle of the direct image of an <em>ℓ</em>-adic sheaf. We verify the requirements for the formula to hold by calculating the dimension of the direct image of a certain closed conical subset of cotangent bundle. We also define an <em>ℓ</em>-adic GKZ-type sheaf whose specialization tensored with a constant sheaf is isomorphic to an <em>ℓ</em>-adic non-confluent GKZ hypergeometric sheaf. On the other hand, the topological model of an <em>ℓ</em>-adic GKZ-type sheaf is isomorphic to the image by the de Rham functor of a non-confluent GKZ hypergeometric <span><math><mi>D</mi></math></span>-module whose characteristic cycle has been calculated. This gives an easier way to determine the characteristic cycle of an <em>ℓ</em>-adic non-confluent GKZ hypergeometric sheaf of certain type.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"267 ","pages":"Pages 1-33"},"PeriodicalIF":0.6,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142328258","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":"On PGL2(F7) and PSL2(F7) number fields ramified at a single prime","authors":"Takeshi Ogasawara ,&nbsp;George J. Schaeffer","doi":"10.1016/j.jnt.2024.08.006","DOIUrl":"10.1016/j.jnt.2024.08.006","url":null,"abstract":"<div><div>We present new examples of <span><math><msub><mrow><mi>PGL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>7</mn></mrow></msub><mo>)</mo></math></span> and <span><math><msub><mrow><mi>PSL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>7</mn></mrow></msub><mo>)</mo></math></span> number fields ramified at a single prime. To find these number fields we employ the following methods: (i) Specializing a modification of Malle's <span><math><msub><mrow><mi>PGL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>7</mn></mrow></msub><mo>)</mo></math></span> polynomial, (ii) Modular method: computation of Katz modular forms of weight one over <span><math><msub><mrow><mover><mrow><mi>F</mi></mrow><mo>‾</mo></mover></mrow><mrow><mn>7</mn></mrow></msub></math></span> with prime level, and (iii) Searching for polynomials with prescribed ramification.</div><div>Method (i) quickly generates many <span><math><msub><mrow><mi>PGL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>7</mn></mrow></msub><mo>)</mo></math></span> number fields unramified at 7 including those fields ramified at only a single prime. Method (ii) can be used to show the existence of <span><math><msub><mrow><mi>PGL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>7</mn></mrow></msub><mo>)</mo></math></span> or <span><math><msub><mrow><mi>PSL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>7</mn></mrow></msub><mo>)</mo></math></span> number fields ramified only at primes that divide the level; we can then use method (iii) to find polynomials for those fields in many cases.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"267 ","pages":"Pages 202-220"},"PeriodicalIF":0.6,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142425201","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 new bound for A(A + A) for large sets","authors":"Aliaksei Semchankau","doi":"10.1016/j.jnt.2024.08.002","DOIUrl":"10.1016/j.jnt.2024.08.002","url":null,"abstract":"<div><div>For a large prime number <em>p</em> and a set <span><math><mi>A</mi><mo>⊂</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> we prove the following:<ul><li><span>(1)</span><span><div>If <span><math><mi>A</mi><mo>(</mo><mi>A</mi><mo>+</mo><mi>A</mi><mo>)</mo></math></span> does not cover all nonzero residues in <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>, then <span><math><mo>|</mo><mi>A</mi><mo>|</mo><mo>⩽</mo><mi>p</mi><mo>/</mo><mn>8</mn><mo>+</mo><mi>o</mi><mo>(</mo><mi>p</mi><mo>)</mo></math></span>.</div></span></li><li><span>(2)</span><span><div>If <em>A</em> is both sum-free and satisfies <span><math><mi>A</mi><mo>=</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>, then <span><math><mo>|</mo><mi>A</mi><mo>|</mo><mo>⩽</mo><mi>p</mi><mo>/</mo><mn>9</mn><mo>+</mo><mi>o</mi><mo>(</mo><mi>p</mi><mo>)</mo></math></span>.</div></span></li><li><span>(3)</span><span><div>If <span><math><mo>|</mo><mi>A</mi><mo>|</mo><mo>≫</mo><mfrac><mrow><mi>log</mi><mo>⁡</mo><mi>log</mi><mo>⁡</mo><mi>p</mi></mrow><mrow><msqrt><mrow><mi>log</mi><mo>⁡</mo><mi>p</mi></mrow></msqrt></mrow></mfrac><mi>p</mi></math></span>, then <span><math><mo>|</mo><mi>A</mi><mo>+</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>|</mo><mo>⩾</mo><mo>(</mo><mn>1</mn><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo><mi>min</mi><mo>⁡</mo><mo>(</mo><mn>2</mn><msqrt><mrow><mo>|</mo><mi>A</mi><mo>|</mo><mi>p</mi></mrow></msqrt><mo>,</mo><mi>p</mi><mo>)</mo></math></span>.</div></span></li></ul> Here the constants 1/8, 1/9, 2 are the best possible. Proofs make use of <em>wrappers</em>, subsets of a finite abelian group <em>G</em>, which ‘wrap’ popular values in convolutions of dense sets <span><math><mi>A</mi><mo>,</mo><mi>B</mi><mo>⊆</mo><mi>G</mi></math></span>. These objects carry certain structural features, making them capable of addressing additive-combinatorial and enumerative problems.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"268 ","pages":"Pages 142-162"},"PeriodicalIF":0.6,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142699245","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":"Recurrence formulae for spectral determinants","authors":"José Cunha ,&nbsp;Pedro Freitas","doi":"10.1016/j.jnt.2024.08.004","DOIUrl":"10.1016/j.jnt.2024.08.004","url":null,"abstract":"<div><div>We develop a unified method to study spectral determinants for several different manifolds, including spheres and hemispheres, and projective spaces. This is a direct consequence of an approach based on deriving recursion relations for the corresponding zeta functions, which we are then able to solve explicitly. Apart from new applications such as hemispheres, we also believe that the resulting formulae in the cases for which expressions for the determinant were already known are simpler and easier to compute in general, when compared to those resulting from other approaches.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"267 ","pages":"Pages 134-175"},"PeriodicalIF":0.6,"publicationDate":"2024-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142328242","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":"Integral points on moduli schemes","authors":"Rafael von Känel","doi":"10.1016/j.jnt.2024.07.005","DOIUrl":"https://doi.org/10.1016/j.jnt.2024.07.005","url":null,"abstract":"The strategy of combining the method of Faltings (Arakelov, Paršin, Szpiro) with modularity and Masser–Wüstholz isogeny estimates allows to explicitly bound the height and the number of the solutions of certain Diophantine equations related to integral points on moduli schemes of abelian varieties. In this paper we survey the development and various applications of this strategy.","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"43 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260799","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}