Pub Date : 2025-10-31DOI: 10.1016/j.jnt.2025.10.006
Ernest X.W. Xia
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.
andrew - merca和Guo-Zeng在研究经典函数的截断和时，提出了Jacobi三重积恒等式的一种特殊情况的截断和的猜想，该猜想已被Mao和Yee独立地证实。2016年，Chan， Ho和Mao研究了由五重乘积恒等式的两个结果引起的截断系列。本文采用不同的截断级数，利用Bailey对，在比Andrews-Merca和Guo-Zeng猜想更强的Jacobi三重积单位的截断和上，建立了一个非负系数的显式级数形式。作为我们的结果的一个推论，我们得到了Jacobi恒等式的一个新的截断和，它蕴涵了Guo和Zeng给出的另一个猜想。此外，我们确定了由五元积恒等式导出的两个著名恒等式的系数符号，这两个恒等式可以被认为是Chan， Ho和Mao证明的定理的伴随结果。
{"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":"2025-10-31","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}
Pub Date : 2025-10-30DOI: 10.1016/j.jnt.2025.10.002
Sean Eberhard
We show that the sequence of ratios of consecutive values of the divisor function attains every positive rational infinitely many times. This confirms a prediction of Erdős.
{"title":"Ratios of consecutive values of the divisor function","authors":"Sean Eberhard","doi":"10.1016/j.jnt.2025.10.002","DOIUrl":"10.1016/j.jnt.2025.10.002","url":null,"abstract":"<div><div>We show that the sequence of ratios <span><math><mi>d</mi><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>/</mo><mi>d</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> of consecutive values of the divisor function attains every positive rational infinitely many times. This confirms a prediction of Erdős.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"281 ","pages":"Pages 426-428"},"PeriodicalIF":0.7,"publicationDate":"2025-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145467143","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}
Pub Date : 2025-10-27DOI: 10.1016/j.jnt.2025.09.024
Kathrin Bringmann , Nikolaos Diamantis
We introduce an extension of the standard cohomology which is characterised by maps that fail to be classical cocycles by products of simpler maps. The construction is motivated by the study of Manin's noncommutative modular symbols and of false theta functions. We use this construction to obtain a cohomological interpretation of important iterated integrals that arise in that study. In another direction, our approach gives modular counterparts to the long-studied relations among multiple zeta values.
{"title":"Iterated integrals and cohomology","authors":"Kathrin Bringmann , Nikolaos Diamantis","doi":"10.1016/j.jnt.2025.09.024","DOIUrl":"10.1016/j.jnt.2025.09.024","url":null,"abstract":"<div><div>We introduce an extension of the standard cohomology which is characterised by maps that fail to be classical cocycles by products of simpler maps. The construction is motivated by the study of Manin's noncommutative modular symbols and of false theta functions. We use this construction to obtain a cohomological interpretation of important iterated integrals that arise in that study. In another direction, our approach gives modular counterparts to the long-studied relations among multiple zeta values.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"281 ","pages":"Pages 397-425"},"PeriodicalIF":0.7,"publicationDate":"2025-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145418340","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}
Pub Date : 2025-10-24DOI: 10.1016/j.jnt.2025.09.027
Sen Xu , Tianping Zhang
Text
We prove a new bound on bilinear forms with generalized Kloosterman sums by using the sum-product phenomenon in , which reached the barrier for the more general situation and complements those obtained by Kowalski, Michel, and Sawin (2020). We also establish new estimates for bilinear forms with two variables from arbitrary subsets of , which has expanded the range of obtained by Xi (2023).
Video
For a video summary of this paper, please visit https://youtu.be/Q472zpufLEs.
本文利用Fp中的和积现象证明了广义Kloosterman和双线性形式的一个新界，该界在更一般的情况下达到了MN>；p34，并补充了Kowalski， Michel, and Sawin（2020）的结果。我们还从Fp的任意子集中建立了具有两个变量的双线性形式的新估计，这扩展了Xi（2023）得到的M，N的范围。观看本文的视频摘要，请访问https://youtu.be/Q472zpufLEs。
{"title":"Bounds on bilinear sums of generalized Kloosterman sums over arbitrary sets","authors":"Sen Xu , Tianping Zhang","doi":"10.1016/j.jnt.2025.09.027","DOIUrl":"10.1016/j.jnt.2025.09.027","url":null,"abstract":"<div><h3>Text</h3><div>We prove a new bound on bilinear forms with generalized Kloosterman sums by using the sum-product phenomenon in <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>, which reached the barrier <span><math><mi>M</mi><mi>N</mi><mo>></mo><msup><mrow><mi>p</mi></mrow><mrow><mfrac><mrow><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></msup></math></span> for the more general situation and complements those obtained by Kowalski, Michel, and Sawin (2020). We also establish new estimates for bilinear forms with two variables from arbitrary subsets of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>, which has expanded the range of <span><math><mi>M</mi><mo>,</mo><mi>N</mi></math></span> obtained by Xi (2023).</div></div><div><h3>Video</h3><div>For a video summary of this paper, please visit <span><span>https://youtu.be/Q472zpufLEs</span><svg><path></path></svg></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"281 ","pages":"Pages 373-396"},"PeriodicalIF":0.7,"publicationDate":"2025-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145418338","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}
Pub Date : 2025-10-22DOI: 10.1016/j.jnt.2025.09.014
Fei Hou , GuangShi Lü
Let be two primes such that . Let f be a Hecke newform of level P, and χ a primitive Dirichlet character modulo M. In this paper, we study the hybrid subconvexity problem for simultaneously in the level and conductor aspects. Among other things, we prove that the hybrid subconvex bound can be achieved, so long as . One of the key ingredients is that we develop the classical level aspect version of the Voronoĭ formula for the symmetric square lift in an alternative way by tracing back to its geometric nature. As the direct applications, we obtained the subconvex bound for simultaneously in the level and conductor aspects and the non-obvious bound for the problem of distinguishing modular forms f and based on their first Fourier coefficients.
{"title":"Hybrid subconvex bounds for GL3 × GL1 twisted L-functions and their applications","authors":"Fei Hou , GuangShi Lü","doi":"10.1016/j.jnt.2025.09.014","DOIUrl":"10.1016/j.jnt.2025.09.014","url":null,"abstract":"<div><div>Let <span><math><mi>P</mi><mo>,</mo><mi>M</mi></math></span> be two primes such that <span><math><mo>(</mo><mi>P</mi><mo>,</mo><mi>M</mi><mo>)</mo><mo>=</mo><mn>1</mn></math></span>. Let <em>f</em> be a Hecke newform of level <em>P</em>, and <em>χ</em> a primitive Dirichlet character modulo <em>M</em>. In this paper, we study the hybrid subconvexity problem for <span><math><mi>L</mi><mo>(</mo><mi>s</mi><mo>,</mo><msup><mrow><mtext>sym</mtext></mrow><mrow><mn>2</mn></mrow></msup><mi>f</mi><mo>⊗</mo><mi>χ</mi><mo>)</mo></math></span> simultaneously in the level and conductor aspects. Among other things, we prove that the hybrid subconvex bound can be achieved, so long as <span><math><msup><mrow><mi>M</mi></mrow><mrow><mi>ε</mi></mrow></msup><mo><</mo><mi>P</mi><mo><</mo><msup><mrow><mi>M</mi></mrow><mrow><mn>3</mn><mo>/</mo><mn>20</mn></mrow></msup></math></span>. One of the key ingredients is that we develop the classical level aspect version of the Voronoĭ formula for the symmetric square lift in an alternative way by tracing back to its geometric nature. As the direct applications, we obtained the subconvex bound for <span><math><mi>L</mi><mo>(</mo><mi>s</mi><mo>,</mo><mi>f</mi><mo>⊗</mo><mi>f</mi><mo>⊗</mo><mi>χ</mi><mo>)</mo></math></span> simultaneously in the level and conductor aspects and the non-obvious bound for the problem of distinguishing modular forms <em>f</em> and <span><math><msup><mrow><mi>f</mi></mrow><mrow><mi>χ</mi></mrow></msup></math></span> based on their first Fourier coefficients.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"281 ","pages":"Pages 283-320"},"PeriodicalIF":0.7,"publicationDate":"2025-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145418334","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}
Pub Date : 2025-10-22DOI: 10.1016/j.jnt.2025.09.020
Sungyoon Cho
In this paper, we prove that there are certain relations among representation densities and provide an efficient way to compute representation densities by using these relations. As an application, we compute certain arithmetic intersection numbers of special cycles on unitary Rapoport-Zink spaces and propose a conjecture on these.
{"title":"On local representation densities of Hermitian forms and special cycles II","authors":"Sungyoon Cho","doi":"10.1016/j.jnt.2025.09.020","DOIUrl":"10.1016/j.jnt.2025.09.020","url":null,"abstract":"<div><div>In this paper, we prove that there are certain relations among representation densities and provide an efficient way to compute representation densities by using these relations. As an application, we compute certain arithmetic intersection numbers of special cycles on unitary Rapoport-Zink spaces and propose a conjecture on these.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"281 ","pages":"Pages 169-204"},"PeriodicalIF":0.7,"publicationDate":"2025-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145418998","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}
Pub Date : 2025-10-22DOI: 10.1016/j.jnt.2025.09.022
Brody Lynch
Let ℓ be prime, and K be a number field containing the ℓ-th roots of unity. We use classical algebraic number theory and some analytic techniques to prove that the Steinitz classes of extensions of K ordered by relative discriminant are equidistributed among realizable classes in the ideal class group of K. For , this was proved by Kable and Wright using the deep theory of prehomogeneous vector spaces. Foster proved that Steinitz classes are uniformly distributed between realizable classes for tamely ramified elementary-m extensions using the theory of Galois modules; our approach eliminates this tameness hypothesis.
{"title":"Equidistribution of realizable Steinitz classes for cyclic Kummer extensions","authors":"Brody Lynch","doi":"10.1016/j.jnt.2025.09.022","DOIUrl":"10.1016/j.jnt.2025.09.022","url":null,"abstract":"<div><div>Let <em>ℓ</em> be prime, and <em>K</em> be a number field containing the <em>ℓ</em>-th roots of unity. We use classical algebraic number theory and some analytic techniques to prove that the Steinitz classes of <span><math><mi>Z</mi><mo>/</mo><mi>ℓ</mi><mi>Z</mi></math></span> extensions of <em>K</em> ordered by relative discriminant are equidistributed among realizable classes in the ideal class group of <em>K</em>. For <span><math><mi>ℓ</mi><mo>=</mo><mn>2</mn></math></span>, this was proved by Kable and Wright using the deep theory of prehomogeneous vector spaces. Foster proved that Steinitz classes are uniformly distributed between realizable classes for tamely ramified elementary-<em>m</em> extensions using the theory of Galois modules; our approach eliminates this tameness hypothesis.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"281 ","pages":"Pages 139-168"},"PeriodicalIF":0.7,"publicationDate":"2025-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145418337","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}
Pub Date : 2025-10-22DOI: 10.1016/j.jnt.2025.09.023
Andreea Iorga
In this paper, we prove that, under a technical assumption, any semi-direct product of a p-group G with a group Φ of order prime to p can appear as the Galois group of a tower of extensions with the property that M is the maximal unramified p-extension of L, and . A consequence of this result is that any local ring admitting a surjection to , or with finite kernel can be realized as a universal everywhere unramified deformation ring.
{"title":"Constructing unramified extensions and Murphy's law for Galois deformation rings","authors":"Andreea Iorga","doi":"10.1016/j.jnt.2025.09.023","DOIUrl":"10.1016/j.jnt.2025.09.023","url":null,"abstract":"<div><div>In this paper, we prove that, under a technical assumption, any semi-direct product of a <em>p</em>-group <em>G</em> with a group Φ of order prime to <em>p</em> can appear as the Galois group of a tower of extensions <span><math><mi>M</mi><mo>/</mo><mi>L</mi><mo>/</mo><mi>K</mi></math></span> with the property that <em>M</em> is the maximal unramified <em>p</em>-extension of <em>L</em>, and <span><math><mi>Gal</mi><mo>(</mo><mi>M</mi><mo>/</mo><mi>L</mi><mo>)</mo><mo>≅</mo><mi>G</mi></math></span>. A consequence of this result is that any local ring admitting a surjection to <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>, <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span> or <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mn>7</mn></mrow></msub></math></span> with finite kernel can be realized as a universal everywhere unramified deformation ring.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"281 ","pages":"Pages 59-95"},"PeriodicalIF":0.7,"publicationDate":"2025-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145365625","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}
Pub Date : 2025-10-22DOI: 10.1016/j.jnt.2025.09.015
Mohamed Mahmoud Chems-Eddin
The main aim of this paper is to investigate Greenberg's conjecture for real biquadratic fields. More precisely, we propose the following problem:
What are real biquadratic number fields k such that ? where is the 2-Iwasawa module of k and is the 2-class group of the first layer of the cyclotomic -extension of k. Moreover, we give several families of real biquadratic fields k such that is trivial or isomorphic to or , where n is a given positive integer. The reader can also find some results concerning the 2-rank of the class group of certain real triquadratic fields.
{"title":"Greenberg's conjecture and Iwasawa module of real biquadratic fields I","authors":"Mohamed Mahmoud Chems-Eddin","doi":"10.1016/j.jnt.2025.09.015","DOIUrl":"10.1016/j.jnt.2025.09.015","url":null,"abstract":"<div><div>The main aim of this paper is to investigate Greenberg's conjecture for real biquadratic fields. More precisely, we propose the following problem:</div><div>What are real biquadratic number fields <em>k</em> such that <span><math><mrow><mi>rank</mi></mrow><mo>(</mo><mi>A</mi><mo>(</mo><msub><mrow><mi>k</mi></mrow><mrow><mo>∞</mo></mrow></msub><mo>)</mo><mo>)</mo><mo>=</mo><mrow><mi>rank</mi></mrow><mo>(</mo><mi>A</mi><mo>(</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>)</mo></math></span>? where <span><math><mi>A</mi><mo>(</mo><msub><mrow><mi>k</mi></mrow><mrow><mo>∞</mo></mrow></msub><mo>)</mo></math></span> is the 2-Iwasawa module of <em>k</em> and <span><math><mi>A</mi><mo>(</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></math></span> is the 2-class group of <span><math><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> the first layer of the cyclotomic <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-extension of <em>k</em>. Moreover, we give several families of real biquadratic fields <em>k</em> such that <span><math><mi>A</mi><mo>(</mo><msub><mrow><mi>k</mi></mrow><mrow><mo>∞</mo></mrow></msub><mo>)</mo></math></span> is trivial or isomorphic to <span><math><mi>Z</mi><mo>/</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup><mi>Z</mi></math></span> or <span><math><mi>Z</mi><mo>/</mo><mn>2</mn><mi>Z</mi><mo>×</mo><mi>Z</mi><mo>/</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup><mi>Z</mi></math></span>, where <em>n</em> is a given positive integer. The reader can also find some results concerning the 2-rank of the class group of certain real triquadratic fields.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"281 ","pages":"Pages 224-266"},"PeriodicalIF":0.7,"publicationDate":"2025-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145418339","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}
Pub Date : 2025-10-21DOI: 10.1016/j.jnt.2025.09.016
Prashant Tiwari , Lalit Vaishya
We prove explicit lower bounds for the natural density of the sets of primes p represented by a reduced form of negative discriminant D such that Siegel eigenvalues of a Cuspidal Siegel eigenforms F of degree 2 satisfy for the real numbers and . A similar result is also proved for the set of primes p represented by a reduced form of negative discriminant D such that . Analogous results are also valid if one replaces natural density by Dirichlet density. Moreover, we deal with various kinds of quantitative results concerning the comparison between the normalized Siegel eigenvalues over the primes p represented by a reduced form of negative discriminant D, of two distinct cuspidal Siegel eigenforms for the full symplectic group of degree 2 which are not Saito–Kurokawa lifts.
{"title":"Natural density of the sets associated to Siegel eigenvalues of a Siegel cusp form of degree 2","authors":"Prashant Tiwari , Lalit Vaishya","doi":"10.1016/j.jnt.2025.09.016","DOIUrl":"10.1016/j.jnt.2025.09.016","url":null,"abstract":"<div><div>We prove explicit lower bounds for the natural density of the sets of primes <em>p</em> represented by a reduced form of negative discriminant <em>D</em> such that Siegel eigenvalues <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mi>F</mi></mrow></msub><mo>(</mo><mi>p</mi><mo>)</mo></math></span> of a Cuspidal Siegel eigenforms <em>F</em> of degree 2 satisfy <span><math><msub><mrow><mi>c</mi></mrow><mrow><mn>1</mn></mrow></msub><mo><</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>F</mi></mrow></msub><mo>(</mo><mi>p</mi><mo>)</mo><mo><</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> for the real numbers <span><math><msub><mrow><mi>c</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>. A similar result is also proved for the set of primes <em>p</em> represented by a reduced form of negative discriminant <em>D</em> such that <span><math><mo>|</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>F</mi></mrow></msub><mo>(</mo><mi>p</mi><mo>)</mo><mo>|</mo><mo>></mo><mi>c</mi></math></span>. Analogous results are also valid if one replaces natural density by Dirichlet density. Moreover, we deal with various kinds of quantitative results concerning the comparison between the normalized Siegel eigenvalues over the primes <em>p</em> represented by a reduced form of negative discriminant <em>D</em>, of two distinct cuspidal Siegel eigenforms for the full symplectic group of degree 2 which are not Saito–Kurokawa lifts.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"281 ","pages":"Pages 344-372"},"PeriodicalIF":0.7,"publicationDate":"2025-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145418335","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}
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