Pub Date : 2026-03-01Epub Date: 2025-09-05DOI: 10.1016/j.jnt.2025.08.007
Pavel Guerzhoy
Several authors have recently proved results which express a cusp form as a p-adic limit of weakly holomorphic modular forms under repeated application of Atkin's U-operator. Initially, these results had a deficiency: one could not rule out the possibility when a certain quantity vanishes and the final result fails to be true. Later on, Ahlgren and Samart [1] found a method to prove the non-vanishing in question in the specific case considered by El-Guindy and Ono [10]. Hanson and Jameson [15] and (independently) Dicks [8] generalized this method to finitely many other cases.
In this paper, we present a different approach which allows us to prove a similar non-vanishing result for an infinite family of similar cases. Our approach also allows us to return back to the original example considered by El-Guindy and Ono [10], where we calculate the (manifestly non-zero) quantity explicitly in terms of Morita's p-adic Γ-function.
{"title":"Non-vanishing of a certain quantity related to the p-adic coupling of mock modular forms with newforms","authors":"Pavel Guerzhoy","doi":"10.1016/j.jnt.2025.08.007","DOIUrl":"10.1016/j.jnt.2025.08.007","url":null,"abstract":"<div><div>Several authors have recently proved results which express a cusp form as a <em>p</em>-adic limit of weakly holomorphic modular forms under repeated application of Atkin's <em>U</em>-operator. Initially, these results had a deficiency: one could not rule out the possibility when a certain quantity vanishes and the final result fails to be true. Later on, Ahlgren and Samart <span><span>[1]</span></span> found a method to prove the non-vanishing in question in the specific case considered by El-Guindy and Ono <span><span>[10]</span></span>. Hanson and Jameson <span><span>[15]</span></span> and (independently) Dicks <span><span>[8]</span></span> generalized this method to finitely many other cases.</div><div>In this paper, we present a different approach which allows us to prove a similar non-vanishing result for an infinite family of similar cases. Our approach also allows us to return back to the original example considered by El-Guindy and Ono <span><span>[10]</span></span>, where we calculate the (manifestly non-zero) quantity explicitly in terms of Morita's <em>p</em>-adic Γ-function.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 191-211"},"PeriodicalIF":0.7,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145096119","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 : 2026-03-01Epub Date: 2025-09-23DOI: 10.1016/j.jnt.2025.08.011
Steven Charlton , Herbert Gangl , Danylo Radchenko
We explicitly reduce the Grassmannian polylogarithm in weight 4 and in weight 5 each to depth 2 iterated integrals. Furthermore, using this reduction in weight 4 we obtain an explicit, albeit complicated, form of the so-called 4-ratio, which gives an expression for the Borel class in continuous cohomology of in terms of .
{"title":"Explicit formulas for Grassmannian polylogarithms in weights 4 and 5","authors":"Steven Charlton , Herbert Gangl , Danylo Radchenko","doi":"10.1016/j.jnt.2025.08.011","DOIUrl":"10.1016/j.jnt.2025.08.011","url":null,"abstract":"<div><div>We explicitly reduce the Grassmannian polylogarithm in weight 4 and in weight 5 each to depth 2 iterated integrals. Furthermore, using this reduction in weight 4 we obtain an explicit, albeit complicated, form of the so-called 4-ratio, which gives an expression for the Borel class in continuous cohomology of <span><math><msub><mrow><mi>GL</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>(</mo><mi>C</mi><mo>)</mo></math></span> in terms of <span><math><msub><mrow><mi>Li</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 537-582"},"PeriodicalIF":0.7,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145220653","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 : 2026-03-01Epub Date: 2025-09-29DOI: 10.1016/j.jnt.2025.08.015
Víctor Hernández Barrios , Santiago Molina Blanco
In this note we define anticyclotomic p-adic measures attached to a modular elliptic curve E over a general number field F, a quadratic extension , and a set of places S of F above p. We study the exceptional zero phenomenon that arises when E has multiplicative reduction at some place in S. In this direction, we obtain p-adic Gross-Zagier formulas relating derivatives of the corresponding p-adic L-functions to the extended Mordell-Weil group of E. Our main result uses the recent construction of plectic points on elliptic curves due to Fornea and Gehrmann and generalizes their main result in [9]. We obtain a formula that computes the r-th derivative of the p-adic L-function, where r is the number of places in S where E has multiplicative reduction, in terms of plectic points and Tate periods of E.
{"title":"Exceptional zero formulas for anticyclotomic p-adic L-functions","authors":"Víctor Hernández Barrios , Santiago Molina Blanco","doi":"10.1016/j.jnt.2025.08.015","DOIUrl":"10.1016/j.jnt.2025.08.015","url":null,"abstract":"<div><div>In this note we define anticyclotomic <em>p</em>-adic measures attached to a modular elliptic curve <em>E</em> over a general number field <em>F</em>, a quadratic extension <span><math><mi>K</mi><mo>/</mo><mi>F</mi></math></span>, and a set of places <em>S</em> of <em>F</em> above <em>p</em>. We study the exceptional zero phenomenon that arises when <em>E</em> has multiplicative reduction at some place in <em>S</em>. In this direction, we obtain <em>p</em>-adic Gross-Zagier formulas relating derivatives of the corresponding <em>p</em>-adic L-functions to the extended Mordell-Weil group of <em>E</em>. Our main result uses the recent construction of plectic points on elliptic curves due to Fornea and Gehrmann and generalizes their main result in <span><span>[9]</span></span>. We obtain a formula that computes the <em>r</em>-th derivative of the <em>p</em>-adic L-function, where <em>r</em> is the number of places in <em>S</em> where <em>E</em> has multiplicative reduction, in terms of plectic points and Tate periods of <em>E</em>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 583-633"},"PeriodicalIF":0.7,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145267059","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 : 2026-03-01Epub Date: 2025-10-03DOI: 10.1016/j.jnt.2025.09.008
Lior Bary-Soroker , Daniele Garzoni , Sasha Sodin
Conditionally on the Riemann hypothesis for certain Dedekind zeta functions, we show that the characteristic polynomial of a class of random tridiagonal matrices of large dimension is irreducible, with probability exponentially close to one; moreover, its Galois group over the rational numbers is either the symmetric or the alternating group. This is the counterpart of the results of Breuillard–Varjú (for polynomials with independent coefficients), and with those of Eberhard and Ferber–Jain–Sah–Sawhney (for full random matrices). We also analyse a related class of random tridiagonal matrices for which the Galois group is much smaller.
{"title":"Irreducibility of the characteristic polynomials of random tridiagonal matrices","authors":"Lior Bary-Soroker , Daniele Garzoni , Sasha Sodin","doi":"10.1016/j.jnt.2025.09.008","DOIUrl":"10.1016/j.jnt.2025.09.008","url":null,"abstract":"<div><div>Conditionally on the Riemann hypothesis for certain Dedekind zeta functions, we show that the characteristic polynomial of a class of random tridiagonal matrices of large dimension is irreducible, with probability exponentially close to one; moreover, its Galois group over the rational numbers is either the symmetric or the alternating group. This is the counterpart of the results of Breuillard–Varjú (for polynomials with independent coefficients), and with those of Eberhard and Ferber–Jain–Sah–Sawhney (for full random matrices). We also analyse a related class of random tridiagonal matrices for which the Galois group is much smaller.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 973-998"},"PeriodicalIF":0.7,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145324770","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 : 2026-03-01Epub Date: 2025-09-23DOI: 10.1016/j.jnt.2025.08.016
Wenxia Qu , Wenston J.T. Zang
Let denote the total number of i hooks in the t-regular partitions of n. Singh and Barman (2024) [14] raised two conjectures on . The first conjecture is on the positivity of for . The second conjecture states that when , for all n except for . In this paper, we confirm the first conjecture. Moreover, we show that for any odd , the second conjecture fails for infinitely many n. Furthermore, we verify that the second conjecture holds for and 6. We also propose a conjecture on the even case k, which is a modification of Singh and Barman's second conjecture.
{"title":"On the hook length biases of the 2- and 3-regular partitions","authors":"Wenxia Qu , Wenston J.T. Zang","doi":"10.1016/j.jnt.2025.08.016","DOIUrl":"10.1016/j.jnt.2025.08.016","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>b</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>i</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> denote the total number of <em>i</em> hooks in the <em>t</em>-regular partitions of <em>n</em>. Singh and Barman (2024) <span><span>[14]</span></span> raised two conjectures on <span><math><msub><mrow><mi>b</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>i</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span>. The first conjecture is on the positivity of <span><math><msub><mrow><mi>b</mi></mrow><mrow><mn>3</mn><mo>,</mo><mn>2</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>−</mo><msub><mrow><mi>b</mi></mrow><mrow><mn>3</mn><mo>,</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> for <span><math><mi>n</mi><mo>≥</mo><mn>28</mn></math></span>. The second conjecture states that when <span><math><mi>k</mi><mo>≥</mo><mn>3</mn></math></span>, <span><math><msub><mrow><mi>b</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>≥</mo><msub><mrow><mi>b</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> for all <em>n</em> except for <span><math><mi>n</mi><mo>=</mo><mi>k</mi><mo>+</mo><mn>1</mn></math></span>. In this paper, we confirm the first conjecture. Moreover, we show that for any odd <span><math><mi>k</mi><mo>≥</mo><mn>3</mn></math></span>, the second conjecture fails for infinitely many <em>n</em>. Furthermore, we verify that the second conjecture holds for <span><math><mi>k</mi><mo>=</mo><mn>4</mn></math></span> and 6. We also propose a conjecture on the even case <em>k</em>, which is a modification of Singh and Barman's second conjecture.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 455-480"},"PeriodicalIF":0.7,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145220655","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 : 2026-03-01Epub Date: 2025-10-02DOI: 10.1016/j.jnt.2025.09.009
Dan Fretwell , Jenny Roberts
Let F be an arbitrary totally real field. Under standard conditions we prove the existence of certain Eisenstein congruences between parallel weight Hilbert eigenforms of level and Hilbert Eisenstein series of level , for arbitrary ideal and prime ideal of . Such congruences have their moduli coming from special values of Hecke L-functions and their Euler factors, and our results allow for the eigenforms to have non-trivial Hecke character. After this, we consider the question of when such congruences can be satisfied by newforms, proving general results about this.
{"title":"Hilbert modular Eisenstein congruences of local origin","authors":"Dan Fretwell , Jenny Roberts","doi":"10.1016/j.jnt.2025.09.009","DOIUrl":"10.1016/j.jnt.2025.09.009","url":null,"abstract":"<div><div>Let <em>F</em> be an arbitrary totally real field. Under standard conditions we prove the existence of certain Eisenstein congruences between parallel weight <span><math><mi>k</mi><mo>≥</mo><mn>3</mn></math></span> Hilbert eigenforms of level <span><math><mi>mp</mi></math></span> and Hilbert Eisenstein series of level <span><math><mi>m</mi></math></span>, for arbitrary ideal <span><math><mi>m</mi></math></span> and prime ideal <span><math><mi>p</mi><mo>∤</mo><mi>m</mi></math></span> of <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>F</mi></mrow></msub></math></span>. Such congruences have their moduli coming from special values of Hecke <em>L</em>-functions and their Euler factors, and our results allow for the eigenforms to have non-trivial Hecke character. After this, we consider the question of when such congruences can be satisfied by newforms, proving general results about this.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 861-896"},"PeriodicalIF":0.7,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145267053","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 : 2026-03-01Epub Date: 2025-09-08DOI: 10.1016/j.jnt.2025.08.010
Özlem Ejder
Given a number field k, and a quadratic rational function , the associated arboreal representation of the absolute Galois group of k is a subgroup of the automorphism group of a regular rooted binary tree. Boston and Jones conjectured that the image of such a representation for contains a dense set of settled elements. An automorphism is settled if the number of its orbits on the nth level of the tree remains small as n goes to infinity.
In this article, we exhibit many quadratic rational functions whose associated Arboreal Galois groups are not densely settled. These examples arise from quadratic rational functions whose critical points lie in a single periodic orbit. To prove our results, we present a detailed study of the iterated monodromy groups (IMG) of f, which also allows us to provide a negative answer to Jones and Levy's question regarding settled pairs.
Furthermore, we study the iterated extension generated by adjoining to all roots of for for a parameter t. We call the intersection of with , the field of constants associated with f. When one of the two critical points of f is the image of the other, we show that the field of constants is contained in the cyclotomic extension of k generated by all 2-power roots of unity. In particular, we prove the conjecture of Ejder, Kara, and Ozman regarding the rational function .
{"title":"Galois theory of quadratic rational functions with periodic critical points","authors":"Özlem Ejder","doi":"10.1016/j.jnt.2025.08.010","DOIUrl":"10.1016/j.jnt.2025.08.010","url":null,"abstract":"<div><div>Given a number field <em>k</em>, and a quadratic rational function <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>∈</mo><mi>k</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span>, the associated arboreal representation of the absolute Galois group of <em>k</em> is a subgroup of the automorphism group of a regular rooted binary tree. Boston and Jones conjectured that the image of such a representation for <span><math><mi>f</mi><mo>∈</mo><mi>Z</mi><mo>[</mo><mi>x</mi><mo>]</mo></math></span> contains a dense set of settled elements. An automorphism is settled if the number of its orbits on the <em>n</em>th level of the tree remains small as <em>n</em> goes to infinity.</div><div>In this article, we exhibit many quadratic rational functions whose associated Arboreal Galois groups are not densely settled. These examples arise from quadratic rational functions whose critical points lie in a single periodic orbit. To prove our results, we present a detailed study of the iterated monodromy groups (IMG) of <em>f</em>, which also allows us to provide a negative answer to Jones and Levy's question regarding settled pairs.</div><div>Furthermore, we study the iterated extension <span><math><mi>k</mi><mo>(</mo><msup><mrow><mi>f</mi></mrow><mrow><mo>−</mo><mo>∞</mo></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>)</mo></math></span> generated by adjoining to <span><math><mi>k</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span> all roots of <span><math><msup><mrow><mi>f</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>t</mi></math></span> for <span><math><mi>n</mi><mo>≥</mo><mn>1</mn></math></span> for a parameter <em>t</em>. We call the intersection of <span><math><mi>k</mi><mo>(</mo><msup><mrow><mi>f</mi></mrow><mrow><mo>−</mo><mo>∞</mo></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>)</mo></math></span> with <span><math><mover><mrow><mi>k</mi></mrow><mrow><mo>¯</mo></mrow></mover></math></span>, the field of constants associated with <em>f</em>. When one of the two critical points of <em>f</em> is the image of the other, we show that the field of constants is contained in the cyclotomic extension of <em>k</em> generated by all 2-power roots of unity. In particular, we prove the conjecture of Ejder, Kara, and Ozman regarding the rational function <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><msup><mrow><mo>(</mo><mi>x</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></math></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 212-245"},"PeriodicalIF":0.7,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145096120","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 : 2026-03-01Epub Date: 2025-09-23DOI: 10.1016/j.jnt.2025.08.014
Stijn Cambie
Letting be the density of the set of integers with exactly one divisor in , Erdős wondered if is unimodular for fixed n. We prove this is false in general, as the sequence has superpolynomially many local extrema. However, we confirm unimodality in the single case for which it occurs; . We also solve the question on unimodality of the density of integers whose prime is p.
{"title":"Resolution of Erdős' problems about unimodularity","authors":"Stijn Cambie","doi":"10.1016/j.jnt.2025.08.014","DOIUrl":"10.1016/j.jnt.2025.08.014","url":null,"abstract":"<div><div>Letting <span><math><msub><mrow><mi>δ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>m</mi><mo>)</mo></math></span> be the density of the set of integers with exactly one divisor in <span><math><mo>(</mo><mi>n</mi><mo>,</mo><mi>m</mi><mo>)</mo></math></span>, Erdős wondered if <span><math><msub><mrow><mi>δ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>m</mi><mo>)</mo></math></span> is unimodular for fixed <em>n</em>. We prove this is false in general, as the sequence <span><math><mo>(</mo><msub><mrow><mi>δ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>m</mi><mo>)</mo><mo>)</mo></math></span> has superpolynomially many local extrema. However, we confirm unimodality in the single case for which it occurs; <span><math><mi>n</mi><mo>=</mo><mn>1</mn></math></span>. We also solve the question on unimodality of the density of integers whose <span><math><msup><mrow><mi>k</mi></mrow><mrow><mi>t</mi><mi>h</mi></mrow></msup></math></span> prime is <em>p</em>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 271-277"},"PeriodicalIF":0.7,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145158355","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 : 2026-03-01Epub Date: 2025-10-01DOI: 10.1016/j.jnt.2025.09.003
Sara Chari , Sergio Ricardo Zapata Ceballos , Erik Holmes , Fatemeh Jalalvand , Rahinatou Yuh Njah Nchiwo , Kelly O'Connor , Fabian Ramirez , Sameera Vemulapalli
There has been a recent surge of interest on distributions of shapes of unit lattices in number fields, due to both their applications to number theory and the lack of known results.
In this work we focus on -quartic fields with signature ; such fields have a rank 2 unit group. Viewing the unit lattice as a point of , we prove that every lattice which arises this way must correspond to a transcendental point on the boundary of a certain fundamental domain of . Moreover, we produce three explicit (algebraic) points of which are limit points of the set of (points associated to) unit lattices of -quartic fields with signature .
{"title":"Unit lattices of D4-quartic number fields with signature (2,1)","authors":"Sara Chari , Sergio Ricardo Zapata Ceballos , Erik Holmes , Fatemeh Jalalvand , Rahinatou Yuh Njah Nchiwo , Kelly O'Connor , Fabian Ramirez , Sameera Vemulapalli","doi":"10.1016/j.jnt.2025.09.003","DOIUrl":"10.1016/j.jnt.2025.09.003","url":null,"abstract":"<div><div>There has been a recent surge of interest on distributions of shapes of unit lattices in number fields, due to both their applications to number theory and the lack of known results.</div><div>In this work we focus on <span><math><msub><mrow><mi>D</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-quartic fields with signature <span><math><mo>(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>; such fields have a rank 2 unit group. Viewing the unit lattice as a point of <span><math><msub><mrow><mi>GL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>Z</mi><mo>)</mo><mo>﹨</mo><mi>h</mi></math></span>, we prove that every lattice which arises this way must correspond to a transcendental point on the boundary of a certain fundamental domain of <span><math><msub><mrow><mi>GL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>Z</mi><mo>)</mo><mo>﹨</mo><mi>h</mi></math></span>. Moreover, we produce three explicit (algebraic) points of <span><math><msub><mrow><mi>GL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>Z</mi><mo>)</mo><mo>﹨</mo><mi>h</mi></math></span> which are limit points of the set of (points associated to) unit lattices of <span><math><msub><mrow><mi>D</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-quartic fields with signature <span><math><mo>(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 761-784"},"PeriodicalIF":0.7,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145267055","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 : 2026-03-01Epub Date: 2025-09-25DOI: 10.1016/j.jnt.2025.08.021
Alexandre Dieguez
For a fixed irrational with a prescribed irrationality measure function, we study the correlation , where Δ is the Dirichlet error term in the divisor problem. When θ has a finite irrationality measure, it is known that decorrelation occurs at a rate expressible in terms of this measure. Strong decorrelation occurs for all positive irrationals, except possibly Liouville numbers. We show that for irrationals with a prescribed irrationality measure function ψ, decorrelation can be quantified in terms of .
{"title":"On certain correlations into the divisor problem","authors":"Alexandre Dieguez","doi":"10.1016/j.jnt.2025.08.021","DOIUrl":"10.1016/j.jnt.2025.08.021","url":null,"abstract":"<div><div>For a fixed irrational <span><math><mi>θ</mi><mo>></mo><mn>0</mn></math></span> with a prescribed irrationality measure function, we study the correlation <span><math><msubsup><mrow><mo>∫</mo></mrow><mrow><mn>1</mn></mrow><mrow><mi>X</mi></mrow></msubsup><mi>Δ</mi><mo>(</mo><mi>x</mi><mo>)</mo><mi>Δ</mi><mo>(</mo><mi>θ</mi><mi>x</mi><mo>)</mo><mi>d</mi><mi>x</mi></math></span>, where Δ is the Dirichlet error term in the divisor problem. When <em>θ</em> has a finite irrationality measure, it is known that decorrelation occurs at a rate expressible in terms of this measure. Strong decorrelation occurs for all positive irrationals, except possibly Liouville numbers. We show that for irrationals with a prescribed irrationality measure function <em>ψ</em>, decorrelation can be quantified in terms of <span><math><msup><mrow><mi>ψ</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span>.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"280 ","pages":"Pages 519-536"},"PeriodicalIF":0.7,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145220654","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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