Research in Number Theory最新文献

Applications of representation theory and of explicit units to Leopoldt's conjecture. 表征理论和显式单位在利奥波德猜想中的应用。

IF 0.8

Research in Number Theory Pub Date : 2026-01-01 Epub Date: 2026-03-17 DOI: 10.1007/s40993-026-00717-2

Fabio Ferri, Henri Johnston

{"title":"Applications of representation theory and of explicit units to Leopoldt's conjecture.","authors":"Fabio Ferri, Henri Johnston","doi":"10.1007/s40993-026-00717-2","DOIUrl":"https://doi.org/10.1007/s40993-026-00717-2","url":null,"abstract":"Let L/K be a Galois extension of number fields and let <math><mrow><mi>G</mi> <mo>=</mo> <mtext>Gal</mtext> <mo>(</mo> <mi>L</mi> <mo>/</mo> <mi>K</mi> <mo>)</mo></mrow> </math> . We show that under certain hypotheses on G, for a fixed prime number p, Leopoldt's conjecture at p for certain proper intermediate fields of L/K implies Leopoldt's conjecture at p for L. We also obtain relations between the Leopoldt defects of intermediate fields of L/K. By applying a result of Buchmann and Sands together with an explicit description of units and a special case of the above results, we show that given any finite set of prime numbers <math><mi>P</mi></math> , there exists an infinite family <math><mi>F</mi></math> of totally real <math><msub><mi>S</mi> <mn>3</mn></msub> </math> -extensions of <math><mi>Q</mi></math> such that Leopoldt's conjecture for F at p holds for every <math><mrow><mi>F</mi> <mo>∈</mo> <mi>F</mi></mrow> </math> and <math><mrow><mi>p</mi> <mo>∈</mo> <mi>P</mi></mrow> </math> .","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"12 2","pages":"32"},"PeriodicalIF":0.8,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12995976/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147487823","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Wild Galois representations: elliptic curves with wild cyclic reduction. 野伽罗瓦表示：带野循环约简的椭圆曲线。

IF 0.8

Research in Number Theory Pub Date : 2026-01-01 Epub Date: 2026-05-04 DOI: 10.1007/s40993-026-00730-5

Nirvana Coppola

引用次数: 0

Adjoint L-functions, congruence ideals, and Selmer groups over GL n. 伴随l函数，同余理想，和GL n上的Selmer群。

IF 0.8

Research in Number Theory Pub Date : 2026-01-01 Epub Date: 2026-02-27 DOI: 10.1007/s40993-026-00721-6

Ho Leung Fong

{"title":"<ArticleTitle xmlns:ns0=\"http://www.w3.org/1998/Math/MathML\">Adjoint L-functions, congruence ideals, and Selmer groups over <ns0:math><ns0:msub><ns0:mtext>GL</ns0:mtext> <ns0:mi>n</ns0:mi></ns0:msub></ns0:math>.","authors":"Ho Leung Fong","doi":"10.1007/s40993-026-00721-6","DOIUrl":"https://doi.org/10.1007/s40993-026-00721-6","url":null,"abstract":"The study of special values of adjoint L-functions and congruence ideals is gradually becoming a classical theme in number theory, driven by the Bloch-Kato conjecture and generalisations of Wiles-Lenstra's numerical criterion. In this paper, we relate <math><mrow><mi>L</mi> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mi>π</mi> <mo>,</mo> <msup> <mrow><mrow><mspace></mspace> <mtext>Ad</mtext> <mspace></mspace></mrow> </mrow> <mo>∘</mo></msup> <mo>)</mo></mrow> </math> to the congruence ideals for cohomological cuspidal automorphic representations <math><mi>π</mi></math> of <math><msub><mtext>GL</mtext> <mi>n</mi></msub> </math> over any number field. We then use this result to deduce relationships between the congruences of automorphic forms and adjoint L-functions. For CM fields, using the existence of Galois representations, we apply the result to obtain a lower bound on the cardinality of certain Selmer groups in terms of <math><mrow><mi>L</mi> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mi>π</mi> <mo>,</mo> <msup> <mrow><mrow><mspace></mspace> <mtext>Ad</mtext> <mspace></mspace></mrow> </mrow> <mo>∘</mo></msup> <mo>)</mo></mrow> </math> . This can be viewed as partial progress on the Bloch-Kato conjecture. The main technical ingredients are a careful study of the cohomology associated with the locally symmetric space of <math><msub><mtext>GL</mtext> <mi>n</mi></msub> </math> , its relation to automorphic representations, and the establishment of some algebraic properties of the congruence ideals. We anticipate that the methods developed here will find further applications in related problems, particularly in the study of congruence modules and their relation to the arithmetic of automorphic forms.","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"12 1","pages":"20"},"PeriodicalIF":0.8,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12948911/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147327608","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Endomorphism algebras of abelian varieties with large cyclic 2-torsion field over a given field. 给定域上具有大循环2-扭转域的阿贝尔变体的自同态代数。

IF 0.8

Research in Number Theory Pub Date : 2026-01-01 Epub Date: 2026-03-23 DOI: 10.1007/s40993-026-00722-5

Pip Goodman

{"title":"Endomorphism algebras of abelian varieties with large cyclic 2-torsion field over a given field.","authors":"Pip Goodman","doi":"10.1007/s40993-026-00722-5","DOIUrl":"10.1007/s40993-026-00722-5","url":null,"abstract":"In this article we study the endomorphism algebras of abelian varieties A defined over a given number field K with large cyclic 2-torsion fields. A key step in doing so is to provide criteria for all the endomorphisms of A to be defined over K(A[2]), the field extension generated by its 2-torsion. When <math><mrow><mi>K</mi> <mo>=</mo> <mi>Q</mi></mrow> </math> and <math><mrow><mtext>Gal</mtext> <mo>(</mo> <mi>Q</mi> <mo>(</mo> <mi>A</mi> <mo>[</mo> <mn>2</mn> <mo>]</mo> <mo>)</mo> <mo>/</mo> <mi>Q</mi> <mo>)</mo></mrow> </math> is cyclic of prime order <math><mrow><mi>p</mi> <mo>=</mo> <mn>2</mn> <mo>dim</mo> <mo>(</mo> <mi>A</mi> <mo>)</mo> <mo>+</mo> <mn>1</mn></mrow> </math> , we prove that there are only finitely many possibilities for the geometric endomorphism algebra <math><mrow><mtext>End</mtext> <mo>(</mo> <mi>A</mi> <mo>)</mo> <mo>⊗</mo> <mi>Q</mi></mrow> </math> . In fact, when <math><mrow><mo>dim</mo> <mo>(</mo> <mi>A</mi> <mo>)</mo> <mo>∉</mo> <mo>{</mo> <mn>3</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>9</mn> <mo>,</mo> <mn>21</mn> <mo>,</mo> <mn>33</mn> <mo>,</mo> <mn>81</mn> <mo>}</mo></mrow> </math> , we show <math><mrow><mtext>End</mtext> <mo>(</mo> <mi>A</mi> <mo>)</mo> <mo>⊗</mo> <mi>Q</mi></mrow> </math> is a proper subfield of the p-th cyclotomic field. In particular, when <math><mrow><mi>g</mi> <mo>=</mo> <mn>2</mn></mrow> </math> , <math><mrow><mtext>End</mtext> <mo>(</mo> <mi>A</mi> <mo>)</mo> <mo>⊗</mo> <mi>Q</mi></mrow> </math> is isomorphic to either <math><mi>Q</mi></math> or <math><mrow><mi>Q</mi> <mo>(</mo> <msqrt><mn>5</mn></msqrt> <mo>)</mo></mrow> </math> .","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"12 2","pages":"34"},"PeriodicalIF":0.8,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13009002/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147515610","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Moments of derivatives of quadratic Dirichlet L-functions with prime conductor. 带素数导体的二次狄利克雷l函数的导数矩。

IF 0.8

Research in Number Theory Pub Date : 2026-01-01 Epub Date: 2026-02-07 DOI: 10.1007/s40993-026-00704-7

Christopher G Best

引用次数: 0

Transcendental Brauer-Manin obstructions on singular K3 surfaces. 奇异K3曲面上的先验Brauer-Manin障碍。

IF 0.6

Research in Number Theory Pub Date : 2025-01-01 Epub Date: 2024-12-18 DOI: 10.1007/s40993-024-00580-z

Mohamed Alaa Tawfik, Rachel Newton

引用次数: 0

A Fourier-Jacobi Dirichlet series for cusp forms on orthogonal groups. 正交群上尖形的Fourier-Jacobi Dirichlet级数。

IF 0.8

Research in Number Theory Pub Date : 2025-01-01 Epub Date: 2025-09-19 DOI: 10.1007/s40993-025-00668-0

Rafail Psyroukis

引用次数: 0

Traces of partition Eisenstein series and almost holomorphic modular forms. 划分爱森斯坦级数的迹与几乎全纯模形式。

IF 0.6

Research in Number Theory Pub Date : 2025-01-01 Epub Date: 2025-04-23 DOI: 10.1007/s40993-025-00615-z

Kathrin Bringmann, Badri Vishal Pandey

引用次数: 0

Constructing families of 3-Selmer companions. 构建由3-Selmer同伴组成的家庭。

IF 0.6

Research in Number Theory Pub Date : 2025-01-01 Epub Date: 2025-07-04 DOI: 10.1007/s40993-025-00647-5

Harry Spencer

引用次数: 0

On p-refined Friedberg-Jacquet integrals and the classical symplectic locus in the GL 2 n eigenvariety. gl2n特征变化中的p-精炼Friedberg-Jacquet积分和经典辛轨迹。

IF 0.6

Research in Number Theory Pub Date : 2025-01-01 Epub Date: 2025-04-25 DOI: 10.1007/s40993-025-00631-z

Daniel Barrera Salazar, Andrew Graham, Chris Williams

{"title":"<ArticleTitle xmlns:ns0=\"http://www.w3.org/1998/Math/MathML\">On p-refined Friedberg-Jacquet integrals and the classical symplectic locus in the <ns0:math> <ns0:msub><ns0:mrow><ns0:mspace /> <ns0:mtext>GL</ns0:mtext> <ns0:mspace /></ns0:mrow> <ns0:mrow><ns0:mn>2</ns0:mn> <ns0:mi>n</ns0:mi></ns0:mrow> </ns0:msub> </ns0:math> eigenvariety.","authors":"Daniel Barrera Salazar, Andrew Graham, Chris Williams","doi":"10.1007/s40993-025-00631-z","DOIUrl":"https://doi.org/10.1007/s40993-025-00631-z","url":null,"abstract":"Friedberg-Jacquet proved that if <math><mi>π</mi></math> is a cuspidal automorphic representation of <math> <mrow><msub><mtext>GL</mtext> <mrow><mn>2</mn> <mi>n</mi></mrow> </msub> <mrow><mo>(</mo> <mi>A</mi> <mo>)</mo></mrow> </mrow> </math> , then <math><mi>π</mi></math> is a functorial transfer from <math><msub><mtext>GSpin</mtext> <mrow><mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn></mrow> </msub> </math> if and only if a global zeta integral <math><msub><mi>Z</mi> <mi>H</mi></msub> </math> over <math><mrow><mi>H</mi> <mo>=</mo> <msub><mtext>GL</mtext> <mi>n</mi></msub> <mo>×</mo> <msub><mtext>GL</mtext> <mi>n</mi></msub> </mrow> </math> is non-vanishing on <math><mi>π</mi></math> . We conjecture a p-refined analogue: that any P-parahoric p-refinement <math> <msup><mover><mi>π</mi> <mo>~</mo></mover> <mi>P</mi></msup> </math> is a functorial transfer from <math><msub><mtext>GSpin</mtext> <mrow><mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn></mrow> </msub> </math> if and only if a P-twisted version of <math><msub><mi>Z</mi> <mi>H</mi></msub> </math> is non-vanishing on the <math> <msup><mover><mi>π</mi> <mo>~</mo></mover> <mi>P</mi></msup> </math> -eigenspace in <math><mi>π</mi></math> . This twisted <math><msub><mi>Z</mi> <mi>H</mi></msub> </math> appears in all constructions of p-adic L-functions via Shalika models. We connect our conjecture to the study of classical symplectic families in the <math><msub><mtext>GL</mtext> <mrow><mn>2</mn> <mi>n</mi></mrow> </msub> </math> eigenvariety, and-by proving upper bounds on the dimensions of such families-obtain various results towards the conjecture.","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"11 2","pages":"51"},"PeriodicalIF":0.6,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12031854/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144028936","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0