Algebra & Number Theory最新文献

{"title":"On Ozaki’s theorem realizing prescribed p-groups as p-class tower groups","authors":"Farshid Hajir, Christian Maire, Ravi Ramakrishna","doi":"10.2140/ant.2024.18.771","DOIUrl":"https://doi.org/10.2140/ant.2024.18.771","url":null,"abstract":"<p>We give a streamlined and effective proof of Ozaki’s theorem that any finite <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-group <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>Γ</mi></math> is the Galois group of the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-Hilbert class field tower of some number field <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> F</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--></math>. Our work is inspired by Ozaki’s and applies in broader circumstances. While his theorem is in the totally complex setting, we obtain the result in any mixed signature setting for which there exists a number field <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi> k</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow><mn>0</mn></mrow></msub></math> with class number prime to <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>. We construct <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> F</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo>∕</mo><msub><mrow><mi>k</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow><mn>0</mn></mrow></msub></math> by a sequence of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ℤ</mi><mo>∕</mo><mi>p</mi></math>-extensions ramified only at finite tame primes and also give explicit bounds on <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo stretchy=\"false\">[</mo><mi>F</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits-->\u0000<mo>:</mo><msub><mrow><mi> k</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow><mn>0</mn></mrow></msub><mo stretchy=\"false\">]</mo></math> and the number of ramified primes of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> F</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo>∕</mo><msub><mrow><mi>k</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow><mn>0</mn></mrow></msub></math> in terms of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>#</mi><mi>Γ</mi></math>. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"142 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139976770","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Wide moments of L-functions I : Twists by class group characters of imaginary quadratic fields","authors":"Asbjørn Christian Nordentoft","doi":"10.2140/ant.2024.18.735","DOIUrl":"https://doi.org/10.2140/ant.2024.18.735","url":null,"abstract":"<p>We calculate certain “wide moments” of central values of Rankin–Selberg <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>L</mi></math>-functions <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>L</mi><mrow><mo fence=\"true\" mathsize=\"1.19em\">(</mo><mrow><mi>π</mi>\u0000<mo>⊗</mo><mi mathvariant=\"normal\">Ω</mi><mo>,</mo> <mfrac><mrow><mn>1</mn></mrow>\u0000<mrow><mn>2</mn></mrow></mfrac></mrow><mo fence=\"true\" mathsize=\"1.19em\">)</mo></mrow></math> where <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>π</mi></math> is a cuspidal automorphic representation of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi> GL</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow><mn>2</mn></mrow></msub></math> over <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ℚ</mi></math> and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi mathvariant=\"normal\">Ω</mi></math> is a Hecke character (of conductor <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>1</mn></math>) of an imaginary quadratic field. This moment calculation is applied to obtain “weak simultaneous” nonvanishing results, which are nonvanishing results for different Rankin–Selberg <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>L</mi></math>-functions where the product of the twists is trivial. </p><p> The proof relies on relating the wide moments of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>L</mi></math>-functions to the usual moments of automorphic forms evaluated at Heegner points using Waldspurger’s formula. To achieve this, a classical version of Waldspurger’s formula for general weight automorphic forms is derived, which might be of independent interest. A key input is equidistribution of Heegner points (with explicit error terms), together with nonvanishing results for certain period integrals. In particular, we develop a soft technique for obtaining the nonvanishing of triple convolution <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>L</mi></math>-functions. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"13 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139976783","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Infinitesimal dilogarithm on curves over truncated polynomial rings","authors":"Sinan Ünver","doi":"10.2140/ant.2024.18.685","DOIUrl":"https://doi.org/10.2140/ant.2024.18.685","url":null,"abstract":"<p>We construct infinitesimal invariants of thickened one dimensional cycles in three dimensional space, which are the simplest cycles that are not in the Milnor range. This generalizes Park’s work on the regulators of additive cycles. The construction also allows us to prove the infinitesimal version of the strong reciprocity conjecture for thickenings of all orders. Classical analogs of our invariants are based on the dilogarithm function and our invariant could be seen as their infinitesimal version. Despite this analogy, the infinitesimal version cannot be obtained from their classical counterparts through a limiting process. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"17 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139976694","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Fundamental exact sequence for the pro-étale fundamental group","authors":"Marcin Lara","doi":"10.2140/ant.2024.18.631","DOIUrl":"https://doi.org/10.2140/ant.2024.18.631","url":null,"abstract":"<p>The pro-étale fundamental group of a scheme, introduced by Bhatt and Scholze, generalizes formerly known fundamental groups — the usual étale fundamental group <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msubsup><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow><mrow><!--mstyle--><mtext> ét</mtext><!--/mstyle--></mrow></msubsup></math> defined in SGA1 and the more general <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msubsup><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>SGA3</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow></msubsup></math>. It controls local systems in the pro-étale topology and leads to an interesting class of “geometric coverings” of schemes, generalizing finite étale coverings. </p><p> We prove exactness of the fundamental sequence for the pro-étale fundamental group of a geometrically connected scheme <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>X</mi></math> of finite type over a field <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi></math>, i.e., that the sequence </p>\u0000<div><math display=\"block\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\u0000<mn>1</mn>\u0000<mo>→</mo> <msubsup><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow><mrow><!--mstyle--><mtext> proét</mtext><!--/mstyle--></mrow></msubsup><mo stretchy=\"false\">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mover accent=\"true\"><mrow>\u0000<mi>k</mi></mrow><mo accent=\"true\">¯</mo></mover></mrow></msub><mo stretchy=\"false\">)</mo>\u0000<mo>→</mo> <msubsup><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow><mrow><!--mstyle--><mtext> proét</mtext><!--/mstyle--></mrow></msubsup><mo stretchy=\"false\">(</mo><mi>X</mi><mo stretchy=\"false\">)</mo>\u0000<mo>→</mo><msub><mrow><mi> Gal</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow>\u0000<mi>k</mi></mrow></msub>\u0000<mo>→</mo> <mn>1</mn>\u0000</math>\u0000</div>\u0000<p> is exact as abstract groups and the map <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msubsup><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow><mrow><!--mstyle--><mtext> proét</mtext><!--/mstyle--></mrow></msubsup><mo stretchy=\"false\">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mover accent=\"true\"><mrow><mi>k</mi></mrow><mo accent=\"true\">¯</mo></mover></mrow></msub><mo stretchy=\"false\">)</mo>\u0000<mo>→</mo> <msubsup><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow><mrow><!--mstyle--><mtext> proét</mtext><!--/mstyle--></mrow></msubsup><mo stretchy=\"false\">(</mo><mi>X</mi><mo stretchy=\"false\">)</mo></math> is a topological embedding. </p><p> On the way, we prove a general van Kampen theorem and the Künneth formula for the pro-étale fundamental group. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"30 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139976765","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Supersolvable descent for rational points","authors":"Yonatan Harpaz, Olivier Wittenberg","doi":"10.2140/ant.2024.18.787","DOIUrl":"https://doi.org/10.2140/ant.2024.18.787","url":null,"abstract":"<p>We construct an analogue of the classical descent theory of Colliot-Thélène and Sansuc in which algebraic tori are replaced with finite supersolvable groups. As an application, we show that rational points are dense in the Brauer–Manin set for smooth compactifications of certain quotients of homogeneous spaces by finite supersolvable groups. For suitably chosen homogeneous spaces, this implies the existence of supersolvable Galois extensions of number fields with prescribed norms, generalising work of Frei, Loughran and Newton. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"57 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139976774","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On Kato and Kuzumaki’s properties for the Milnor K2 of function fields of p-adic curves","authors":"Diego Izquierdo, Giancarlo Lucchini Arteche","doi":"10.2140/ant.2024.18.815","DOIUrl":"https://doi.org/10.2140/ant.2024.18.815","url":null,"abstract":"<p>Let <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>K</mi></math> be the function field of a curve <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>C</mi></math> over a <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-adic field <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi></math>. We prove that, for each <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi><mo>,</mo><mi>d</mi>\u0000<mo>≥</mo> <mn>1</mn></math> and for each hypersurface <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>Z</mi></math> in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msubsup><mrow><mi>ℙ</mi></mrow><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math> of degree <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>d</mi></math> with <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msup>\u0000<mo>≤</mo>\u0000<mi>n</mi></math>, the second Milnor <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>K</mi></math>-theory group of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>K</mi></math> is spanned by the images of the norms coming from finite extensions <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>L</mi></math> of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>K</mi></math> over which <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>Z</mi></math> has a rational point. When the curve <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>C</mi></math> has a point in the maximal unramified extension of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi></math>, we generalize this result to hypersurfaces <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>Z</mi></math> in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msubsup><mrow><mi>ℙ</mi></mrow><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math> of degree <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>d</mi></math> with <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>d</mi>\u0000<mo>≤</mo>\u0000<mi>n</mi></math>. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"135 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139976857","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A categorical Künneth formula for constructible Weil sheaves","authors":"Tamir Hemo, Timo Richarz, Jakob Scholbach","doi":"10.2140/ant.2024.18.499","DOIUrl":"https://doi.org/10.2140/ant.2024.18.499","url":null,"abstract":"<p>We prove a Künneth-type equivalence of derived categories of lisse and constructible Weil sheaves on schemes in characteristic <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi>\u0000<mo>&gt;</mo> <mn>0</mn></math> for various coefficients, including finite discrete rings, algebraic field extensions <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>E</mi>\u0000<mo>⊃</mo> <msub><mrow><mi>ℚ</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math>, <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ℓ</mi><mo>≠</mo><mi>p</mi></math>, and their rings of integers <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi mathvariant=\"bold-script\">𝒪</mi></mrow><mrow><mi>E</mi></mrow></msub></math>. We also consider a variant for ind-constructible sheaves which applies to the cohomology of moduli stacks of shtukas over global function fields. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"22 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139898776","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Quotients of admissible formal schemes and adic spaces by finite groups","authors":"Bogdan Zavyalov","doi":"10.2140/ant.2024.18.409","DOIUrl":"https://doi.org/10.2140/ant.2024.18.409","url":null,"abstract":"<p>We give a self-contained treatment of finite group quotients of admissible (formal) schemes and adic spaces that are locally topologically finite type over a locally strongly noetherian adic space. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"185 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139898799","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Subconvexity bound for GL(3) × GL(2) L-functions : Hybrid level aspect","authors":"Sumit Kumar, Ritabrata Munshi, Saurabh Kumar Singh","doi":"10.2140/ant.2024.18.477","DOIUrl":"https://doi.org/10.2140/ant.2024.18.477","url":null,"abstract":"<p>Let <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>F</mi></math> be a <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> GL</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><mn>3</mn><mo stretchy=\"false\">)</mo></math> Hecke–Maass cusp form of prime level <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msub></math> and let <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>f</mi></math> be a <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> GL</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><mn>2</mn><mo stretchy=\"false\">)</mo></math> Hecke–Maass cuspform of prime level <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msub></math>. We will prove a subconvex bound for the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> GL</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><mn>3</mn><mo stretchy=\"false\">)</mo>\u0000<mo>×</mo><mi> GL</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><mn>2</mn><mo stretchy=\"false\">)</mo></math> Rankin–Selberg <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>L</mi></math>-function <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>L</mi><mo stretchy=\"false\">(</mo><mi>s</mi><mo>,</mo><mi>F</mi>\u0000<mo>×</mo>\u0000<mi>f</mi><mo stretchy=\"false\">)</mo></math> in the level aspect for certain ranges of the parameters <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msub></math> and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msub></math>. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"185 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139898784","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Generalized Igusa functions and ideal growth in nilpotent Lie rings","authors":"Angela Carnevale, Michael M. Schein, Christopher Voll","doi":"10.2140/ant.2024.18.537","DOIUrl":"https://doi.org/10.2140/ant.2024.18.537","url":null,"abstract":"<p>We introduce a new class of combinatorially defined rational functions and apply them to deduce explicit formulae for local ideal zeta functions associated to the members of a large class of nilpotent Lie rings which contains the free class-2-nilpotent Lie rings and is stable under direct products. Our results unify and generalize a substantial number of previous computations. We show that the new rational functions, and thus also the local zeta functions under consideration, enjoy a self-reciprocity property, expressed in terms of a functional equation upon inversion of variables. We establish a conjecture of Grunewald, Segal, and Smith on the uniformity of normal zeta functions of finitely generated free class-2-nilpotent groups. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"232 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139898768","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}