Pub Date : 2024-03-22DOI: 10.1017/s0305004124000069
ZHIYOU WU
We prove that the Hodge–Tate spectral sequence of a proper smooth rigid analytic variety can be reconstructed from its infinitesimal $mathbb{B}_{text{dR}}^+$ -cohomology through the Bialynicki–Birula map. We also give a new proof of the torsion-freeness of the infinitesimal $mathbb{B}_{text{dR}}^+$ -cohomology independent of Conrad–Gabber spreading theorem, and a conceptual explanation that the degeneration of Hodge–Tate spectral sequences is equivalent to that of Hodge–de Rham spectral sequences.
{"title":"A note on Hodge–Tate spectral sequences","authors":"ZHIYOU WU","doi":"10.1017/s0305004124000069","DOIUrl":"https://doi.org/10.1017/s0305004124000069","url":null,"abstract":"We prove that the Hodge–Tate spectral sequence of a proper smooth rigid analytic variety can be reconstructed from its infinitesimal <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0305004124000069_inline1.png\" /> <jats:tex-math> $mathbb{B}_{text{dR}}^+$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-cohomology through the Bialynicki–Birula map. We also give a new proof of the torsion-freeness of the infinitesimal <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0305004124000069_inline2.png\" /> <jats:tex-math> $mathbb{B}_{text{dR}}^+$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-cohomology independent of Conrad–Gabber spreading theorem, and a conceptual explanation that the degeneration of Hodge–Tate spectral sequences is equivalent to that of Hodge–de Rham spectral sequences.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"30 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140197514","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 : 2024-03-12DOI: 10.1017/s0305004124000021
EDGAR ASSING
In this paper we take up the classical sup-norm problem for automorphic forms and view it from a new angle. Given a twist minimal automorphic representation $pi$ we consider a special small $mathrm{GL}_2(mathbb{Z}_p)$-type V in $pi$ and prove global sup-norm bounds for an average over an orthonormal basis of V. We achieve a non-trivial saving when the dimension of V grows.
在本文中，我们从一个新的角度探讨了自动形式的经典超规范问题。给定一个扭转最小自形表示 $pi$ ，我们考虑 $pi$ 中一个特殊的小 $mathrm{GL}_2(mathbb{Z}_p)$ 型 V，并证明 V 的正交基础上的平均的全局超规范边界。
{"title":"The sup-norm problem beyond the newform","authors":"EDGAR ASSING","doi":"10.1017/s0305004124000021","DOIUrl":"https://doi.org/10.1017/s0305004124000021","url":null,"abstract":"<p>In this paper we take up the classical sup-norm problem for automorphic forms and view it from a new angle. Given a twist minimal automorphic representation <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240311161759314-0086:S0305004124000021:S0305004124000021_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$pi$</span></span></img></span></span> we consider a special small <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240311161759314-0086:S0305004124000021:S0305004124000021_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$mathrm{GL}_2(mathbb{Z}_p)$</span></span></img></span></span>-type <span>V</span> in <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240311161759314-0086:S0305004124000021:S0305004124000021_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$pi$</span></span></img></span></span> and prove global sup-norm bounds for an average over an orthonormal basis of <span>V</span>. We achieve a non-trivial saving when the dimension of <span>V</span> grows.</p>","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"22 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140106381","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 : 2024-03-08DOI: 10.1017/s030500412400001x
SHIHOKO ISHII
We study a pair consisting of a smooth 3-fold defined over an algebraically closed field and a “general” ${Bbb R}$ -ideal. We show that the minimal log discrepancy (“mld” for short) of every such a pair is computed by a prime divisor obtained by at most two weighted blow-ups. This bound is regarded as a weighted blow-up version of Mustaţă–Nakamura’s conjecture. We also show that if the mld of such a pair is not less than 1, then it is computed by at most one weighted blow-up. As a consequence, ACC of mld holds for such pairs.
{"title":"A bound of the number of weighted blow-ups to compute the minimal log discrepancy for smooth 3-folds","authors":"SHIHOKO ISHII","doi":"10.1017/s030500412400001x","DOIUrl":"https://doi.org/10.1017/s030500412400001x","url":null,"abstract":"We study a pair consisting of a smooth 3-fold defined over an algebraically closed field and a “general” <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S030500412400001X_inline1.png\" /> <jats:tex-math> ${Bbb R}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-ideal. We show that the minimal log discrepancy (“mld” for short) of every such a pair is computed by a prime divisor obtained by at most two weighted blow-ups. This bound is regarded as a weighted blow-up version of Mustaţă–Nakamura’s conjecture. We also show that if the mld of such a pair is not less than 1, then it is computed by at most one weighted blow-up. As a consequence, ACC of mld holds for such pairs.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"5 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140074760","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 : 2024-03-07DOI: 10.1017/s0305004124000057
DAVID MASSER
Given any polynomial in two variables of degree at most three with rational integer coefficients, we obtain a new search bound to decide effectively if it has a zero with rational integer coefficients. On the way we encounter a natural problem of estimating singular points. We solve it using elementary invariant theory but an optimal solution would seem to be far from easy even using the full power of the standard Height Machine.
{"title":"How to solve a binary cubic equation in integers","authors":"DAVID MASSER","doi":"10.1017/s0305004124000057","DOIUrl":"https://doi.org/10.1017/s0305004124000057","url":null,"abstract":"<p>Given any polynomial in two variables of degree at most three with rational integer coefficients, we obtain a new search bound to decide effectively if it has a zero with rational integer coefficients. On the way we encounter a natural problem of estimating singular points. We solve it using elementary invariant theory but an optimal solution would seem to be far from easy even using the full power of the standard Height Machine.</p>","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"66 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140055480","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 : 2024-03-04DOI: 10.1017/s0305004124000033
RICCARDO ONTANI, JACOPO STOPPA
We prove an equality, predicted in the physical literature, between the Jeffrey–Kirwan residues of certain explicit meromorphic forms attached to a quiver without loops or oriented cycles and its Donaldson–Thomas type invariants.
In the special case of complete bipartite quivers we also show independently, using scattering diagrams and theta functions, that the same Jeffrey–Kirwan residues are determined by the the Gross–Hacking–Keel mirror family to a log Calabi–Yau surface.
{"title":"Log Calabi–Yau surfaces and Jeffrey–Kirwan residues","authors":"RICCARDO ONTANI, JACOPO STOPPA","doi":"10.1017/s0305004124000033","DOIUrl":"https://doi.org/10.1017/s0305004124000033","url":null,"abstract":"<p>We prove an equality, predicted in the physical literature, between the Jeffrey–Kirwan residues of certain explicit meromorphic forms attached to a quiver without loops or oriented cycles and its Donaldson–Thomas type invariants.</p><p>In the special case of complete bipartite quivers we also show independently, using scattering diagrams and theta functions, that the same Jeffrey–Kirwan residues are determined by the the Gross–Hacking–Keel mirror family to a log Calabi–Yau surface.</p>","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"10 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140026267","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 : 2024-03-01DOI: 10.1017/s0305004124000045
FATMA ÇİÇEK, STEVEN M. GONEK
<p>On the assumption of the Riemann hypothesis and a spacing hypothesis for the nontrivial zeros <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240229125249422-0748:S0305004124000045:S0305004124000045_inline1.png"><span data-mathjax-type="texmath"><span>$1/2+igamma$</span></span></img></span></span> of the Riemann zeta function, we show that the sequence <span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240229125249422-0748:S0305004124000045:S0305004124000045_eqnU1.png"><span data-mathjax-type="texmath"><span>begin{equation*}Gamma_{[a, b]} =Bigg{ gamma : gamma>0 quad mbox{and} quad frac{ logbig(| zeta^{(m_{gamma })} (frac12+ i{gamma }) | / (!log{{gamma }} )^{m_{gamma }}big)}{sqrt{frac12loglog {gamma }}} in [a, b] Bigg},end{equation*}</span></span></img></span>where the <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240229125249422-0748:S0305004124000045:S0305004124000045_inline2.png"><span data-mathjax-type="texmath"><span>${gamma }$</span></span></img></span></span> are arranged in increasing order, is uniformly distributed modulo one. Here <span>a</span> and <span>b</span> are real numbers with <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240229125249422-0748:S0305004124000045:S0305004124000045_inline3.png"><span data-mathjax-type="texmath"><span>$a<b$</span></span></img></span></span>, and <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240229125249422-0748:S0305004124000045:S0305004124000045_inline4.png"><span data-mathjax-type="texmath"><span>$m_gamma$</span></span></img></span></span> denotes the multiplicity of the zero <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240229125249422-0748:S0305004124000045:S0305004124000045_inline5.png"><span data-mathjax-type="texmath"><span>$1/2+i{gamma }$</span></span></img></span></span>. The same result holds when the <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240229125249422-0748:S0305004124000045:S0305004124000045_inline6.png"><span data-mathjax-type="texmath"><span>${gamma }$</span></span></img></span></span>’s are restricted to be the ordinates of simple zeros. With an extra hypothesis, we are also able to show an equidistribution result for the scaled numbers <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240229125249422-0748:S0305004124000045:S0305004124000045_inline7.png"><span data-mathjax-type="texmath
{"title":"The uniform distribution modulo one of certain subsequences of ordinates of zeros of the zeta function","authors":"FATMA ÇİÇEK, STEVEN M. GONEK","doi":"10.1017/s0305004124000045","DOIUrl":"https://doi.org/10.1017/s0305004124000045","url":null,"abstract":"<p>On the assumption of the Riemann hypothesis and a spacing hypothesis for the nontrivial zeros <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240229125249422-0748:S0305004124000045:S0305004124000045_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$1/2+igamma$</span></span></img></span></span> of the Riemann zeta function, we show that the sequence <span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240229125249422-0748:S0305004124000045:S0305004124000045_eqnU1.png\"><span data-mathjax-type=\"texmath\"><span>begin{equation*}Gamma_{[a, b]} =Bigg{ gamma : gamma>0 quad mbox{and} quad frac{ logbig(| zeta^{(m_{gamma })} (frac12+ i{gamma }) | / (!log{{gamma }} )^{m_{gamma }}big)}{sqrt{frac12loglog {gamma }}} in [a, b] Bigg},end{equation*}</span></span></img></span>where the <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240229125249422-0748:S0305004124000045:S0305004124000045_inline2.png\"><span data-mathjax-type=\"texmath\"><span>${gamma }$</span></span></img></span></span> are arranged in increasing order, is uniformly distributed modulo one. Here <span>a</span> and <span>b</span> are real numbers with <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240229125249422-0748:S0305004124000045:S0305004124000045_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$a<b$</span></span></img></span></span>, and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240229125249422-0748:S0305004124000045:S0305004124000045_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$m_gamma$</span></span></img></span></span> denotes the multiplicity of the zero <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240229125249422-0748:S0305004124000045:S0305004124000045_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$1/2+i{gamma }$</span></span></img></span></span>. The same result holds when the <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240229125249422-0748:S0305004124000045:S0305004124000045_inline6.png\"><span data-mathjax-type=\"texmath\"><span>${gamma }$</span></span></img></span></span>’s are restricted to be the ordinates of simple zeros. With an extra hypothesis, we are also able to show an equidistribution result for the scaled numbers <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240229125249422-0748:S0305004124000045:S0305004124000045_inline7.png\"><span data-mathjax-type=\"texmath","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"252 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140009119","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 : 2023-12-20DOI: 10.1017/s0305004123000610
{"title":"PSP volume 176 issue 1 Cover and Front matter","authors":"","doi":"10.1017/s0305004123000610","DOIUrl":"https://doi.org/10.1017/s0305004123000610","url":null,"abstract":"","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"51 2","pages":"f1 - f2"},"PeriodicalIF":0.8,"publicationDate":"2023-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139168877","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 : 2023-12-20DOI: 10.1017/s0305004123000622
{"title":"PSP volume 176 issue 1 Cover and Back matter","authors":"","doi":"10.1017/s0305004123000622","DOIUrl":"https://doi.org/10.1017/s0305004123000622","url":null,"abstract":"","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"38 2","pages":"b1 - b2"},"PeriodicalIF":0.8,"publicationDate":"2023-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139169293","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 : 2023-11-24DOI: 10.1017/s030500412300052x
DAVID DE BOER, PJOTR BUYS, LORENZO GUERINI, HAN PETERS, GUUS REGTS
The independence polynomial originates in statistical physics as the partition function of the hard-core model. The location of the complex zeros of the polynomial is related to phase transitions, and plays an important role in the design of efficient algorithms to approximately compute evaluations of the polynomial. In this paper we directly relate the location of the complex zeros of the independence polynomial to computational hardness of approximating evaluations of the independence polynomial. We do this by moreover relating the location of zeros to chaotic behaviour of a naturally associated family of rational functions; the occupation ratios.
{"title":"Zeros, chaotic ratios and the computational complexity of approximating the independence polynomial","authors":"DAVID DE BOER, PJOTR BUYS, LORENZO GUERINI, HAN PETERS, GUUS REGTS","doi":"10.1017/s030500412300052x","DOIUrl":"https://doi.org/10.1017/s030500412300052x","url":null,"abstract":"The independence polynomial originates in statistical physics as the partition function of the hard-core model. The location of the complex zeros of the polynomial is related to phase transitions, and plays an important role in the design of efficient algorithms to approximately compute evaluations of the polynomial. In this paper we directly relate the location of the complex zeros of the independence polynomial to computational hardness of approximating evaluations of the independence polynomial. We do this by moreover relating the location of zeros to chaotic behaviour of a naturally associated family of rational functions; the occupation ratios.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"8 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138532081","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 : 2023-10-16DOI: 10.1017/s0305004123000580
VLADIMIR DOTSENKO, OISÍN FLYNN-CONNOLLY
Abstract We give explicit combinatorial descriptions of three Schur functors arising in the theory of pre-Lie algebras. The first of them leads to a functorial description of the underlying vector space of the universal enveloping pre-Lie algebra of a given Lie algebra, strengthening the Poincaré-Birkhoff-Witt (PBW) theorem of Segal. The two other Schur functors provide functorial descriptions of the underlying vector spaces of the universal multiplicative enveloping algebra and of the module of Kähler differentials of a given pre-Lie algebra. An important consequence of such descriptions is an interpretation of the cohomology of a pre-Lie algebra with coefficients in a module as a derived functor for the category of modules over the universal multiplicative enveloping algebra.
{"title":"Three Schur functors related to pre-Lie algebras","authors":"VLADIMIR DOTSENKO, OISÍN FLYNN-CONNOLLY","doi":"10.1017/s0305004123000580","DOIUrl":"https://doi.org/10.1017/s0305004123000580","url":null,"abstract":"Abstract We give explicit combinatorial descriptions of three Schur functors arising in the theory of pre-Lie algebras. The first of them leads to a functorial description of the underlying vector space of the universal enveloping pre-Lie algebra of a given Lie algebra, strengthening the Poincaré-Birkhoff-Witt (PBW) theorem of Segal. The two other Schur functors provide functorial descriptions of the underlying vector spaces of the universal multiplicative enveloping algebra and of the module of Kähler differentials of a given pre-Lie algebra. An important consequence of such descriptions is an interpretation of the cohomology of a pre-Lie algebra with coefficients in a module as a derived functor for the category of modules over the universal multiplicative enveloping algebra.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"251 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136112794","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}
$pi$ we consider a special small 
$mathrm{GL}_2(mathbb{Z}_p)$-type V in 
$pi$ and prove global sup-norm bounds for an average over an orthonormal basis of V. We achieve a non-trivial saving when the dimension of V grows.
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