In this paper we first prove that the maximal ideal of the universal affine vertex operator algebra $V^k(sl_n)$ for $k=-n+frac{n-1}{q}$ is generated by two singular vectors of conformal weight $3q$ if $n=3$, and by one singular vector of conformal weight $2q$ if $ngeq 4$. We next determine the associated varieties of the simple vertex operator algebras $L_k(sl_3)$ for all the non-admissible levels $k=-3+frac{2}{2m+1}$, $mgeq 0$. The varieties of the associated simple affine $W$-algebras $W_k(sl_3,f)$, for nilpotent elements $f$ of $sl_3$, are also determined.
{"title":"Associated varieties of simple affine VOAs $L_k(sl_3)$ and $W$-algebras $W_k(sl_3,f)$","authors":"Cuipo Jiang, Jingtian Song","doi":"arxiv-2409.03552","DOIUrl":"https://doi.org/arxiv-2409.03552","url":null,"abstract":"In this paper we first prove that the maximal ideal of the universal affine\u0000vertex operator algebra $V^k(sl_n)$ for $k=-n+frac{n-1}{q}$ is generated by\u0000two singular vectors of conformal weight $3q$ if $n=3$, and by one singular\u0000vector of conformal weight $2q$ if $ngeq 4$. We next determine the associated\u0000varieties of the simple vertex operator algebras $L_k(sl_3)$ for all the\u0000non-admissible levels $k=-3+frac{2}{2m+1}$, $mgeq 0$. The varieties of the\u0000associated simple affine $W$-algebras $W_k(sl_3,f)$, for nilpotent elements $f$\u0000of $sl_3$, are also determined.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187343","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Thomas Creutzig, Vladimir Kovalchuk, Andrew R. Linshaw
The universal $2$-parameter vertex algebra $W_{infty}$ of type $W(2,3,4,dots)$ serves as a classifying object for vertex algebras of type $W(2,3,dots,N)$ for some $N$ in the sense that under mild hypothesis, all such vertex algebras arise as quotients of $W_{infty}$. There is an $mathbb{N} times mathbb{N}$ family of such $1$-parameter vertex algebras known as $Y$-algebras. They were introduced by Gaiotto and Rapv{c}'ak and are expected to be the building blocks for all $W$-algebras in type $A$, i.e., every $W$-(super) algebra in type $A$ is an extension of a tensor product of finitely many $Y$-algebras. Similarly, the orthosymplectic $Y$-algebras are $1$-parameter quotients of a universal $2$-parameter vertex algebra $W^{text{ev}}_{infty}$ of type $W(2,4,6,dots)$, which is a classifying object for vertex algebras of type $W(2,4,dots, 2N)$ for some $N$. Unlike type $A$, these algebras are not all the building blocks for $W$-algebras of types $B$, $C$, and $D$. In this paper, we construct a new universal $2$-parameter vertex algebra of type $W(1^3, 2, 3^3, 4, 5^3,6,dots)$ which we denote by $W^{mathfrak{sp}}_{infty}$ since it contains a copy of the affine vertex algebra $V^k(mathfrak{sp}_2)$. We identify $8$ infinite families of $1$-parameter quotients of $W^{mathfrak{sp}}_{infty}$ which are analogues of the $Y$-algebras. We regard $W^{mathfrak{sp}}_{infty}$ as a fundamental object on equal footing with $W_{infty}$ and $W^{text{ev}}_{infty}$, and we give some heuristic reasons for why we expect the $1$-parameter quotients of these three objects to be the building blocks for all $W$-algebras of classical types. Finally, we prove that $W^{mathfrak{sp}}_{infty}$ has many quotients which are strongly rational. This yields new examples of strongly rational $W$-superalgebras.
{"title":"Building blocks for $W$-algebras of classical types","authors":"Thomas Creutzig, Vladimir Kovalchuk, Andrew R. Linshaw","doi":"arxiv-2409.03465","DOIUrl":"https://doi.org/arxiv-2409.03465","url":null,"abstract":"The universal $2$-parameter vertex algebra $W_{infty}$ of type\u0000$W(2,3,4,dots)$ serves as a classifying object for vertex algebras of type\u0000$W(2,3,dots,N)$ for some $N$ in the sense that under mild hypothesis, all such\u0000vertex algebras arise as quotients of $W_{infty}$. There is an $mathbb{N}\u0000times mathbb{N}$ family of such $1$-parameter vertex algebras known as\u0000$Y$-algebras. They were introduced by Gaiotto and Rapv{c}'ak and are expected\u0000to be the building blocks for all $W$-algebras in type $A$, i.e., every\u0000$W$-(super) algebra in type $A$ is an extension of a tensor product of finitely\u0000many $Y$-algebras. Similarly, the orthosymplectic $Y$-algebras are\u0000$1$-parameter quotients of a universal $2$-parameter vertex algebra\u0000$W^{text{ev}}_{infty}$ of type $W(2,4,6,dots)$, which is a classifying\u0000object for vertex algebras of type $W(2,4,dots, 2N)$ for some $N$. Unlike type\u0000$A$, these algebras are not all the building blocks for $W$-algebras of types\u0000$B$, $C$, and $D$. In this paper, we construct a new universal $2$-parameter\u0000vertex algebra of type $W(1^3, 2, 3^3, 4, 5^3,6,dots)$ which we denote by\u0000$W^{mathfrak{sp}}_{infty}$ since it contains a copy of the affine vertex\u0000algebra $V^k(mathfrak{sp}_2)$. We identify $8$ infinite families of\u0000$1$-parameter quotients of $W^{mathfrak{sp}}_{infty}$ which are analogues of\u0000the $Y$-algebras. We regard $W^{mathfrak{sp}}_{infty}$ as a fundamental\u0000object on equal footing with $W_{infty}$ and $W^{text{ev}}_{infty}$, and we\u0000give some heuristic reasons for why we expect the $1$-parameter quotients of\u0000these three objects to be the building blocks for all $W$-algebras of classical\u0000types. Finally, we prove that $W^{mathfrak{sp}}_{infty}$ has many quotients\u0000which are strongly rational. This yields new examples of strongly rational\u0000$W$-superalgebras.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"59 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187165","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, Whittaker modules are studied for a subalgebra $mathfrak{q}_{epsilon}$ of the $emph{N}$=2 superconformal algebra. The Whittaker modules are classified by central characters. Additionally, criteria for the irreducibility of the Whittaker modules are given.
{"title":"Whittaker modules for a subalgebra of N=2 superconformal algebra","authors":"Naihuan Jing, Pengfa Xu, Honglian Zhang","doi":"arxiv-2409.03159","DOIUrl":"https://doi.org/arxiv-2409.03159","url":null,"abstract":"In this paper, Whittaker modules are studied for a subalgebra\u0000$mathfrak{q}_{epsilon}$ of the $emph{N}$=2 superconformal algebra. The Whittaker modules are classified by central characters. Additionally, criteria for the irreducibility of the Whittaker modules are\u0000given.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187137","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study sheaves of modules for the Lie algebra of vector fields with the action of the algebra of functions, compatible via the Leibniz rule. A crucial role in this theory is played by the virtual jets of vector fields - jets that evaluate to a zero vector field under the anchor map. Virtual jets of vector fields form a vector bundle $mathcal{L}_+$ whose fiber is Lie algebra $widehat{L}_+$ of vanishing at zero derivations of power series. We show that a sheaf of $AV$-modules is characterized by two ingredients - it is a module for $mathcal{L}_+$ and an $mathcal{L}_+$-charged $D$-module. For each rational finite-dimensional representation of $widehat{L}_+$, we construct a bundle of jet $AV$-modules. We also show that Rudakov modules may be realized as tensor products of jet modules with a $D$-module of delta functions.
{"title":"Sheaves of AV-modules on quasi-projective varieties","authors":"Yuly Billig, Emile Bouaziz","doi":"arxiv-2409.02677","DOIUrl":"https://doi.org/arxiv-2409.02677","url":null,"abstract":"We study sheaves of modules for the Lie algebra of vector fields with the\u0000action of the algebra of functions, compatible via the Leibniz rule. A crucial\u0000role in this theory is played by the virtual jets of vector fields - jets that\u0000evaluate to a zero vector field under the anchor map. Virtual jets of vector\u0000fields form a vector bundle $mathcal{L}_+$ whose fiber is Lie algebra\u0000$widehat{L}_+$ of vanishing at zero derivations of power series. We show that\u0000a sheaf of $AV$-modules is characterized by two ingredients - it is a module\u0000for $mathcal{L}_+$ and an $mathcal{L}_+$-charged $D$-module. For each rational finite-dimensional representation of $widehat{L}_+$, we\u0000construct a bundle of jet $AV$-modules. We also show that Rudakov modules may\u0000be realized as tensor products of jet modules with a $D$-module of delta\u0000functions.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187145","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $mathcal{F}$ be a saturated fusion system on a $p$-group $S$. We study the ring $R(mathcal{F})$ of $mathcal{F}$-stable characters by exploiting a new connection to the modular characters of a finite group $G$ with $mathcal{F} = mathcal{F}_S(G)$. We utilise this connection to find the rank of the $mathcal{F}$-stable character ring over fields with positive characteristic. We use this theory to derive a decomposition of the regular representation for a fixed basis $B$ of the ring of complex $mathcal{F}$-stable characters and give a formula for the absolute value of the determinant of the $mathcal{F}$-character table with respect to $B$ (the matrix of the values taken by elements of $B$ on each $mathcal{F}$-conjugacy class) for a wide class of saturated fusion systems, including all non-exotic fusion systems, and prove this value squared is a power of $p$ for all saturated fusion systems.
{"title":"Representation Rings of Fusion Systems and Brauer Characters","authors":"Thomas Lawrence","doi":"arxiv-2409.03007","DOIUrl":"https://doi.org/arxiv-2409.03007","url":null,"abstract":"Let $mathcal{F}$ be a saturated fusion system on a $p$-group $S$. We study\u0000the ring $R(mathcal{F})$ of $mathcal{F}$-stable characters by exploiting a\u0000new connection to the modular characters of a finite group $G$ with\u0000$mathcal{F} = mathcal{F}_S(G)$. We utilise this connection to find the rank\u0000of the $mathcal{F}$-stable character ring over fields with positive\u0000characteristic. We use this theory to derive a decomposition of the regular\u0000representation for a fixed basis $B$ of the ring of complex\u0000$mathcal{F}$-stable characters and give a formula for the absolute value of\u0000the determinant of the $mathcal{F}$-character table with respect to $B$ (the\u0000matrix of the values taken by elements of $B$ on each $mathcal{F}$-conjugacy\u0000class) for a wide class of saturated fusion systems, including all non-exotic\u0000fusion systems, and prove this value squared is a power of $p$ for all\u0000saturated fusion systems.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"59 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187141","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We survey several recent examples of derived structures emerging in connection with the Langlands correspondence. Cases studies include derived Galois deformation rings, derived Hecke algebras, derived Hitchin stacks, and derived special cycles. We also highlight some open problems that we expect to be important for future progress.
{"title":"Derived structures in the Langlands Correspondence","authors":"Tony Feng, Michael Harris","doi":"arxiv-2409.03035","DOIUrl":"https://doi.org/arxiv-2409.03035","url":null,"abstract":"We survey several recent examples of derived structures emerging in\u0000connection with the Langlands correspondence. Cases studies include derived\u0000Galois deformation rings, derived Hecke algebras, derived Hitchin stacks, and\u0000derived special cycles. We also highlight some open problems that we expect to\u0000be important for future progress.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187144","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We conjecture an algorithm to construct spin multipartitions, prove that the reduced signature is well-defined and give evidence to support our choice of the combinatorics of the spin multipartitions.
{"title":"Spin Multipartitions","authors":"Ola Amara-Omari, Mary Schaps","doi":"arxiv-2409.02801","DOIUrl":"https://doi.org/arxiv-2409.02801","url":null,"abstract":"We conjecture an algorithm to construct spin multipartitions, prove that the\u0000reduced signature is well-defined and give evidence to support our choice of\u0000the combinatorics of the spin multipartitions.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187149","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we extend the results of Grantcharov and Robitaille in 2021 on mixed tensor products and Capelli determinants to the superalgebra setting. Specifically, we construct a family of superalgebra homomorphisms $varphi_R : U(mathfrak{gl}(m+1|n)) rightarrow mathcal{D}'(m|n) otimes U(mathfrak{gl}(m|n))$ for a certain space of differential operators $mathcal{D}'(m|n)$ indexed by a central element $R$ of $mathcal{D}'(m|n) otimes U(mathfrak{gl}(m|n))$. We then use this homomorphism to determine the image of Gelfand generators of the center of $U(mathfrak{gl}(m+1|n))$. We achieve this by first relating $varphi_R$ to the corresponding Harish-Chandra homomorphisms and then proving a super-analog of Newton's formula for $mathfrak{gl}(m)$ relating Capelli generators and Gelfand generators. We also use the homomorphism $varphi_R$ to obtain representations of $U(mathfrak{gl}(m+1|n))$ from those of $U(mathfrak{gl}(m|n))$, and find conditions under which these inflations are simple. Finally, we show that for a distinguished central element $R_1$ in $mathcal{D}'(m|n)otimes U(mathfrak{gl}(m|n))$, the kernel of $varphi_{R_1}$ is the ideal of $U(mathfrak{gl}(m+1|n))$ generated by the first Gelfand invariant $G_1$.
{"title":"Mixed Tensor Products, Capelli Berezinians, and Newton's Formula for $mathfrak{gl}(m|n)$","authors":"Sidarth Erat, Arun S. Kannan, Shihan Kanungo","doi":"arxiv-2409.02422","DOIUrl":"https://doi.org/arxiv-2409.02422","url":null,"abstract":"In this paper, we extend the results of Grantcharov and Robitaille in 2021 on\u0000mixed tensor products and Capelli determinants to the superalgebra setting.\u0000Specifically, we construct a family of superalgebra homomorphisms $varphi_R :\u0000U(mathfrak{gl}(m+1|n)) rightarrow mathcal{D}'(m|n) otimes\u0000U(mathfrak{gl}(m|n))$ for a certain space of differential operators\u0000$mathcal{D}'(m|n)$ indexed by a central element $R$ of $mathcal{D}'(m|n)\u0000otimes U(mathfrak{gl}(m|n))$. We then use this homomorphism to determine the\u0000image of Gelfand generators of the center of $U(mathfrak{gl}(m+1|n))$. We\u0000achieve this by first relating $varphi_R$ to the corresponding Harish-Chandra\u0000homomorphisms and then proving a super-analog of Newton's formula for\u0000$mathfrak{gl}(m)$ relating Capelli generators and Gelfand generators. We also\u0000use the homomorphism $varphi_R$ to obtain representations of\u0000$U(mathfrak{gl}(m+1|n))$ from those of $U(mathfrak{gl}(m|n))$, and find\u0000conditions under which these inflations are simple. Finally, we show that for a\u0000distinguished central element $R_1$ in $mathcal{D}'(m|n)otimes\u0000U(mathfrak{gl}(m|n))$, the kernel of $varphi_{R_1}$ is the ideal of\u0000$U(mathfrak{gl}(m+1|n))$ generated by the first Gelfand invariant $G_1$.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"34 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187146","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We conjecture a precise relationship between Lusztig $q$-weight multiplicities for type $C$ and Kirillov-Reshetikhin crystals. We also define $mathfrak{gl}_n$-version of $q$-weight multiplicity for type $C$ and conjecture the positivity.
{"title":"Combinatorial description of Lusztig $q$-weight multiplicity","authors":"Seung Jin Lee","doi":"arxiv-2409.02341","DOIUrl":"https://doi.org/arxiv-2409.02341","url":null,"abstract":"We conjecture a precise relationship between Lusztig $q$-weight\u0000multiplicities for type $C$ and Kirillov-Reshetikhin crystals. We also define\u0000$mathfrak{gl}_n$-version of $q$-weight multiplicity for type $C$ and\u0000conjecture the positivity.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187147","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We construct a geometric realization of representations for $text{PSL}(2, mathbb{F}_p)$ by the defining ideals of rational models $mathcal{L}(X(p))$ of modular curves $X(p)$ over $mathbb{Q}$. Hence, for the irreducible representations of $text{PSL}(2, mathbb{F}_p)$, whose geometric realizations can be formulated in three different scenarios in the framework of Weil's Rosetta stone: number fields, curves over $mathbb{F}_q$ and Riemann surfaces. In particular, we show that there exists a correspondence among the defining ideals of modular curves over $mathbb{Q}$, reducible $mathbb{Q}(zeta_p)$-rational representations $pi_p: text{PSL}(2, mathbb{F}_p) rightarrow text{Aut}(mathcal{L}(X(p)))$ of $text{PSL}(2, mathbb{F}_p)$, and $mathbb{Q}(zeta_p)$-rational Galois representations $rho_p: text{Gal}(overline{mathbb{Q}}/mathbb{Q}) rightarrow text{Aut}(mathcal{L}(X(p)))$ as well as their modular and surjective realization. This leads to a new viewpoint on the last mathematical testament of Galois by Galois representations arising from the defining ideals of modular curves, which leads to a connection with Klein's elliptic modular functions. It is a nonlinear and anabelian counterpart of the global Langlands correspondence among the $ell$-adic '{e}tale cohomology of modular curves over $mathbb{Q}$, i.e., Grothendieck motives ($ell$-adic system), automorphic representations of $text{GL}(2, mathbb{Q})$ and $ell$-adic representations.
{"title":"Geometric realizations of representations for $text{PSL}(2, mathbb{F}_p)$ and Galois representations arising from defining ideals of modular curves","authors":"Lei Yang","doi":"arxiv-2409.02589","DOIUrl":"https://doi.org/arxiv-2409.02589","url":null,"abstract":"We construct a geometric realization of representations for $text{PSL}(2,\u0000mathbb{F}_p)$ by the defining ideals of rational models $mathcal{L}(X(p))$ of\u0000modular curves $X(p)$ over $mathbb{Q}$. Hence, for the irreducible\u0000representations of $text{PSL}(2, mathbb{F}_p)$, whose geometric realizations\u0000can be formulated in three different scenarios in the framework of Weil's\u0000Rosetta stone: number fields, curves over $mathbb{F}_q$ and Riemann surfaces.\u0000In particular, we show that there exists a correspondence among the defining\u0000ideals of modular curves over $mathbb{Q}$, reducible\u0000$mathbb{Q}(zeta_p)$-rational representations $pi_p: text{PSL}(2,\u0000mathbb{F}_p) rightarrow text{Aut}(mathcal{L}(X(p)))$ of $text{PSL}(2,\u0000mathbb{F}_p)$, and $mathbb{Q}(zeta_p)$-rational Galois representations\u0000$rho_p: text{Gal}(overline{mathbb{Q}}/mathbb{Q}) rightarrow\u0000text{Aut}(mathcal{L}(X(p)))$ as well as their modular and surjective\u0000realization. This leads to a new viewpoint on the last mathematical testament\u0000of Galois by Galois representations arising from the defining ideals of modular\u0000curves, which leads to a connection with Klein's elliptic modular functions. It\u0000is a nonlinear and anabelian counterpart of the global Langlands correspondence\u0000among the $ell$-adic '{e}tale cohomology of modular curves over $mathbb{Q}$,\u0000i.e., Grothendieck motives ($ell$-adic system), automorphic representations of\u0000$text{GL}(2, mathbb{Q})$ and $ell$-adic representations.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"121 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187148","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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