Let E be a Galois extension of ? of degree l, not necessarily solvable. In this paper we first prove that the L-function L(s, π) attached to an automorphic cuspidal representation π of $ GL_m (E_\mathbb{A} ) $ canno...Let E be a Galois extension of ? of degree l, not necessarily solvable. In this paper we first prove that the L-function L(s, π) attached to an automorphic cuspidal representation π of $ GL_m (E_\mathbb{A} ) $ cannot be factored nontrivially into a product of L-functions over E.Next, we compare the n-level correlation of normalized nontrivial zeros of L(s, π1)…L(s, π k ), where π j , j = 1,…, k, are automorphic cuspidal representations of $ GL_{m_j } (\mathbb{Q}_\mathbb{A} ) $ , with that of L(s,π). We prove a necessary condition for L(s, π) having a factorization into a product of L-functions attached to automorphic cuspidal representations of specific $ GL_{m_j } (\mathbb{Q}_\mathbb{A} ) $ , j = 1,…,k. In particular, if π is not invariant under the action of any nontrivial σ ∈ Gal E/?, then L(s, π) must equal a single L-function attached to a cuspidal representation of $ GL_{m\ell } (\mathbb{Q}_\mathbb{A} ) $ and π has an automorphic induction, provided L(s, π) can factored into a product of L-functions over ?. As E is not assumed to be solvable over ?, our results are beyond the scope of the current theory of base change and automorphic induction.Our results are unconditional when m,m 1,…,m k are small, but are under Hypothesis H and a bound toward the Ramanujan conjecture in other cases.展开更多
In this paper,we survey our work on period,poles or special values of certain automorphic Lfunctions and their relations to certain types of Langlands functorial transfers.We outline proofs for some results and leave ...In this paper,we survey our work on period,poles or special values of certain automorphic Lfunctions and their relations to certain types of Langlands functorial transfers.We outline proofs for some results and leave others as conjectures.展开更多
Motivated by T-tilting theory developed by T. Adachi, O. Iyama, I. Reiten, for a finite-dimensional algebra A with action by a finite group G, we introduce the notion of G-stable support τ-tilting modules. Then we es...Motivated by T-tilting theory developed by T. Adachi, O. Iyama, I. Reiten, for a finite-dimensional algebra A with action by a finite group G, we introduce the notion of G-stable support τ-tilting modules. Then we establish bijections among G-stable support τ-tilting modules over ∧, G-stable two-term silting complexes in the homotopy category of bounded complexes of finitely generated projective ∧-modules, and G-stable functorially finite torsion classes in the category of finitely generated left ∧-modules. In the case when ∧ is the endomorphism of a G-stable cluster-tilting object T over a Horn-finite 2-Calabi- Yau triangulated category L with a G-action, these are also in bijection with G-stable cluster-tilting objects in L. Moreover, we investigate the relationship between stable support τ-tilitng modules over ∧ and the skew group algebra ∧G.展开更多
Let A be a finite-dimensional algebra over an algebraically closed field k,ε the category of all exact sequences in A-rood, Mp （respectively, Ml） the full subcategory of C consisting of those objects with projecti...Let A be a finite-dimensional algebra over an algebraically closed field k,ε the category of all exact sequences in A-rood, Mp （respectively, Ml） the full subcategory of C consisting of those objects with projective （respectively, injective） middle terms. It is proved that Mp （respectively, MI） is contravariantly finite （respectively, covariantly finite） in ε. As an application, it is shown that Mp = MI is functorially finite and has Auslander-Reiten sequences provided A is selfinjective. Keywords category of exact sequences, contravariantly finite subcategory, functorially finite subcategory Auslander-Reiten sequences, selfinjective algebra展开更多
Let C be a triangulated category which has Auslander-Reiten triangles, and Ra functorially finite rigid subcategory of C. It is well known that there exist Auslander-Reiten sequences in rood R. In this paper, we give ...Let C be a triangulated category which has Auslander-Reiten triangles, and Ra functorially finite rigid subcategory of C. It is well known that there exist Auslander-Reiten sequences in rood R. In this paper, we give explicitly the relations between the Auslander-Reiten translations, sequences in mod R and the Auslander-Reiten functors, triangles in C, respectively. Furthermore, if T is a cluster-tilting subcategory of C and mod T- is a Frobenius category, we also get the Auslander-Reiten functor and the translation functor of mod T- corresponding to the ones in C. As a consequence, we get that if the quotient of a d-Calabi-Yau triangulated category modulo a cluster tilting subcategory is Probenius, then its stable category is （2d-1）-Calabi-Yau. This result was first proved by Keller and Reiten in the case d= 2, and then by Dugas in the general case, using different methods. 2010 Mathematics Subject Classification： 16G20, 16G70展开更多
基金supported by the National Basic Research Program of China, the National Natural Science Foundation of China (Grant No. 10531060)Ministry of Education of China (Grant No. 305009)+1 种基金The second author was supported by the National Security Agency (Grant No. H98230-06-1-0075)The United States Government is authorized to reproduce and distribute reprints notwithstanding any copyright notation herein
文摘Let E be a Galois extension of ? of degree l, not necessarily solvable. In this paper we first prove that the L-function L(s, π) attached to an automorphic cuspidal representation π of $ GL_m (E_\mathbb{A} ) $ cannot be factored nontrivially into a product of L-functions over E.Next, we compare the n-level correlation of normalized nontrivial zeros of L(s, π1)…L(s, π k ), where π j , j = 1,…, k, are automorphic cuspidal representations of $ GL_{m_j } (\mathbb{Q}_\mathbb{A} ) $ , with that of L(s,π). We prove a necessary condition for L(s, π) having a factorization into a product of L-functions attached to automorphic cuspidal representations of specific $ GL_{m_j } (\mathbb{Q}_\mathbb{A} ) $ , j = 1,…,k. In particular, if π is not invariant under the action of any nontrivial σ ∈ Gal E/?, then L(s, π) must equal a single L-function attached to a cuspidal representation of $ GL_{m\ell } (\mathbb{Q}_\mathbb{A} ) $ and π has an automorphic induction, provided L(s, π) can factored into a product of L-functions over ?. As E is not assumed to be solvable over ?, our results are beyond the scope of the current theory of base change and automorphic induction.Our results are unconditional when m,m 1,…,m k are small, but are under Hypothesis H and a bound toward the Ramanujan conjecture in other cases.
基金supported in part by the Natural Science Foundation of USA (Grant Nos.DMS-0653742,DMS-1001672) and by the Chinese Academy of Sciences
文摘In this paper,we survey our work on period,poles or special values of certain automorphic Lfunctions and their relations to certain types of Langlands functorial transfers.We outline proofs for some results and leave others as conjectures.
基金The authors would like to thank Dong Yang and Yuefei Zheng for their helpful discussion. This work was partially supported by the National Natural Science Foundation of China （Grant No. 11571164） and a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.
文摘Motivated by T-tilting theory developed by T. Adachi, O. Iyama, I. Reiten, for a finite-dimensional algebra A with action by a finite group G, we introduce the notion of G-stable support τ-tilting modules. Then we establish bijections among G-stable support τ-tilting modules over ∧, G-stable two-term silting complexes in the homotopy category of bounded complexes of finitely generated projective ∧-modules, and G-stable functorially finite torsion classes in the category of finitely generated left ∧-modules. In the case when ∧ is the endomorphism of a G-stable cluster-tilting object T over a Horn-finite 2-Calabi- Yau triangulated category L with a G-action, these are also in bijection with G-stable cluster-tilting objects in L. Moreover, we investigate the relationship between stable support τ-tilitng modules over ∧ and the skew group algebra ∧G.
基金supported by National Natural Science Foundation of China(Grant No.11271257)National Science Foundation of Shanghai Municiple(Granted No.13ZR1422500)
文摘Let A be a finite-dimensional algebra over an algebraically closed field k,ε the category of all exact sequences in A-rood, Mp （respectively, Ml） the full subcategory of C consisting of those objects with projective （respectively, injective） middle terms. It is proved that Mp （respectively, MI） is contravariantly finite （respectively, covariantly finite） in ε. As an application, it is shown that Mp = MI is functorially finite and has Auslander-Reiten sequences provided A is selfinjective. Keywords category of exact sequences, contravariantly finite subcategory, functorially finite subcategory Auslander-Reiten sequences, selfinjective algebra
文摘Let C be a triangulated category which has Auslander-Reiten triangles, and Ra functorially finite rigid subcategory of C. It is well known that there exist Auslander-Reiten sequences in rood R. In this paper, we give explicitly the relations between the Auslander-Reiten translations, sequences in mod R and the Auslander-Reiten functors, triangles in C, respectively. Furthermore, if T is a cluster-tilting subcategory of C and mod T- is a Frobenius category, we also get the Auslander-Reiten functor and the translation functor of mod T- corresponding to the ones in C. As a consequence, we get that if the quotient of a d-Calabi-Yau triangulated category modulo a cluster tilting subcategory is Probenius, then its stable category is （2d-1）-Calabi-Yau. This result was first proved by Keller and Reiten in the case d= 2, and then by Dugas in the general case, using different methods. 2010 Mathematics Subject Classification： 16G20, 16G70
