LIN WENG | Last modified date：2020.06.25 |

Professor /
Math /
Department of Mathematics /
Faculty of Mathematics

**Papers**

1. | Lin Weng, A result on bicanonical maps of surfaces of general type, Osaka Journal of Mathematics, 32, 2, 467-473, 1995. |

2. | Lin Weng, Singular moduli and the arakelov intersection, tohoku mathematical journal, second series, 10.2748/tmj/1178225521, 47, 3, 345-356, 1995.01, The values of the modular j-function at imaginary quadratic arguments in the upper half plane are usually called singular moduli. In this paper, we use the Arakelov intersection to give the prime factorizations of a certain combination of singular moduli, coming from the Hecke correspondence. Such a result may be considered as a degenerate one of Gross and Zagier on Heegner points and derivatives of L-series, and is parellel to the result of Gross and Zagier on singular moduli.. |

3. | Lin Weng, Singular moduli and the arakelov intersection, Tohoku Mathematical Journal, 10.2748/tmj/1178225521, 47, 3, 345-356, 1995.01, The values of the modular y-function at imaginary quadratic arguments in the upper half plane are usually called singular moduli. In this paper, we use the Arakelov intersection to give the prime factorizations of a certain combination of singular moduli, coming from the Hecke correspondence. Such a result may be considered as a degenerate one of Gross and Zagier on Heegner points and derivatives of L-series, and is parellel to the result of Gross and Zagier on singular moduli.. |

4. | Lin Weng, Yuching You, Analytic torsions of spheres, International Journal of Mathematics, 10.1142/S0129167X96000074, 7, 1, 109-125, 1996.02. |

5. | Wing Keung To, Lin Weng, The asymptotic behavior of Green's functions for quasi-hyperbolic metrics on degenerating Riemann surfaces, Manuscripta Mathematica, 93, 4, 465-480, 1997.08, In this article, we consider a family of compact Riemann surfaces of genus q ≥ 2 degenerating to a Riemann surface of genus q-1 with a non-separating node. We show that the Green's functions associated to a continuous family of quasi-hyperbolic metrics on such degenerating Riemann surfaces simply degenerate to that on the smooth part of the noded Riemann surface.. |

6. | Lin Weng, Standard modules of level 1 for sl̂_{2} in terms of virasoro algebra representations, Communications in Algebra, 10.1080/00927879808826151, 26, 2, 613-625, 1998.01. |

7. | Lin Weng, Don Zagier, Higher-rank zeta functions and SL_{n}-zeta functions for curves, Proceedings of the National Academy of Sciences of the United States of America, 10.1073/pnas.1912501117, 117, 12, 6398-6408, 2020.03, In earlier papers L.W. introduced two sequences of higher-rank zeta functions associated to a smooth projective curve over a finite field, both of them generalizing the Artin zeta function of the curve. One of these zeta functions is defined geometrically in terms of semistable vector bundles of rank n over the curve and the other one group-theoretically in terms of certain periods associated to the curve and to a split reductive group G and its maximal parabolic subgroup P. It was conjectured that these two zeta functions coincide in the special case when G = SLn and P is the parabolic subgroup consisting of matrices whose final row vanishes except for its last entry. In this paper we prove this equality by giving an explicit inductive calculation of the group-theoretically defined zeta functions in terms of the original Artin zeta function (corresponding to n = 1) and then verifying that the result obtained agrees with the inductive determination of the geometrically defined zeta functions found by Sergey Mozgovoy and Markus Reineke in 2014.. |

8. | Lin Weng, Don Zagier, Higher-rank zeta functions for elliptic curves, Proceedings of the National Academy of Sciences of the United States of America, 10.1073/pnas.1912023117, 117, 9, 4546-4558, 2020.03, In earlier work by L.W., a nonabelian zeta function was defined for any smooth curve X over a finite field Fq and any integer n ≥ 1 by ζ_{X}/_{Fq,n}(s) = ^{X |H0(X, V})r{0^{}|} q^{−}deg(V)^{s} ((s) > 1), |Aut(V)| [V] where the sum is over isomorphism classes of Fq-rational semistable vector bundles V of rank n on X with degree divisible by n. This function, which agrees with the usual Artin zeta function of X/Fq if n = 1, is a rational function of q^{−s} with denominator (1 − q^{−ns})(1 − q^{n}−^{ns}) and conjecturally satisfies the Riemann hypothesis. In this paper we study the case of genus 1 curves in detail. We show that in that case the Dirichlet series 1 X/_{Fq}(s) = ^{X} _{[}V_{] |}Aut(V)_{|} q^{−}rank(V)^{s} ((s) > 0), where the sum is now over isomorphism classes of Fq-rational semistable vector bundles V of degree 0 on X, is equal to ^{Q∞} _{k}=_{1} ζ_{X}/_{Fq}(s + k), and use this fact to prove the Riemann hypothesis for ζ_{X} _{,n}(s) for all n.. |

9. | ＃Lin WENG（九大の教員),
他＠Don Zagier, Higher-rank zeta functions and $SL_n$-zeta functions for curves, 米国科学アカデミー紀要(PNAS), PNAS March 24, 2020 117 (12), PNAS March 24, 2020 117 (12), 6398-6408, 2020.03, [URL], In earlier papers L.Weng introduced two sequences of higher-rank zeta functions associated to a smooth projective curve over a finitefield, both of them generalizing the Artin zeta function of the curve. One of these zeta functions is defined geometrically in terms ofsemistable vector bundles of rank n over the curve, and the other one group-theoretically in terms of certain periods associated to thecurve and to a split reductive group G and its maximal parabolic subgroup P. It was conjectured that these two zeta functions coincidein the special case when G = $SL_n$ and $P$ is the parabolic subgroup consisting of matrices whose final row vanishes except for its lastentry. In this paper we prove this equality by giving an explicit inductive calculation of the group-theoretically defined zeta functions interms of the original Artin zeta function (corresponding to n = 1) and then verifying that the result obtained agrees with the inductivedetermination of the geometrically defined zeta functions found by Sergey Mozgovoy and Markus Reineke in 2014.. |

10. | L. Weng, D. Zagier, Higher rank zeta functions for elliptic curves, 米国科学アカデミー紀要(PNAS), PNAS March 3, 2020 117 (9), PNAS March 3, 2020 117 (9), 4546-4558, 2020.02, [URL], The rank n non-abelian zeta functions for curves over finite fields was defined by the first author in 2005, using moduli stacks of rank n semi-stable vector bundles on curves. These zeta functions satisfy the standard zeta properties such as the rationality, the functional equation and are conjectured to satisfy the Riemann Hypothesis. In particular, when n=1, these zeta functions coincide with the famous Artin zeta functions.In this paper, we prove the Riemann hypothesis for non-abelian zeta functions of elliptic curves.. |

11. | LIN WENG, Parabolic reduction, stability and the mass I: Special Linear Groups, RIMS Kôkyûroku 1826, 168-179, 2013.03. |

12. | Indranil Biswas, Georg Schumacher, Lin Weng, Deligne pairing and determinant bundle, Electronic Research Announcements in Mathematical Sciences, 10.3934/era.2011.18.91, 18, 91-96, 2011.12, Let X → S be a smooth projective surjective morphism, where X and S are integral schemes over ℂ. Let L_{0},L_{1}, · · ·,L_{n-1},L_{n} be line bun- dles over X. There is a natural isomorphism of the Deligne pairing 〈L_{0}, · · ·,L_{n}〉 with the determinant line bundle Det(⊕ n/i=0(L_{i} - O_{X})).. |

13. | Lin WENG, Stability and arithmetic: an extract of essence, Algebraic number theory and related topics 2008, 187–220, RIMS Kôkyûroku Bessatsu, B19, Res. Inst. Math. Sci. (RIMS), Kyoto, 2010, 187-220, 2011.10. |

14. | Lin WENG, Symmetries and the Riemann Hypothesis, Advanced Studies in Pure Mathematics, 58, 173-223, 2010.06. |

15. | Lin WENG, Stability and Arithmetic, Advanced Studies in Pure Mathematics , 58, 225-359, 2010.06. |

16. | M. Suzuki and L. Weng, Zeta Functions for G_2 and Their Zeros, International Mathematics Research Notice (IMRN), 2009, 241-290, 2009.01. |

17. | Lin WENG, Zeta Functions for Sp(2n), appendix to The Riemann hypothesis for Weng's zeta function of Sp(4) over Q by M. Suzuki, Journal of Number Theory, 129 (3) (2009), 569-579, 2009.01. |

18. | Kunio Obitsu, Wing Keung To, Lin Weng, The asymptotic behavior of the Takhtajan-Zograf metric, Communications in Mathematical Physics, 10.1007/s00220-008-0520-7, 284, 1, 227-261, 2008.11, We obtain the asymptotic behavior of the Takhtajan-Zograf metric on the Teichmüller space of punctured Riemann surfaces.. |

19. | L. Weng, D. Zagier, Deligne products of line bundles over moduli spaces of curves, Communications in Mathematical Physics, 10.1007/s00220-008-0494-5, 281, 3, 793-803, 2008.08, We study Deligne products for forgetful maps between moduli spaces of marked curves by offering a closed formula for tautological line bundles associated to marked points. In particular, we show that the Deligne products for line bundles on the total spaces corresponding to "forgotten" marked points are positive integral multiples of the Weil-Petersson bundles on the base moduli spaces.. |

20. | K. Obitsu, W.-K. To and L. Weng, The asymptotic behavior of the Takhtajan-Zograf metric, Comm. Math. Phys, 284 (2008), no. 1, 227--261, 2008.11. |

21. | L. Weng and D. Zagier, Deligne Products of line Bundles over Moduli Spaces of Curves, Comm of Math Phys, 281, 793-803, 2008.08, [URL]. |

22. | Henry H. Kim, Lin Weng, Volume of truncated fundamental domains, Proceedings of the American Mathematical Society, 10.1090/S0002-9939-07-08784-9, 135, 6, 1681-1688, 2007.06. |

23. | Kim, Henry H.; Weng, Lin, Volumes of Truncated Fundamental Domains, Proc. of AMS, 135, no.6, 1681-1688, 2007.05, [URL]. |

24. | Weng, Lin, A Geometric Approach to L-Functions, The Conference on L-Functions, 219-370, 2007.02, [URL]. |

25. | Weng, Lin, A Rank Two Zeta and Its Zeros, J. Ramanujan Math. Soc., 21, no.3, 205-266, 2006.10, [URL]. |

26. | Weng, Lin, Geometric Arithmetic: A Program, Arithmetic Geometry and Number Theory, 211-400, 2006.08, [URL]. |

27. | Weng, Lin, Non-Abelian Zeta Functions for Function Fields, Amer. J. Math., 127, 5, 973-1017, 127(2005), 973-1017, 2005.10, [URL]. |

28. | Lin Weng, Non-abelian zeta functions for function fields, American Journal of Mathematics, 10.1353/ajm.2005.0035, 127, 5, 973-1017, 2005.10, In this paper we initiate a geometrically oriented construction of non-abelian zeta functions for curves defined over finite fields. More precisely, We first introduce new yet genuine non-abelian zeta functions for curves defined over finite fields, by a "weighted count" on rational points over the corresponding moduli spaces of semi-stable vector bundles using moduli interpretation of these points. Then we define non-abelian L-functions for curves over finite fields using integrations of Eisenstein series associated to L^{2}-automorphic forms over certain generalized moduli spaces.. |

29. | To, Wing-Keung; Weng, Lin, L2-Metrics, Projective Flatness and Families of Polarized Abelian Varieties, Trans. Amer. Math. Soc., 10.1090/S0002-9947-03-03488-3, 356, 7, 2685-2707, 356, no. 7, 2685--2707, 2004.03, [URL]. |

30. | Weng, Lin, Refined Brill-Noether Locus Non-Abelian Zeta Functions for Elliptic Curves, Algebraic geometry in East Asia, 245--262, World Sci. Publishing, River Edge, NJ, 2002., 2002.10. |

31. | Matsumoto, K.; Weng, Lin, Zeta Functions defined by Two Polynomials, Number Theoretic Methods, 233--262, Dev. Math., 8, Kluwer Acad. Publ., 2002.10. |

32. | Weng, Lin, Omega-Admissible Theory (II), Math. Ann., 320, no. 2, 239--283., 2001.10, [URL]. |

33. | Lin Weng, Ω-admissible theory II. Deligne pairings over moduli spaces of punctured Riemann surfaces, Mathematische Annalen, 10.1007/PL00004473, 320, 2, 239-283, 2001.01, In Part I, Deligne-Riemann-Roch isometry is generalized for punctured Riemann surfaces equipped with quasi-hyperbolic metrics. This is achieved by proving the Mean Value Lemmas, which explicitly explain how metrized Deligne pairings for ω-admissible metrized line bundles depend on ω. In Part II, we first introduce several line bundles over Knudsen-Deligne-Mumford compactification of the moduli space (or rather the algebraic stack) of stable N-pointed algebraic curves of genus g, which are rather natural and include Weil-Petersson, Takhtajan-Zograf and logarithmic Mumford line bundles. Then we use Deligne-Riemann-Roch isomorphism and its metrized version (proved in Part I) to establish some fundamental relations among these line bundles. Finally, we compute first Chern forms of the metrized Weil-Petersson, Takhtajan-Zograf and logarithmic Mumford line bundles by using results of Wolpert and Takhtajan-Zograf, and show that the so-called Takhtajan-Zograf metric on the moduli space is algebraic.. |

34. | Wing Keung To, Lin Weng, Admissible Hermitian metrics on families of line bundles over certain degenerating Riemann surfaces, Pacific Journal of Mathematics, 10.2140/pjm.2001.197.441, 197, 2, 441-489, 2001.02, We show that a family of line bundles of degree zero over a plumbing family of Riemann surfaces with a separating (resp. non-separating) node p admits a nice (resp. almost nice) family of flat p-singular Hermitian metrics. As a consequence, we give necessary and sufficient conditions for a family of line bundles over such families of Riemann surfaces to admit an (almost) nice family of p-singular Hermitian metrics which are admissible with respect to the canonical/hyperbolic (1,1)-forms on the Riemann surfaces.. |

35. | Wing Keung To, Lin Weng, Green's Functions for Quasi-Hyperbolic Metrics on Degenerating Riemann Surfaces with a Separating Node, Annals of Global Analysis and Geometry, 10.1023/A:1006506623667, 17, 3, 239-265, 1999.01, In this article, we consider a family of compact Riemann surfaces of genus q ≥ 2 degenerating to a Riemann surface with a separating node and many non-separating nodes. We obtain the asymptotic behavior of Green's functions associated to a continuous family of quasi-hyperbolic metrics on such degenerating Riemann surfaces.. |

36. | Lin Weng, Ω-Admissible theory, Proceedings of the London Mathematical Society, 10.1112/S0024611599011995, 79, 3, 481-510, 1999.01. |

37. | Wing Keung To, Lin Weng, Curvature of the L^{2}-metric on the direct image of a family of Hermitian-Einstein vector bundles, American Journal of Mathematics, 120, 3, 649-661, 1998.06, For a holomorphic family of simple Hermitian-Einstein holomorphic vector bundles over a compact Kähler manifold, the locally free part of the associated direct image sheaf over the parameter space forms a holomorphic vector bundle, and it is endowed with a Hermitian metric given by the L^{2} pairing using the Hermitian-Einstein metrics. Our main result in this paper is to compute the curvature of the L^{2}-metric. In the case of a family of Hermitian holomorphic line bundles with fixed positive first Chern form and under certain curvature conditions, we show that the L^{2}-metric is conformally equivalent to a Hermitian-Einstein metric. As applications, this proves the semi-stability of certain Picard bundles, and it leads to an alternative proof of a theorem of Kempf.. |

38. | To, Wing-Keung; Weng, Lin, Admissible Hermitian Metrics on Families of Line Bundles over Certain Degenerating Riemann Surfaces, Pacific J. Math., 197, no. 2, 441--489., 2001.08. |

39. | Weng, Lin, Omega Admissible Theory, Proc. London Math. Soc., 79(3), 481-510, 1999.08. |

40. | To, Wing-Keung; Weng, Lin, Green's Functions for Quasi-Hyperbolic Metrics on Degenerating Riemann Surfaces with a Separating Node, Ann. Global Anal. Geo., 17, no.3, 239-265, 1999.12. |

41. | Weng, Lin; You, Yuching, Standard Modules of Level l for SL2 in terms of Virasoro Algebra Representations, Comm. Algebra, 26, no.2, 613-625, 1998.08. |

42. | To, Wing-Keung; Weng, Lin, Curvature of the L2-Metric on the Direct Image of a Family of Hermitian-Einstein Vector Bundles, Amer. J. Math., 120 (1998), 649-661, 1998.06. |

43. | To, Wing-Keung; Weng, Lin, The Asymptotic Behavior of Green's Functions for Quasi-Hyperbolic Metrics on Degenerating Riemann Surfaces, Manuscripta Math., 93, no.4, 465-480, 1997.12. |

44. | Weng, Lin; You, Yuching, Analytic Torsions of Spheres, Internat. J. Math., 7, no.1, 109-125, 1996.02. |

45. | Weng, Lin, A Result on Bicanonical Maps of Surfaces of General Type, Osaka J. Math., 32, no.2, 467-473, 1995.06. |

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