This web page contains links to some of my papers, sorted by topic in reverse chronological order. Provided are also brief descriptions of these papers.
Geometric Arithmetic 


Stability and Arithmetic
Lin WENG Algebraic and Arithmetic Structures of Moduli Spaces Advanced Studies in Pure Mathematics 58 Math. Soc. Japan, (2010), 225359. 
Stability plays a central role in arithmetic.
In this article, we explain some basic ideas and present certain constructions for such studies.
It consists of two parts:
(I) A general Class Field Theory for both Riemann surfaces using semistable parabolic bundles and for padic number fields using what we call semistable filtered $(\varphi,N;\omega)$modules; and (II) Nonabelian zeta functions for both function fields over finite fields using semistable bundles and for number fields using semistable lattices. 

Geometric Arithmetic: A Program
Lin WENG Arithmetic Geometry and Number Theory World Sci., (2006), 211400 
We originate a program for what I call Geometric Arithmetic. Such a
program would consist of four parts, if I were able to properly
understand the essentials now. Namely, (1) NonAbelian Class
Field Theory; (2) GeoAri Cohomology Theory; (3) New NonAbelian
Zeta Functions; and (4) Riemann Hypothesis. However, here I
could only provide the reader with [1+(1/2+1/2)+1]/4 of them. To be
more precise, discussed in this article are the following particulars:
(A) Representation of Galois Group, Stability and Tannakian Category; (B) Moduli Spaces, RiemannRoch, and New NonAbelian Zeta Function; and (C) Explicit Formula, Functional Equation and GeoAri Intersection. 
Zeta Functions 


Symmetries and the Riemann Hypothesis
Lin WENG Algebraic and Arithmetic Structures of Moduli Spaces Advanced Studies in Pure Mathematics 58 Math. Soc. Japan, (2010), 173223 
Associated to reductive groups and their maximal parabolic subgroups are genuine zeta functions. Naturally related to Riemann's zeta function and governed by symmetries, including that of Weyl, these zeta functions are expected to satisfy the Riemann hypothesis.
\[\]\({\bullet\ \ \color{Red}{\mathrm{Zetas\ for} (G,P)/\mathbb Q}}\) \(\displaystyle{ \widehat\zeta^{G/P}_{\mathbb Q}(s):= {\mathrm{Norm}}( {\mathrm{Res}}_{{\langle \lambda\rho,\beta_j^\vee\rangle=0, 1\leq j\leq P}}\,( \sum_{w\in W} \frac{e^{\langle \lambda,T\rangle}} {\prod_{\alpha\in\Delta}\langle \lambda\rho,\alpha^\vee\rangle}\cdot \prod_{\alpha>0, w\alpha<0} \frac{\widehat\zeta(\langle\lambda,\alpha^\vee\rangle)} {\widehat\zeta(\langle\lambda,\alpha^\vee\rangle+1)} ))}\) \({\bullet\ \ \color{Blue}{\mathrm{Functional\ Equation:}} \qquad \boxed{\exists\qquad {constant}\quad c_{G/P},\quad \mathrm{s.t.} \qquad \widehat\zeta^{G/P}_{\mathbb Q}(c_{P/Q}s)= \widehat\zeta^{G/P}_{\mathbb Q}(s)}}\) \({\bullet\ \ \color{Blue}{\mathrm{Riemann\ Hypothesis:}} \qquad\boxed{\widehat\zeta^{G/P}_{\mathbb Q}(s)=0\qquad\Rightarrow\qquad\mathrm{Re}(s)=\frac{c_{G/P}}{2}}}\) 

Zeta Functions for $\small{G_2}$ and Their Zeros
Masatoshi SUZUKI, Lin WENG International Mathematics Research Notice 2009 (2009), 241290 
The exceptional group $\small{G_2}$ has two maximal parabolic subgroups $\small{P_{\mathrm{long}}}$ and $\small{P_{\mathrm{short}}}$ corresponding to the socalled long root and short root. In this paper, the second named author introduces two zeta functions associated with $\small{P_{\mathrm{long}}}$ and $\small{P_{\mathrm{short}}}$ respectively, and the first named author proves that these zetas satisfy the Riemann hypothesis.  
Zeta Functions for $\small{Sp(2n)}$
Lin WENG appendix to: The Riemann hypothesis for Weng's zeta function of $\small{Sp(4)}$ over $\small{\mathbb Q}$, $\quad$ Masatoshi SUZUKI Journal of Number Theory, 129(2009), 569579 
Recently, Weng defined more general new abelian zeta function associated to a pair of a semisimple reductive algebraic group and its maximal parabolic subgroup. This new zeta function is expected to satisfy standard properties of zeta functions. In this paper, we prove that the Riemann hypothesis for Weng's zeta function attached to the symplectic group of degree four. This paper includes an appendix written by L. Weng, in which he explains a general construction for zeta functions associated to $\small{Sp(2n)}$.  
Geometric Approach to LFunctions
Lin WENG The Conference on LFunctions World Sci., (2007), 219370 
We initiate a geometric approach to the theory of Lfunctions. It consists of two parts: Chapters 14, used to be called 'NonAbelian LFunctions for Number Fields', reveal a general theory, and Chapters 59 give a detailed account of ranks two and three zetas. In particular, as a generalization of the Dedekind zeta function, Weng defined the high rank zeta functions and proved that they have standard properties of zeta functions, namely, meromorphic continuation, functional equation, and having only two simple poles. The rank one zeta function is the Dedekind zeta function.  
A Rank Two Zeta and its Zeros
Lin WENG J. Ramanujan Math. Soc., 21(2006), 205266

In this paper, we first reveal an intrinsic relation between nonabelian zeta functions and Epstein zeta functions for algebraic number fields. Then, we expose a fundamental relation between stability of lattices and distance to cusps. Next, using these two relations, we explicitly express rank two zeta functions in terms of the wellknown Dedekind zeta functions. Finally, based on such an expression, we show that all zeros of rank two nonabelian zeta functions are entirely sitting on the critical line whose real part equals to 1/2. This is an integrated part of our GeoArithmetic Program.  
NonAbelian Zeta Functions for Function Fields
Lin WENG Amer. J. Math., 127(2005), 9731017 
In this paper we initiate a geometrically oriented construction of nonabelian zeta functions for curves defined over finite fields. More precisely, we first introduce new yet genuine nonabelian zeta functions for curves defined over finite fields, by a "weighted count" on rational points over the corresponding moduli spaces of semistable vector bundles using moduli interpretation of these points. Then we define nonabelian Lfunctions for curves over finite fields using integrations of Eisenstein series associated to $\small{L^2}$automorphic forms over certain generalized moduli spaces. 
Arakelov Theory 


Deligne Products of line Bundles over Moduli Spaces of Curves
Lin WENG, Don ZAGIER Comm. Math. Phys., 281 (2008), 793803 
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 WeilPetersson bundles on the base moduli spaces.  
OmegaAdmissible Theory II: Deligne pairings over
Moduli spaces of punctured Rimann surfaces Lin WENG Math. Ann., 320 (2001), 239283 
For singular metrics, there is no Quillen metric formalism on cohomology determinant. In this paper, we develop an admissible theory, with which the arithmetic DeligneRiemannRoch isometry can be established for singular metrics. As an application, we first study WeilPetersson metrics and TakhtajanZograf metrics on moduli spaces of punctured Riemann surfaces, and then give a more geometric interpretation of our determinant metrics in terms of Selberg zeta functions. We end this paper by proposing an arithmetic factorization for WeilPetersson metrics, cuspidal metrics and Selberg zeta functions.  
OmegaAdmissible Theory
Lin WENG Proc. London Math. Soc., 79 (1999), 481510 
Arakelov and Faltings developed an admissible theory on regular arithmetic surfaces by using Arakelov canonical volume forms on the associated Riemann surfaces. Such volume forms are induced from the associated Kaehler forms of the flat metric on the corresponding Jacobians. So this admissible theory is in the nature of Euclidean geometry, and hence is not quite compatible with the moduli theory of Riemann surfaces. In this paper, we develop a general admissible theory for arithmetic surfaces (associated with stable curves) with respect to any volume form. In particular, we have a theory of arithmetic surfaces in the nature of hyperbolic geometry by using hyperbolic volume forms on the associated Riemann surfaces. Our theory is proved to be useful as well: we have a very natural Weil function on the moduli space of Riemann surfaces, and show that in order to solve the arithmetic BogomolovMiyaokaYau inequality, it is sufficient to give an estimation for Petersson norms of some modular forms.  
Relative Botthern Secondary Characteristic Classes and Arithmetic GrothendieckRiemannRoch Theorem I, II
Lin WENG Preprints of MaxPlanck Institute for Mathematics, 4732(9179), 4986(9451), 4987(9452)(1991, 1994, 1994), pp.24, 261, 206

In these two volumes (and an original short announcement), we introduce six axioms for relative BottChern secondary characteristic classes and prove the uniqueness and existence theorem for them. Such a work provides us a natural way to understand and hence to prove the arithmetic GrothendieckRiemannRoch theorem.  
Hyperbolic Metrics, Selberg Zeta Functions and
Arakelov Theory for Punctured Riemann Surfaces Lin WENG Lecture Note Series in Mathematics Osaka Univ. 6(1998) 
Hyperbolic Metrics, Selberg Zeta Functions and Arakelov Theory for Punctured Riemann Surfaces 
Geometry 


The Asymptotic Behavior of the TakhtajanZograf Metric
Kunio OBITSU, WingKeung TO, Lin WENG Comm. Math. Phys., 284 (2008), 227261 
We consider a universal degenerating family of punctured hyperbolic Riemann surfaces of signature (g,N). In this paper, we obtain the asymptotic behavior of the TakhtajanZograf metric on the Teichmueller space of punctured Riemann surfaces.  
$\small{L^2}$Metrics, Projective Flatness and Families of Polarized Abelian Varieties
WingKeung To, Lin WENG Trans. Amer. Math. Soc., 356 (2004), 26852707 
We compute the curvature of the $\small{L^2}$metric on the direct image of a family of Hermitian holomorphic vector bundles over a family of compact Kaehler manifolds. As an application, we show that the $\small{L^2}$metric on the direct image of a family of ample line bundles over a family of abelian varieties and equipped with a family of canonical Hermitian metrics is always projectively flat. When the parameter space is a compact Kaehler manifold, this leads to the polystability of the direct image with respect to any Kaehler form on the parameter space.  
Admissible Hermitian Metrics on Families of Line Bundles over Certain Degenerating Riemann Surfaces
WingKeung TO, Lin WENG Pacific J. Math., 197 (2001), 441489 
We show that a family of line bundles of degree zero over a plumbing family of Riemann surfaces with a separating (resp. nonseparating) node p admits a nice (resp. almost nice) family of flat psingular 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 psingular Hermitian metrics which are admissible with respect to the canonical/hyperbolic (1,1) forms onthe Riemannsurfaces.  
Green's Functions for QuasiHyperbolic Metrics on Degenerating Riemann Surfaces with a Separating Node
WingKeung TO, Lin WENG Ann. Global Anal. Geo., 17 (1999), 239265 
We consider a family of compact Riemann surfaces of genus q>1 degenerating to a Riemann surface with a separating node and many nonseparating nodes. We obtain the asymptotic behavior of Green's functions associated to a continuous family of quasihyperbolic metrics on such degenerating Riemann surfaces.  
Curvature of the $\small{L^2}$Metric on the Direct Image of a Family of HermitianEinstein Vector Bundles
WingKeung TO, Lin WENG Amer. J. Math., 120 (1998), 649661 
For a holomorphic family of simple HermitianEinstein holomorphic vector bundles over a compact Kaehler 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 $\small{L^2}$ pairing using the HermitianEinstein metrics. Our main result in this paper is to compute the curvature of the $\small{L^2}$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 $\small{L^2}$metric is conformally equivalent to a HermitianEinstein metric. As applications, this proves the semistability of certain Picard bundles, and it leads to an alternative proof of a theorem of Kempf.  
The Asymptotic Behavior of Green's Functions for QuasiHyperbolic Metrics on Degenerating Riemann Surfaces
WingKeung TO, Lin WENG Manuscripta Math., 93 (1997), 465480 
We consider a family of compact Riemann surfaces of genus g>1 degenerating to a Riemann surface of genus g1 with a nonseparating node. We show that the Green's function associated to a continuous family of quasihyperbolic metrics on such degenerating Riemann surfaces simply degenerate to that on the smooth part of the noded Riemann surfaces.  
Analytic Torions of Spheres
Lin WENG, Yuching YOU International J. Math., 93 (1997), 465480 
The main result of this paper is to give the analytic torisons
of unit spheres in
Euclidean spaces with the standard Riemannian metric: \[\]\(Analytic\ torsion\ for\ the\ sphere\ {\small{{\mathbb S}^{2m1}}}\ in\ the\ standard\ Euclidean\) \(space\ {\small{{\mathbb R}^{2m}}}\ is\ given\ by\) \[\small{\frac{2\pi^m}{(m1)!}=\frac{1}{2^{m1}(m1)!}\cdot e^{2m\zeta'(0)}}\] 
Classical Algebraic Geometry 


A Result on Bicanonical Maps of Surfaces of General Type
Lin WENG Osaka J. Math., 32 (1995), 467473 
\(\mathrm{Pluricanonical\ maps\ of\ surfaces\ of\ general\ type\ are\ left\ open\ only}\) \(\mathrm{for\ bicanonical\ maps\ with\ small}\ K^2(\leq 4), \mathrm{and\ for\ canonical\ maps.}\) \(\mathrm{For\ bicanonical\ maps,\ the\ nontrivial\ cases\ are\ these\ surfaces}\) \(\mathrm{with}\ p_g= 0\ \mathrm{and}\ K^2= 3,4.\ \mathrm{Our\ result\ is:}\) \[\] \({Let}\ S{\ be\ a\ minimal\ surface\ of\ general\ type\ with}\ p_g(S) = 0\ {and}\) \( K^2_S= 3{\ or\ 4.\ Then\ any}\ (2)curve\ in\ S \ \mathrm{cannot}\ be\ an\ irreducible\) \(component\ of\ the\ fixed\ part\ of\ the\ bicanonical\ {map}\ \phi_{2K_S.}\) 
Others 


Automorphic Forms, Eisenstein Series and Spectral Decomposition
Lin WENG Arithmetic Geometry and Number Theory World Sci., (2006), 123210 
This note is prepared for the reader who wants to learn Langlands' fundamental
results about Eisenstein series and spectral decompositions, say using Moeglin
and Waldspurger' Cambridge tract. It results from my six lectures given at a
special seminar on Automorphic Forms and Eisenstein Series at Department of
Mathematics, University of Toronto in 2005. Day One: Basics of Automorphic Forms Day Two: Eisenstein Series Day Three: PseudoEisenstein Series Day Four: Spectrum Decomposition (I): Residual Process Day Five: Eisenstein Systems and Spectral Decomposition (II) Day Six: Arthur's Truncation and Meromorphic Continuation 

Standard Modules of Level l for $\small{\widehat{sl_2}}$ in terms of
Virasoro Algebra Representations Lin WENG, Yuching YOU Comm. Alg., 26 (1998), 613625 
We give the irreducible decomposition of the level one integrable highest weight modules of $\small{\widehat{sl_2}}$ associated to $\small{sl_2({\mathbb C})}$ for the action of the associated Virasoro algebra. In fact, using a realization of the modules, we will explicitly construct the highest weight vectors for the Virasoro algebra. As a byproduct, we obtain the norm of these highest weight vectors for the inner product introduced by Garland.  