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Zeta Functions 


Zeta Functions of Reductive Groups and Their Zeros L. WENG 
Contents Part 1. NonAbelian Zeta Functions Part 2. Rank Two Zeta Functions Part 3. Eisenstein Periods and Multiple Zeta Functions Part 4. Zeta Functions for Reductive Groups Part 5. Algebraic, Analytic Structures and the Riemann Hypothesis Part 6. Geometric Structures and the Riemann Hypothesis Part 7. Five Essays on Arithmetic Cohomology (with K. Sugahara) To be published in Feb 2018 

HIGHER RANK ZETA FUNCTIONS FOR ELLIPTIC CURVES L. WENG and D. Zagier 
Nonabelian zeta function is defined for any smooth curve $X$ over a finite field $\mathbb F_q$ and any integer $n\ge 1$ by $$ \zeta_{X/\mathbb F_q,n}(s) \,=\, \sum_{[\mathcal E]}\frac{H^0(X,\mathcal E)\!\smallsetminus\!\{0\}} {\mathrm{Aut}(\mathcal E)}\;q^{\text{deg}(\mathcal E)s} \qquad(\Re(s)>1)\,,$$ where the sum is over moduli stack of $\mathbb F_q$rational semistable vector bundles $\mathcal E$ of rank $n$ on $X$ with degree divisible by $n$. This function, which agrees with the usual Artin zeta function of $X/\mathbb F_q$ if $n=1$, is a rational function of $q^{s}$ with denominator $(1q^{ns})(1q^{nns})$ 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 $$\mathcal Z_{X/\mathbb F_q}(s) \,=\, \sum_{[\mathcal E]}\frac1{\mathrm{Aut}(\mathcal E)}\;q^{\text{rank}(\mathcal E)s} \qquad(\Re(s)>0)\,,$$ where the sum is now over isomorphism classes of $\mathbb F_q$rational semistable vector bundles $\mathcal E$ of degree 0 on $X$, is equal to $\prod_{k=1}^\infty\zeta_{X/\mathbb F_q}(s+k)\,$, and use this fact to deduce the validity of the Riemann hypothesis for $\zeta_{X,n}(s)$ for all $n$.  
HIGHER RANK ZETA FUNCTIONS AND $\mathrm{SL}_n$ZETA FUNCTIONS FOR CURVES L. WENG and D. Zagier 
A different approach to zeta functions for curves leads to the socalled group zeta function ${\widehat\zeta}_X^{G/P}(s)$ associated to a connected split algebraic reductive group $G$ and its maximal parabolic subgroup $P$. We will be interested in the $SL_n$zeta ${\widehat\zeta}_X^{\mathrm{SL}_n}(s):={\widehat\zeta}_X^{SL_n/P_{n1,1}}(s)$. Our main result will be a proof of the following theorem on the conjecture of special uniformity of zetas: $$\widehat\zeta_{X,n}(s)\,=\,{\widehat\zeta}_X^{\mathrm{SL}_n}(s).$$ This theorem should be seen as a joint result of the present authors and of Mozgovoy and Reineke, because it is proved by comparing a formula established here for the $\mathrm{SL}_n$zeta with a formula given in their work for rank $n$ nonabelian zeta using Hall algebra and wallcrossing.  
Zeros of Zeta Functions for Exceptional Groups of Type $E$ Lin WENG 
A selection of 128 elements form Weyl groups for exceptional groups of type $E_8$ with 696,729,600 elements is given. As a direct consequence, we then prove the weak Riemann Hypothesis for the socalled Weng zeta functions of $E_8$. Moreover, we explain what is the meaning of these $128\,(=2^7)$ elements in terms of the masses of semistable principal $E_8$lattices, following our earlier conjecture on Stability, Parabolic Reduction, and the Masses.
To be appeared as Chapter 13 of Part 5 in Zeta Functions of Reductive Groups and Their Zeros 

Distributions of Zeros for NonAbelian Zeta Functions Lin WENG 
Two levels of fine structures on distributions of nonabelian zeta zeros are exposed. For
pair correlations of zeros for rank $n$ nonabelian zeta $\widehat\zeta_{\mathbb Q,n}(s)$
$\ (n\geq 2)$, we have, for $\gamma_{n,k}$ the $k$th zero,
$$
\gamma_{n,k}=\frac{2\,\pi}{n}\frac{k}{\log k}\big(1+O\big(\frac{1}{\log k}\big)\big),\ \ \qquad
\delta_{n,k}=1+O\big(\frac{1}{\log k}\big),
$$
where
$
\delta_{n,k}
:=\big(\frac{n}{2\,\pi}(\gamma_{n,k+1}\gamma_{n,k})\big)\cdot \log \big(\frac{n}{2\,\pi}\gamma_{n,k}\big).
$
For GUE, we introduce
$$
\Delta_{n,k}:=\big(\delta_{n,k}1 \big)\cdot\log\big(\frac{n}{2\,\pi}\gamma_{n,k}\big),
$$
and conjecture that distributions of these big Delta are closely related with GUE.
Supportive evidences from numerical calculations are provided. Also treated are
zeta functions for reductive groups and their maximal parabolic subgroups.
Paper with pictures.
To be appeared as Chapter 18 of Part 6 in Zeta Functions of Reductive Groups and Their Zeros In memory of Gang XIAO. 

Motivic Euler Product and Its Aplications Lin WENG 
We begin with a construction of nonabelian motivic zeta functions for curves over any base field, using moduli stacks of semistable bundles. As an application, we define motivic Euler products. Then, we introduce genuine zeta functions for Riemann surfaces and establish their convergences, based on the theory of RaySinger analytic torsions. To understand common features of these zetas, we next introduce natural motivic measures for the associated adelic spaces and hence obtain a motivic SiegelWeil formula for the total mass of $\mathcal G$torsors in terms of special values of motivic zetas, using the newly defined motivic Euler product. Moreover, we, using parabolic reduction and stability, obtain natural decompositions for moduli stacks of $\mathcal G$torsors, and prove the parabolic reduction, stability and the mass conjecture for $\mathcal G$torsors relating the total mass and the semistable masses. Finally, with AtiyahBott's analogue between Riemann surfaces and curves over finite fields and the conformal field theory in mind, we conjecture that our analytic zeta functions for Riemann surfaces are motivic, and hence unify our algebraic and analytic zetas. 
Arithmetic Cohomology Theory 


Arithmetic central extensions and tame reciprocity laws for arithmetic surfaces K. Sugahara and L. WENG 
About 50 years ago, Tate developed a theory of residues for curves using traces and adelic cohomologies. In 1989, Tate's work was integrated with the $K_2$ central extensions by ArbarelloDe ConciniKac. Later, in 2003, Osipov, based on Kapranov's dimension theory, constructed dimension two central extensions and hence established the reciprocity law for algebraic surfaces using Parshin's adelic theory. In essence, Osipov's construction may be viewed as $K_2$ type theory of central extensions developed by BrylinskiDeligne. However, for arithmetic surfaces, $K_2$ theory does not work. Instead, we first develop a new theory of central extensions based on Arakelov theory, then we use our adelic cohomology theory to establish the reciprocity law for arithmetic surfaces. Reciprocity Law version, Reciprocty Law+Central Extension version To be appeared as Appendix E in Zeta Functions of Reductive Groups and Their Zeros 

Arithmetic Cohomology Groups K. Sugahara and L. WENG 
We first introduce global arithmetic cohomology groups for quasicoherent sheaves on arithmetic varieties, adopting an adelic approach. Then, we establish fundamental properties, such as topological duality and inductive long exact sequences, for these cohomology groups on arithmetic surfaces. Finally, we expose basic structures for indpro topologies on adelic spaces of arithmetic surfaces. In particular, we show that these adelic spaces are topologically selfdual. This is a revised version dated on 27 November, 2014. Newly developed is a theory of indpro topologies over adelic spaces for arithmetic surfaces. To be appeared as Appendix A in Zeta Functions of Reductive Groups and Their Zeros 