This is an integrated part of our Geo-Arithmetic Program.
In this paper we initiate a geometrically oriented construction of non-abelian zeta functions for curves defined over finite fields by a weighted count of semi-stable bundles. Basic properties such as rationality and functional equation are established. Examples of rank two zetas over genus two curves are given as well. Based on this and motivated by our study for non-abelian zetas of number fields, general non-abelian $L$ functions for function fields are defined and studied using Langlands and Morris' theory of Eisenstein series.
This paper gives a natural extension of Frobenius-Stickelberger formula and Kiepert formula to Abelian functions for "Purely Trigonal Curves", especially, of degree four. A description on the theory of Abelian functions for general trigonal curves of degree four is also included.
Among abelian extensions of a congruence function field, an asymptotic relation of class number and genus is established. The proof is classical, employing well-known results from congruence function field theory.
We survey Colmez's theory and conjecture about the Faltings height and a product formula for the periods of abelian varieties with complex multiplication, along with the function field analog developed by the authors. In this analog, abelian varieties are replaced by Drinfeld modules and $A$-motives. We also explain the necessary background on abelian varieties, Drinfeld modules and $A$-motives, including their cohomology theories and comparison isomorphisms and their theory of complex multiplication.
We first study geometrically oriented truncation associated with stability along the line of Arthur's analytic truncation. Then, we give a detailed discussion on the so-called Abelian Parts of non-abelian L functions, using an advanced version of Rankin-Selberg method. All this is based on Jacquet-Lapid-Rogawski's regularized integrals over cones. This is an integrated part of our Program for Geometric Arithmetic.
Let $\cac$ be a smooth projective curve defined over a number field $k$, $A/k(\cac)$ an abelian variety and $(τ,B)$ the $k(\cac)/k$-trace of $A$. We estimate how the rank of $A(k(\cac))/τB(k)$ varies when we take a finite cover $π:\cac'\to\cac$ defined over $k$ geometrically abelian.
Our goal in this note is to give a number of examples of abelian varieties over function fields k(t) which have bounded ranks in towers of extensions such as k(t^{1/d}) for varying d. Along the way we prove some new results on Fermat curves which may be of independent interest.
We prove that for a large class of subvarieties of abelian varieties over global function fields, the Brauer-Manin condition on adelic points cuts out exactly the rational points. This result is obtained from more general results concerning the intersection of the adelic points of a subvariety with the adelic closure of the group of rational points of the abelian variety.
The goal of this paper is to explain how a simple but apparently new fact of linear algebra together with the cohomological interpretation of L-functions allows one to produce many examples of L-functions over function fields vanishing to high order at the center point of their functional equation. The main application is that for every prime p and every integer g>0 there are absolutely simple abelian varieties of dimension g over Fp(t) for which the BSD conjecture holds and which have arbitrarily large rank.
In this paper, we first reveal an intrinsic relation between non-abelian 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 well-known Dedekind zeta functions. Finally, based on such an expression, we show that all zeros of rank two non-abelian zeta functions are entirely sitting on the critical line whose real part equals to 1/2. This is an integrated part of our Geo-Arithmetic Program.
Let $A$ be an abelian variety over the function field of a smooth projective curve $C$ over an algebraically closed field $k$. We compute the $l$-adic cohomology groups of $C$ with coefficients in the locally constant sheaf associated to $H^1(\bar A,\mathbf{Q}_l)$ in terms of arithmetico-geometric invariants of $A$. We apply this, when $k$ is the algebraic closure of a finite field, to a motivic computation of the $L$-function of $A$.
Let $A$ be a non-isotrivial ordinary abelian surface over a global function field with good reduction everywhere. Suppose that $A$ does not have real multiplication by any real quadratic field with discriminant a multiple of $p$. We prove that there are infinitely many places modulo which $A$ is isogenous to the product of two elliptic curves.
The paper starts out from pseudomeasures (in the sense of Serre) which hold the arithmetic properties of the abelian $l$-adic Artin $L$-functions over totally real number fields. In order to generalize to non-abelian $l$-adic $L$-functions, these abelian pseudomeasures must satisfy congruences which are introduced but not yet known to be true. The relation to the ``equivariant main conjecture'' of Iwasawa theory is discussed.
We show that the coefficients of rational 2-functions are contained in an abelian number field. More precisely, we show that the poles of such functions are poles of order one and given by roots of unity and rational residue.
In this article we consider some questions raised by F. Benoist, E. Bouscaren and A. Pillay. We prove that infinitely $p$-divisible points on abelian varieties defined over function fields of transcendence degree one over a finite field are necessarily torsion points. We also prove that when the endomorphism ring of the abelian variety is $\mZ$ then there are no infinitely $p$-divisible points of order a power of $p$.
We prove the Iwasawa main conjecture over the arithmetic $\mathbb{Z}_p$-extension for semistable abelian varieties over function fields of characteristic $p>0$.
Initially motivated by the relations between Anabelian Geometry and Artin's L-functions of the associated Galois-representations, here we study the list of zeta-functions of genus two abelian coverings of elliptic curves over finite fields. Our goal is to provide a complete description of such a list.
In this paper, we consider Abelian varieties over function fields that arise as twists of Abelian varieties by cyclic covers of irreducible quasi-projective varieties. Then, in terms of Prym varieties associated to the cyclic covers, we prove a structure theorem on their Mordell-Weil group. Our results give an explicit method for construction of elliptic curves, hyper- and super-elliptic Jacobians that have large ranks over function fields of certain varieties.
Let $F$ be a global function field of characteristic $p>0$, $K/F$ an $\ell$-adic Lie extension ($\ell\neq p$) and $A/F$ an abelian variety. We provide Euler characteristic formulas for the $Gal(K/F)$-module $Sel_A(K)_\ell$.
We study a geometric analogue of the Iwasawa Main Conjecture for constant ordinary abelian varieties over $\ZZ_p^d$-extensions of function fields ramifying at a finite set of places.
We give an exact formula for the number of $G$-extensions of local function fields $\mathbb{F}_q((t))$ for finite abelian groups $G$ up to a conductor bound. As an application we give a lower bound for the corresponding counting problem by discriminant.
For an abelian extension of number fields we show that the Stark conjecture for all Artin L-functions with zero of order r is equivalent to existence of a special element in the rational span of the r-th exterior power of the Galois module of units of the bigger field.
Let $a(n)$ denote the number of non-isomorphic abelian groups with $n$ elements. In 1991 Ivić proved an asymptotic formula of the sum $\sum_{n\leq x} a(n+ a(n)).$ In this paper, we will prove a sharper asymptotic formula for this sum.
We study a geometric analogue of the Iwasawa Main Conjecture for abelian varieties in the two following cases: constant ordinary abelian varieties over $Z_p^d$-extensions of function fields ($d\geq 1$) ramified at a finite set of places, and semistable abelian varieties over the arithmetic $Z_p$-extension of a function field. One of the tools we use in our proof is a pseudo-isomorphism relating the duals of the Selmer groups of $A$ and its dual abelian variety $A^t$. This holds as well over number fields and is a consequence of a quite general algebraic functional equation.
Let $F$ be a global function field of characteristic $p>0$ and $A/F$ an abelian variety. Let $K/F$ be an $ł$-adic Lie extension ($ł\neq p$) unramified outside a finite set of primes $S$ and such that $\Gal(K/F)$ has no elements of order $ł$. We shall prove that, under certain conditions, $Sel_A(K)_ł^\vee$ has no nontrivial pseudo-null submodule.
Let $K$ be an imaginary quadratic field and $K_\infty$ be the $\mathbf{Z}_p^2$-extension of $K$. Answering a question of Ahmed and Lim, we show that the Pontryagin dual of the Selmer group associated to a supersingular polarized abelian variety admits an algebraic functional equation. The proof uses the theory of $Γ$-system developed by Lai, Longhi, Tan and Trihan. We also show the algebraic functional equation holds for Sprung's chromatic Selmer groups of supersingular elliptic curves along $K_\infty$.
I associate to a global field K a Lax-Phillips scattering which has the property of causality if and only if the Riemann Hypothesis holds for all the abelian L-functions of K. As a Hilbert space closure problem this provides an adelic variation on a theme initiated by Nyman and Beurling. The adelic aspects are related to previous work by Tate, Iwasawa and Connes.
Following Deligne and Ribet (`Values of abelian $L$-functions at negative integers over totally real fields.' Invent. Math. 59 (1980), 227-286) we prove that the `torsion congruences' (as introduced in our paper `Non-abelian pseudomeasures and congruences between abelian Iwasawa $L$-functions.' To appear in Pure and Applied Mathematics Quarterly) hold and so reduce the `main conjecture' of equivariant Iwasawa theory to the integrality of the logarithmic pseudomeasure.
We prove new congruences between special values of Rankin-Selberg $L$-functions for $\mathrm{GL}(n+1)\times\mathrm{GL}(n)$ over arbitrary number fields. This allows us to control the behavior of $p$-adic $L$-functions under Tate twists and to prove the existence of non-abelian $p$-adic $L$-functions for Hida families on $\mathrm{GL}(n+1)\times\mathrm{GL}(n)$. As an application, we prove strong non-vanishing results for central $L$-values: We give sufficient local conditions for twisted central Rankin-Selberg $L$-values to be generically non-zero.
We study Ruelle's type zeta and $L$-functions for a torsion free abelian group $\G$ of rank $\n\ge 2$ defined via an Euler product. It is shown that the imaginary axis is a natural boundary of this zeta function when $\n=2,4$ and 8, and in particular, such a zeta function has no determinant expression. Thus, conversely, expressions like Euler's product for the determinant of the Laplacians of the torus $\bR^{\n}/\G$ defined via zeta regularizations are investigated. Also, the limit behavior of an arithmetic function arising from the Ruelle type zeta function is observed.
In this work we introduce new folding axioms involving easy 3D manoeuvres with the aim to push forward the arithmetic limits of the Huzita-Justin axioms. Those 3D axioms involve the use of a flat surface and the rigidity property of convex polyhedra. Using those folding moves, we show that we can construct all Abelian numbers, and numbers whose Galois group is not solvable.
For a semi-abelian variety over a global function field which is isogenous to an isotrivial one, we show that on the product of local points of a subvariety satisfying a minor condition, the topological closure of a finitely generated subgroup of %the group of its global points cuts out exactly the global points of the subvariety lying in this subgroup. As a corollary, on every non-isotrivial super-singular curve of genus two over a global function field, we conclude that the Brauer-Manin condition cuts out exactly the set of its rational points.
For a finite group $G$, we consider the zeta function $ζ_G(s) = \sum_{H} \abs{H}^{-s}$, where $H$ runs over the subgroups of $G$. First we give simple examples of abelian $p$-group $G$ and non-abelian $p$-group $G'$ of order $p^m, \; m \geq 3$ for odd $p$ (resp. $2^m, \; m \geq 4$) for which $ζ_G(s) = ζ_{G'}(s)$. Hence we see there are many non-abelian groups whose zeta functions have symmetry and Euler product, like the case of abelian groups. On the other hand, we show that $ζ_G(s)$ determines the isomorphism class of $G$ within abelian groups, by estimating the number of subgroups of abelia…
We consider some classical maps from the theory of abelian varieties and their moduli spaces and prove their definability, on restricted domains, in the o-minimal structure $\Rae$. In particular, we prove that the embedding of moduli space of principally polarized ableian varierty, $Sp(2g,\Z)\backslash \CH_g$, is definable in $\Rae$, when restricted to Siegel's fundamental set $\fF_g$. We also prove the definability, on appropriate domains, of embeddings of families of abelian varieties into projective space.
