In this paper we construct an algebraic invariant attached to Galois representations over number fields. This invariant, which we call an Artin symmetric function, lives in a certain ring we introduce called the ring of arithmetic symmetric functions. This ring is built from a family of symmetric functions rings indexed by prime ideals of the base field. We prove many necessary basic results for the ring of arithmetic symmetric functions as well as introduce the analogues of some standard number-theoretic objects in this setting. We prove that the Artin symmetric functions satisfy the same algebraic properties that the Artin L-functions do with respect to induction, inflation, and direct summation of representations. The expansion coefficients of these symmetric functions in different natural bases are shown to be character values of representations of a compact group related to the original Galois group. In the most interesting case, the expansion coefficients into a specialized Hall-Littlewood basis come from new representations built from the original Galois representation using polynomial functors corresponding to modified Hall-Littlewood polynomials. Using a special case of the Satake isomorphism in type GL, as formulated by Macdonald, we show that the Artin symmetric functions yield families of functions in the (finite) global spherical Hecke algebras in type GL which exhibit natural stability properties. We compute the Mellin transforms of these functions and relate them to infinite products of shifted Artin L-functions. We then prove some analytic properties of these Dirichlet series and give an explicit expansion of these series using the Hall-Littlewood polynomial functors.
The author reviews results and conjectures of Selberg on a class of Dirichlet series functions which share properties with the Riemann zeta function, and he relates this work to the theory of Artin L-functions.
We study the distribution of the zeroes of the L-functions of curves in the Artin-Schreier family. We consider the number of zeroes in short intervals and obtain partial results which agree with a random unitary matrix model.
Let $K/{\mathbb Q}$ be a finite Galois extension, and let $s_0\neq 1$ be a complex number. We present two new criteria for the Artin L-functions to be holomorphic at $s_0$.
We study the special value $u=1$ of Artin-Ihara $L$-functions associated to characters of the automorphism group of abelian covers of multigraphs. In particular, we show an annihilation statement analogous to a classical conjecture of Brumer on annihilation of class groups for abelian extensions of number fields and we also calculate the index of an ideal analogous to the classical Stickelberger ideal in algebraic number theory. Along the way, we make some observations about the number of spanning trees in abelian multigraph coverings that may be of independent interest.
We study Artin $L$-functions on a finite $2$-dimensional complex $X_Γ$ arising from PGL$_3$ attached to finite-dimensional representations $ρ$ of its fundamental group. Some key properties, such as rationality, functional equation, and invariance under induction, of these functions are proved. Moreover, using a cohomological argument, we establish a connection between the Artin $L$- functions and the $L$-function of $ρ$, extending the identity on zeta functions of $X_Γ$ obtained in [KL,KLW].
Let $π$ be an irreducible admissible (complex) representation of $GL(2)$ over a non-archimedean characteristic zero local field with odd residual characteristic. In this paper we prove the equality between the local symmetric square $L$-function associated to $π$ arising from integral representations and the corresponding Artin $L$-function for its Langlands parameter through the local Langlands correspondence. With this in hand, we show the stability of local symmetric $γ$-factors attached to $π$ under highly ramified twists.
We study various families of Artin $L$-functions attached to geometric parametrizations of number fields. In each case we find the Sato-Tate measure of the family and determine the symmetry type of the distribution of the low-lying zeros.
If $K/\mathbb Q$ is a finite Galois extension with an almost monomial Galois group and if $s_0\in\mathbb C\setminus\{1\}$ is not a common zero for any two Artin L-functions associated to distinct complex irreducible characters of the Galois group then all Artin L-functions of $K/\mathbb Q$ are holomorphic at $s_0$. We present examples and basic properties of almost monomial groups.
We determine all irreducible representations of primary quasi-cyclotomic fields in this paper. The methods can be applied to determine the irreducible representations of any quasi-cyclotomic field. We also compute the Artin L-functions for a class of quasi-cyclotomic fields.
In this paper we present an approach to study arithmetical properties of global function fields by working with Artin L-functions. In particular we recall and then extend a criteria of two function fields to be arithmetically equivalent in terms of Artin L-functions of representations associated to the common normal closure of those fields. We provide few examples of such non-isomorphic fields and also discuss an algorithm to construct many such examples by using torsion points on elliptic curves. Finally, we will show how to apply our results in order to distinguish two global fields by a fin…
Under GRH, we establish a version of Duke's short-sum theorem for entire Artin $L$-functions. This yields corresponding bounds for residues of Dedekind zeta functions. All numerical constants in this work are explicit.
We compute the Artin $L$-function of a diagonal hypersurface D_λ over a finite field associated to a character of a finite group acting on D_λ , and under some condition, express it in terms of hypergeometric functions and Jacobi sums over the finite field. As an application, we consider the Dwork hypersurfaces and obtain relations among certain hypergeometric functions over different finite fields.
A dynamically affine map is a finite quotient of an affine morphism of an algebraic group. We determine the rationality or transcendence of the Artin-Mazur zeta function of a dynamically affine self-map of $\mathbb{P}^1(k)$ for $k$ an algebraically closed field of positive characteristic.
We study the local statistics of zeros of $L$-functions attached to Artin-Scheier curves over finite fields. We consider three families of Artin-Schreier $L$-functions: the ordinary, polynomial (the $p$-rank 0 stratum) and odd-polynomial families. We compute the 1-level zero-density of the first and third families and the 2-level density of the second family for test functions with Fourier transform supported in a suitable interval. In each case we obtain agreement with a unitary or symplectic random matrix model.
Let $K/\mathbb Q$ be a finite Galois extension, $s_0\in \mathbb C\setminus \{1\}$, ${\it Hol}(s_0)$ the semigroup of Artin L-functions holomorphic at $s_0$. If the Galois group is almost monomial then Artin's L-functions are holomorphic at $s_0$ if and only if $ {\it Hol}(s_0)$ is factorial. This holds also if $s_0$ is a zero of an irreducible L-function of dimension $\leq 2$, without any condition on the Galois group.
Let $k$ be a number field, $K/k$ a finite Galois extension with Galois group $G$, $χ$ a faithful character of $G$. We prove that the Artin L-function $L(s,χ,K/k)$ determines the Galois closure of $K$ over $\Q$. In the special case $k=\Q$ it also determines the character $χ$.
For a global function field K of positive characteristic p, we show that Artin conjecture for L-functions of geometric p-adic Galois representations of K is true in a non-trivial p-adic disk but is false in the full p-adic plane. In particular, we prove the non-rationality of the geometric unit root L-functions.
We study the problem of finding the Artin L-functions with the smallest conductor for a given Galois type. We adapt standard analytic techniques to our novel situation of fixed Galois type and get much improved lower bounds on the smallest conductor. For small Galois types we use complete tables of number fields to determine the actual smallest conductor.
We show that the sum of the traces of Frobenius elements of Artin $L$-functions in a family of $G$-fields satisfies the Gaussian distribution under certain counting conjectures. We prove the counting conjectures for $S_4$ and $S_5$-fields. We also show central limit theorem for modular form $L$-functions with the trivial central character with respect to congruence subgroups as the level goes to infinity.
In this paper we prove that any Artin--Schreier extension of a congruence rational function field is contained in the composite of a cyclotomic function field and a constant field extension that are explicitly prescribed.
We give a new derivation of an identity due to Z. Rudnick and P. Sarnak about the $n$-level correlations of eigenvalues of random unitary matrices as well as a new proof of a formula due to M. Diaconis and P. Shahshahani expressing averages of trace products over the unitary matrix ensemble. Our method uses the zero statistics of Artin-Schreier L-functions and a deep equidistribution result due to N. Katz and P. Sarnak.
We study the rationality of the Artin-Mazur zeta function of a dynamical system defined by a polynomial self-map of A^1(k), where k is the algebraic closure of the finite field F_p. The zeta functions of the maps f(x)=x^m for (p,m)=1 and f(x)=x^{p^m}+ax for nonzero a in F_{p^m}, p odd, are shown to be transcendental.
Let E be an elliptic curve and ρan Artin representation, both defined over the rational numbers. Let p be a prime at which E has good reduction. We prove that there exists an infinite set of Dirichlet characters χ, ramified only at p, such that the Artin-twisted L-values L(E,ρχ,β) are non-zero when βlies in a specified region in the critical strip (assuming the conjectural continuations and functional equations for these L-functions). The new contribution of our paper is that we may choose our characters to be ramified only at one prime, which may divide the conductor of ρ.
The aim of this paper is twofold: on one hand we study the invariants of traces of quadratic forms over a finite field of characteristic two. On the other hand, we give results about the zeta functions of certain curves studied by van der Geer and ven der Vlugt.
We discuss how one could study asymptotics of cyclotomic quantities via the mean values of certain multiplicative functions and their Dirichlet series using a theorem of Delange. We show how this could provide a new approach to Artin's conjecture on primitive roots. We focus on whether a fixed prime has a certain order modulo infinitely many other primes. We also give an estimate for the mean value of one such Dirichlet series.
Given a pair of abelian varieties defined over a number field k and isogenous over a finite Galois extension L/k, we define a rational Artin representation of the group Gal(L/k) that shows a global relation between the L-functions of each variety and provides certain information about their decomposition up to isogeny over L. We study several properties of these Artin representations. As an application, for each curve C' in a family of twists of a certain genus 3 curve C, we explicitly compute the Artin representation attached to the Jacobians of C and C' and show how this Artin representation…
This note gives a simple proof that certain values of Artin's $L$-function, for a representation $ρ$ with character $χ_ρ$, are stable under twisting by an even Dirichlet character $χ$, up to an element generated over $\mathbb Q$ by the values of $χ$ and $χ_ρ$, and a product with a power of the Gauss sum $τ(χ)$ equal to the dimension of $ρ$. This extends a result due to J. Coates and S. Lichtenbaum.
Let $K/\mathbb Q$ be a finite Galois extension. Let $χ_1,\ldots,χ_r$ be $r\geq 1$ distinct characters of the Galois group with the associated Artin L-functions $L(s,χ_1),\ldots, L(s,χ_r)$. Let $m\geq 0$. We prove that the derivatives $L^{(k)}(s,χ_j)$, $1\leq j\leq r$, $0\leq k\leq m$, are linearly independent over the field of meromorphic functions of order $<1$. From this it follows that the L-functions corresponding to the irreducible characters are algebraically independent over the field of meromorphic functions of order $<1$.
We define and study a $p$-adic analogue of the incomplete gamma function related to Morita's $p$-adic gamma function. We also discuss a combinatorial identity related to the Artin-Hasse series, which is a special case of the exponential principle in combinatorics. From this we deduce a curious $p$-adic property of $|\mathrm{Hom} (G,S_n)|$ for a topologically finitely generated group $G$, using a characterization of $p$-adic continuity for certain functions $f \colon \mathbb Z_{>0} \to \mathbb Q_p$ due to O'Desky-Richman. In the end, we give an exposition of some standard properties of the Arti…
The corresponding Hasse-Weil bound was a major breakthrough in history of mathematics. It has found many applications in mathematics, coding theory and theoretical computer science. In general, the Hasse-Weil bound is tight and cannot be improved. However, the Hasse-Weil bound is no longer tight when it is applied to some specific classes of curves. One of the examples where the Hasse-Weil bound is not tight is the family of Artin-Schreier curves. Due to various applications of Artin-Schreier curves to coding, cryptography and theoretical computer science, researchers have made great effort to…
This paper describes a class of Artin-Schreier curves, generalizing results of Van der Geer and Van der Vlugt to odd characteristic. The automorphism group of these curves contains a large extraspecial group as a subgroup. Precise knowledge of this subgroup makes it possible to compute the zeta functions of the curves in the class over the field of definition of all automorphisms in the subgroup. As a consequence, we obtain new examples of maximal curves.
This article is the first of a pair of articles dealing with the Iwasawa theory of modular forms of weight 1 and, more generally, of Artin representations satisfying certain conditions. The main results in this part analyze the structure of certain Selmer groups for the Artin representation. In particular, it is shown that the Selmer groups are co-torsion as $Λ$-modules. For each Selmer group, we consider a generator of the characteristic ideal of its Pontrjagin dual. We call that the algebraic p-adic L-function. We also construct an analytic $p$-adic L-function via deformation to higher weigh…
For Kleinian groups acting on hyperbolic three-space, we prove factorization formulas for both the Selberg zeta-function and the automorphic scattering matrix. We extend results of Venkov and Zograf from Fuchsian groups, to Kleinian groups, and we give a proof that is simple and extendable to more general groups.
In this paper, we study a certain Artin--Schreier family of elliptic curves over the function field $\mathbb{F}_q(t)$. We prove an asymptotic estimate on the size of the special value of their $L$-function in terms of the degree of their conductor; loosely speaking, we show that the special values are "asymptotically as large as possible". We also provide an explicit expression for the $L$-function of the elliptic curves in the family. The proof of the main result uses this expression and a detailed study of the distribution of some character sums related to Kloosterman sums. Via the BSD conje…
We prove an explicit upper bound for the k-th prime ideal with fixed Artin symbol, under the assumption of the validity of the Riemann hypothesis for the Dedekind zeta functions.
We study abelian varieties defined over function fields of curves in positive characteristic $p$, focusing on their arithmetic within the system of Artin-Schreier extensions. First, we prove that the $L$-function of such an abelian variety vanishes to high order at the center point of its functional equation under a parity condition on the conductor. Second, we develop an Artin-Schreier variant of a construction of Berger. This yields a new class of Jacobians over function fields for which the Birch and Swinnerton-Dyer conjecture holds. Third, we give a formula for the rank of the Mordell-Weil…
Selmer group for an Artin representation over totally real fields was studied by Greenberg and Vatsal. In this paper we study the Selmer groups for an Artin representation over a totally complex field. We establish an algebraic function of the characteristic ideal of the Selmer group associated to Artin representation over the cyclotomic $\Z_p$- extension of the rational numbers under certain mild hypotheses and construct several examples to illustrate our result. We also prove that in this situation $μ$-invariant of the dual Selmer group is independent of the choice of the lattice.
Let $Γ\backslash\mathbb H^3$ be a finite-volume quotient of the upper-half space, where $Γ\subset {\rm SL}(2,\mathbb C)$ is a discrete subgroup. To a finite dimensional unitary representation $χ$ of $Γ$ one associates the Selberg zeta function $Z(s;Γ;χ)$. In this paper we prove the Artin formalism for the Selberg zeta function. Namely, if $\tildeΓ$ is a finite index group extension of $Γ$ in ${\rm SL}(2,\mathbb C)$, and $π={\rm Ind}_Γ^{\tildeΓ}χ$ is the induced representation, then $Z(s;Γ;χ)=Z(s;\tildeΓ;π)$. In the second part of the paper we prove by a direct method the analogous identity for…
This is an updated version of ANT-0253. Let F be a number field with absolute Galois group G. We associate, to each continuous, solvable C-representation of G of GO(4)-type, an automorphic form P of GL(4)/F with the same L-function. As a consequence we exhibit an infinite class of primitive, 16-dimensional representations for which the Artin conjecture holds.
We study a tower of function fields of Artin-Schreier type over a finite field with $2^s$ elements. The study of the asymptotic behavior of this tower was left as an open problem by Beelen, García and Stichtenoth in $2006$. We prove that this tower is asymptotically good for $s$ even and asymptotically bad for $s$ odd.
Effective Brauer-Siegel theorems for Artin $L$-functions
Ulsan National Institute of Science and Technology（蔚山科学技术院）；University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Tufts University（塔夫茨大学）；University of Toronto（多伦多大学）
This is an investigation into the possible existence and consequences of a Birch-Swinnerton-Dyer-type formula for L-functions of elliptic curves twisted by Artin representations. We translate expected properties of L-functions into purely arithmetic predictions for elliptic curves, and show that these force some peculiar properties of the Tate-Shafarevich group, which do not appear to be tractable by traditional Selmer group techniques. In particular we exhibit settings where the different p-primary components of the Tate-Shafarevich group do not behave independently of one another. We also gi…
We present a group-theoretic criterion under which one may verify the Artin conjecture for some (non-monomial) Galois representations, up to finite height in the complex plane. In particular, the criterion applies to S5 and A5 representations. Under more general conditions, the technique allows for the possibility of verifying the Riemann hypothesis for Dedekind zeta functions of non-abelian extensions of Q. In addition, we discuss two methods for locating zeros of arbitrary L-functions. The first uses the explicit formula and techniques (developed jointly with Andreas Strombergsson) for compu…
Let $K$ be an imaginary quadratic field. In this article, we study the eigenvariety for $\mathrm{GL}_2/K$, proving an étaleness result for the weight map at non-critical classical points and a smoothness result at base-change classical points. We give three main applications of this; let $f$ be a $p$-stabilised newform of weight $k \geq 2$ without CM by $K$. Suppose $f$ has finite slope at $p$ and its base-change $f_{/K}$ to $K$ is $p$-regular. Then: (1) We construct a two-variable $p$-adic $L$-function attached to $f_{/K}$ under assumptions on $f$ that conjecturally always hold, in particular…
We study the distribution of the zeroes of the zeta functions of the family of Artin-Schreier covers of the projective line over $\mathbb{F}_q$ when $q$ is fixed and the genus goes to infinity. We consider both the global and the mesoscopic regimes, proving that when the genus goes to infinity, the number of zeroes with angles in a prescribed non-trivial subinterval of $[-π,π)$ has a standard Gaussian distribution (when properly normalized).
