Hasse-Weil Zeta Function Research Articles

Weil zeta functions of group representations over finite fields

In this article we define and study a zeta function ζG\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\zeta _G$$\\end{document}—similar to the Hasse-Weil zeta function—which enumerates absolutely irreducible representations over finite fields of a (profinite) group G. This Weil representation zeta function converges on a complex half-plane for all UBERG groups and admits an Euler product decomposition. Our motivation for this investigation is the observation that the reciprocal value ζG(k)-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\zeta _G(k)^{-1}$$\\end{document} at a sufficiently large integer k coincides with the probability that k random elements generate the completed group ring of G. The explicit formulas obtained so far suggest that ζG\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\zeta _G$$\\end{document} is rather well-behaved. A central object of this article is the Weil abscissa, i.e., the abscissa of convergence a(G) of ζG\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\zeta _G$$\\end{document}. We calculate the Weil abscissae for free abelian, free abelian pro-p, free pro-p, free pronilpotent and free prosoluble groups. More generally, we obtain bounds (and sometimes explicit values) for the Weil abscissae of free pro-C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathfrak {C}}$$\\end{document} groups, where C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathfrak {C}}$$\\end{document} is a class of finite groups with prescribed composition factors. We prove that every real number a≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a \\ge 1$$\\end{document} is the Weil abscissa a(G) of some profinite group G. In addition, we show that the Euler factors of ζG\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\zeta _G$$\\end{document} are rational functions in p-s\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p^{-s}$$\\end{document} if G is virtually abelian. For finite groups G we calculate ζG\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\zeta _G$$\\end{document} using the rational representation theory of G.

Lifting low-gonal curves for use in Tuitman’s algorithm

Consider a smooth projective curve $\overline{C}$ over a finite field $\mathbb{F}_q$, equipped with a simply branched morphism $\overline{C} \to \mathbb{P}^1$ of degree $d \leq 5$. Assume char$\, \mathbb{F}_q > 2$ if $d \leq 4$, and char$\, \mathbb{F}_q > 3$ if $d=5$. In this paper we describe how to efficiently compute a lift of $\overline{C}$ to characteristic zero, such that it can be fed as input to Tuitman's algorithm for computing the Hasse-Weil zeta function of $\overline{C} / \mathbb{F}_q$. Our method relies on the parametrizations of low rank rings due to Delone-Faddeev and Bhargava.

Noncommutative motives in positive characteristic and their applications

Let k be a base field of positive characteristic. Making use of topological periodic cyclic homology, we start by proving that the category of noncommutative numerical motives over k is abelian semi-simple, as conjectured by Kontsevich. Then, we establish a far-reaching noncommutative generalization of the Weil conjectures, originally proven by Dwork and Grothendieck. In the same vein, we establish a far-reaching noncommutative generalization of the cohomological interpretations of the Hasse-Weil zeta function, originally proven by Hesselholt. As a third main result, we prove that the numerical Grothendieck group of every smooth proper dg category is a finitely generated free abelian group, as claimed (without proof) by Kuznetsov. Then, we introduce the noncommutative motivic Galois (super-)groups and, following an insight of Kontsevich, relate them to their classical commutative counterparts. Finally, we explain how the motivic measure induced by Berthelot's rigid cohomology can be recovered from the theory of noncommutative motives.

Geometry of the arithmetic site

We introduce the Arithmetic Site: an algebraic geometric space deeply related to the noncommutative geometric approach to the Riemann Hypothesis. We prove that the non-commutative space quotient of the adèle class space of the field of rational numbers by the maximal compact subgroup of the idèle class group, which we had previously shown to yield the correct counting function to obtain the complete Riemann zeta function as Hasse–Weil zeta function, is the set of geometric points of the arithmetic site over the semifield of tropical real numbers. The action of the multiplicative group of positive real numbers on the adèle class space corresponds to the action of the Frobenius automorphisms on the above geometric points. The underlying topological space of the arithmetic site is the topos of functors from the multiplicative semigroup of non-zero natural numbers to the category of sets. The structure sheaf is made by semirings of characteristic one and is given globally by the semifield of tropical integers. In spite of the countable combinatorial nature of the arithmetic site, this space admits a one parameter semigroup of Frobenius correspondences obtained as sub-varieties of the square of the site. This square is a semi-ringed topos whose structure sheaf involves Newton polygons. Finally, we show that the arithmetic site is intimately related to the structure of the (absolute) point in non-commutative geometry.

Bruhat Order on Partial Fixed Point Free Involutions

The order complex of inclusion poset $PF_n$ of Borel orbit closures in skew-symmetric matrices is investigated. It is shown that $PF_n$ is an EL-shellable poset, and furthermore, its order complex triangulates a ball. The rank-generating function of $PF_n$ is computed and the resulting polynomial is contrasted with the Hasse-Weil zeta function of the variety of skew-symmetric matrices over finite fields.

Modular parametrizations of certain elliptic curves

AbstractKaneko and Sakai [11] recently observed that certain elliptic curves whose associatednewforms (by the modularity theorem) are given by the eta-quotients can be character-ized by a particular differential equation involving modular forms and Ramanujan-Serredifferential operator.In this paper, we study certain properties of modular parametrization associated tothe elliptic curves over Q, and as a consequence we generalize and explain some of theirfindings.1. IntroductionBy the modularity theorem [4, 8], an elliptic curve Eover Qadmits a modular parametrizationΦ E : X 0 (N) → Efor some integer N. If Nis the smallest such integer, then it is equal to theconductor of Eand the pullback of the N´eron differential of Eunder Φ E is a rational multipleof 2πif E (τ), where f E (τ) ∈ S 2 (Γ 0 (N)) is a newform with rational Fourier coefficients. The factthat the L-function of f E (τ) coincides with the Hasse-Weil zeta function of E(which followsfrom Eichler-Shimura theory) is central to the proof of Fermat’s last theorem, and is related tothe Birch and Swinnerton-Dyer conjecture. In addition to this, modular parametrization is usedfor constructing rational points on elliptic curves, and appears in the Gross-Zagier formula [9].In this paper, we study some general properties of Φ

MOTIVIC L-FUNCTION IDENTITIES FROM CFT AND ARITHMETIC MIRROR SYMMETRY

Exactly solvable mirror pairs of Calabi–Yau threefolds of hypersurface type exist in the class of Gepner models that include nondiagonal affine invariants. Motivated by the string theoretic automorphy established previously for models in this class, it is natural to ask whether the arithmetic structure of mirror pairs varieties reflects the fact that as conformal field theories, they are isomorphic. Mirror symmetry in particular predicts that the L-functions of the Ω-motives of such pairs are identical. In this paper this prediction is confirmed by showing that the Ω-motives of exactly solvable mirror pairs are isomorphic. This follows as a corollary of the proof of a more general result establishing an isomorphism between nondiagonally and diagonally induced motives in this class of varieties. The motivic approach formulated here circumvents the difficulty that no mirror construction of the Hasse–Weil zeta function is known.

Characterizing abelian varieties by the reduction of the Mordell–Weil group

Let $A$ be an abelian variety defined over a number field $K$. If $\mathfrak{p}$ is a prime of $K$ of good reduction for $A$, let $A(K)_\mathfrak{p}$ denote the image of the Mordell-Weil group via reduction modulo $\mathfrak{p}$. We prove in particular that the size of $A(K)_\mathfrak{p}$, by varying $\mathfrak{p}$, encodes enough information to determine the $K$-isogeny class of $A$, provided that the following necessary condition is satisfied: $B(K)$ has positive rank for every non-trivial abelian subvariety $B$ of $A$. This is the analogue to a result by Faltings of 1983 considering instead the Hasse-Weil zeta function of the special fibers $A_\mathfrak{p}$.

Horizontal Factorizations of Certain Hasse-Weil Zeta Functions - a Remark on a Paper by Taniyama

The paper [T] by Taniyama contains several results in the arithmetic theory of CM abelian varieties which have become well known. Moreover the notion of a compatible system ρ = (ρl) of l-adic representations is introduced there for the first time. Under suitable conditions on ρ, Taniyama proves an interesting formula for the alternating product of the L-functions of the exterior powers of ρ: it is given by an infinite product of Artin L-functions each of which modified by a change of finitely many Euler factors. As an application, Taniyama obtains such a formula for the Hasse–Weil zeta function of an abelian scheme over a localized number ring. These infinite product formulas of Taniyama seem to be little known but they motivated the works [JY] and [JR]. In [JY] Joshi and Yogananda show that the Hasse–Weil zeta function ζGm(s) = ζ(s−1)/ζ(s) of Gm = specZ[T, T −1] is an infinite product of Dirichlet L-series. This is a special case of Taniyama’s formulas but they give an elementary direct argument. In [JR] Joshi and Raghunathan express quotients of very general Dirichlet series with Euler products as infinite products of “twisted Dirichlet series”. The method is purely local. It applies in particular to the Hasse–Weil zeta function of X×Gm where X is a scheme of finite type over Z. In this case the formula has a geometric proof which was suggested by Serre, c.f. [JR] § 3. Joshi and Raghunathan also obtain certain infinite product formulas for quotients of automorphic L-functions. Recently these have been related to canonical bases by Kim and Lee, [KL] Remark 2.21.

On the cohomology of compact unitary group Shimura varieties at ramified split places

In this article, we prove results about the cohomology of compact unitary group Shimura varieties at split places. In nonendoscopic cases, we are able to give a full description of the cohomology, after restricting to integral Hecke operators at p p on the automorphic side. We allow arbitrary ramification at p p ; even the PEL data may be ramified. This gives a description of the semisimple local Hasse-Weil zeta function in these cases. We also treat cases of nontrivial endoscopy. For this purpose, we give a general stabilization of the expression given in the article http://dx.doi.org/ 10.1090/S0894-0347-2012-00753-X, following the stabilization given by Kottwitz. This introduces endoscopic transfers of the functions ϕ τ , h \phi _{\tau ,h} introduced in the above article. We state a general conjecture relating these endoscopic transfers with Langlands parameters. We verify this conjecture in all cases of EL type and deduce new results about the endoscopic part of the cohomology of Shimura varieties. This allows us to simplify the construction of Galois representations attached to conjugate self-dual regular algebraic cuspidal automorphic representations of G L n \mathrm {GL}_n .

Hasse-Weil zeta function of absolutely irreducible 𝑆𝐿₂-representations of the figure 8 knot group

Weil-type zeta functions defined by the numbers of absolutely irreducible S L 2 \mathrm {SL}_2 -representations of the figure 8 8 knot group over finite fields are computed explicitly. They are expressed in terms of the congruence zeta functions of reductions of a certain elliptic curve defined over the rational number field. Then the Hasse-Weil type zeta function of the figure 8 8 knot group is also studied. Its central value is written in terms of the Mahler measures of the Alexander polynomial of the figure 8 8 knot and a certain family of elliptic curves.

Proof of R. Salvati Manni and J. Top's conjectures on Siegel modular forms and Abelian surfaces

This paper gives complete proofs of R. Salvati Manni and J. Top's conjectures which are contained in the paper, "Cusp forms of weight 2 for the group Γ(4, 8), Amer. J. Math . 115 (1993), 455-486." One of the conjectures claims that the Hasse-Weil zeta function corresponding to the Jacobian variety defined over [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] of the hyper-elliptic curve y 2 = x 5 - x equals the Andrianov L -function of Siegel cusp form Θ of degree 2 and weight 2 which is four products of the Igusa theta constants. Regarding the Θ as a function obtained by the Yoshida lifting from a pair of elliptic modular forms, we prove the conjecture.

Computing the alternating trace of Frobenius on the sheaves of nearby cycles on local models for GL4 and GL5

Consider a Shimura variety defined over some number field, and assume we have a model over the ring of integers at some prime of bad reduction. It is then interesting to know the alternating trace of Frobenius on the invariants under the inertia group of the sheaf of nearby cycles, since these traces are related to the local factor of the Hasse–Weil zeta function. In this article we compute the semi-simple alternating trace of Frobenius for Shimura varieties associated to the groups GU(2,2) and GU(3,2), with level structure of Iwahori type, by investigating the equations of the local model defined by Rapoport and Zink and by performing explicit blowing ups.

A singular K3 surface related to sums of consecutive cubes

We study the surface arising from the diophantine equation $m^3+(m+1)^3+...+(m+k-1)^3=l^2$. It turns out that this is a $K3$ surface with Picard number 20. We stduy its aritmetic properties in detail. We construct elliptic fibrations on it, and we find a parametric solution to the original equation. Also, we determine the Hasse-Weil zeta function of the surface over $Q$.

$\ep$-constants and Arakelov Euler characteristics

Let X be a regular scheme projective and flat over Spec(Z), equidimensional of relative dimension d. Consider the Hasse-Weil zeta function of X, ζ(X, s) = ∏ x(1 −N(x) −s)−1 where x ranges over the closed points of X and N(x) is the order of the residue field of x. Denote by L(X, s) the zeta function with Γ-factors L(X, s) = ζ(X, s)Γ(X, s). The L-function conjecturally satisfies a functional equation

Coates-Wiles towers in dimension two

Let K be an imaginary quadratic field and E an elliptic curve defined over K with complex multiplication by the ring of integers of K. In 1977 Coates and Wiles proved that if E(K) has positive rank, then the Hasse-Weil zeta function L(E/K, s) of E has a zero at s = 1 [2]. Such curves exist precisely when the class number of K is one. (Subsequent work [1] extended the result to a wider class of CM curves. Another proof emerged from Rubin's important work on the Tate-Shafarevich group [13, 14]. See also [10-12].) Coates and Wiles used formal groups and Iwasawa-theoretic techniques to relate the special value of L(E/F, s) to elliptic units. Employing more classical techniques, Stark and Gupta combined to give a different proof of the theorem [21, 4]. Although the two proofs have different flavors, they include the same ingredients: Let p be one of infinitely many suitably chosen primes of K, and K, the field obtained by adjoining the p-torsion of E to K. There is a unique prime p, of K, lying over p. Let Q be a point of infinite order in E(K), Q C pE(K), and (5) L,(Q) = K Q . Let v 2 be the Gr6ssencharakter of E, and f2 a fundamental

On the Birch and Swinnerton-Dyer conjecture

Let F be a finite extension of the rational field Q. If E is an elliptic curve defined over F, then the Mordell-Weil group E(F) of points on E with coordinates in F is a finitely generated abelian group. Let L(E/F, s) be the HasseWeil zeta function of E over F which, at least in the case where E has complex multiplication, is known to be an analytic function on the entire complex plane. Birch and Swinnerton-Dyer have conjectured that L(E/F,s) has a zero at s = l of order precisely equal to the rank of E(F) over Z. The strongest result known in support of this conjecture is the following theorem of Coates and Wiles [4].