Function Field Case Research Articles

Tori over number fields and special values at $s=1$

We define a Weil-étale complex with compact support for duals (in the sense of the Bloch dualizing cycles complex \mathbb{Z}^c ) of a large class of \mathbb{Z} -constructible sheaves on an integral1-dimensional proper arithmetic scheme flat over Spec( \mathbb{Z} ). This complex can be thought of as computing Weil-étale homology. For those Z-constructible sheaves that are moreover tamely ramified, we define an “additive” complex which we think of as the Lie algebra of the dual of the \mathbb{Z} -constructible sheaf. The product of the determinants of the additive and Weil-étale complex is called the fundamental line. We prove a duality theorem which implies that the fundamental line has a natural trivialization, giving a multiplicative Euler characteristic. We attach a natural L -function to the dual of a \mathbb{Z} -constructible sheaf; up to a finite number of factors, this L -function is an Artin L -function at s = 1 . Our main theorem contains a vanishing order formula at s = 0 for the L -function and states that, in the tamely ramified case, the special value at s = 0 is given up to sign by the Euler characteristic. This generalizes the analytic class number formula for the special value at s = 1 of the Dedekind zeta function. In the function field case, this is a theorem of Geisser–Suzuki.

On the Tits indices of absolutely almost simple algebraic groups over local and global fields

We present a new, detailed and uniform proof of the validity of those parts of the classification results regarding absolutely almost simple algebraic groups over p -adic and number fields, announced earlier by Tits, which also covers the case of local and global function fields. The main tools used in this paper are Prasad–Rapinchuk's reciprocity law for Tits indices over global fields and also some (new) Hasse principles for homogeneous spaces over global function fields and for forms in characteristic 2.

A Gallery Model for Affine Flag Varieties via Chimney Retractions

This paper provides a unified combinatorial framework to study orbits in certain affine flag varieties via the associated Bruhat–Tits buildings. We first formulate, for arbitrary affine buildings, the notion of a chimney retraction. This simultaneously generalizes the two well-known notions of retractions in affine buildings: retractions from chambers at infinity and retractions from alcoves. We then present a recursive formula for computing the images of certain minimal galleries in the building under chimney retractions, using purely combinatorial tools associated to the underlying affine Weyl group. Finally, for Bruhat–Tits buildings in the function field case, we relate these retractions and their effect on minimal galleries to double coset intersections in the corresponding affine flag variety.

Lifting and automorphy of reducible mod p Galois representations over global fields

We prove the modularity of most reducible, odd representations \({\bar{\rho }}: \Gamma _{{\mathbb {Q}}} \rightarrow \mathrm {GL}_2(k)\) with k a finite field of characteristic an odd prime p. This is an analogue of Serre’s celebrated modularity conjecture (which concerned irreducible, odd representations \({\bar{\rho }}: \Gamma _{{\mathbb {Q}}} \rightarrow \mathrm {GL}_2(k)\)) for reducible, odd representations. Our proof lifts \({\bar{\rho }}\) to an irreducible geometric p-adic representation \(\rho \) which is known to arise from a newform by results of Skinner–Wiles and Pan. We likewise prove automorphy of many reducible representations \({\bar{\rho }}:\Gamma _{F} \rightarrow \mathrm {GL}_n(k)\) when F is a global function field of characteristic different from p, by establishing a p-adic lifting theorem and invoking the work of L. Lafforgue. Crucially, in both cases we show that the actual representation \({\bar{\rho }}\), rather than just its semisimplification, arises from reduction of the geometric representation attached to a cuspidal automorphic representation. Our main theorem establishes a geometric lifting result for mod p representations \({\bar{\rho }}:\Gamma _{F} \rightarrow G(k)\) of Galois groups of global fields F, valued in reductive groups G(k), and assumed to be odd when F is a number field. Thus we find that lifting theorems, combined with automorphy lifting results pioneered by Wiles in the number field case and the results in the global Langlands correspondence proved by Drinfeld and L. Lafforgue in the function field case, give the only known method to access modularity of mod p Galois representations both in reducible and irreducible cases.

The relative trace formula approach for spherical varieties

We present a general framework for the relative trace formula approach to the conjecture of Sakellaridis and Venkatesh on period integrals for spherical varieties over global fields.This approach is somehow well-known for experts but it seems lack of literature.Our approach is based on the spectrum isolation technique developed by Beuzart-Plessis et al. (2019) for number fields,which avoids the difficulty from the refined spectral expansion. This technique is also available in the function field case.

Zeta-like Multizeta Values for higher genus curves

We prove or conjecture several relations between the multizeta values for positive genus function fields of class number one, focusing on the zeta-like values, namely those whose ratio with the zeta value of the same weight is rational (or conjecturally equivalently algebraic). These are the first known relations between multizetas, which are not with prime field coefficients. We seem to have one universal family. We also find that interestingly the mechanism with which the relations work is quite different from the rational function field case, raising interesting questions about the expected motivic interpretation in higher genus. We provide some data in support of the guesses.

Isolation of the cuspidal spectrum: The function field case

Isolating cuspidal automorphic representations from the whole automorphic spectrum is a basic problem in the trace formula approach. For example, matrix coefficients of supercuspidal representations can be used as test functions for this. However, they kill a large class of interesting cuspidal automorphic representations. For the case of number fields, multipliers of the Schwartz algebra are used in the recent work (see Beuzart-Plessis et al. (2019)) to isolate all the cuspidal spectrum. In particular, they are suitable for the comparison of orbital integrals. These multipliers are then applied to the proof of the Gan-Gross-Prasad conjecture for unitary groups (see Beuzart-Plessis et al. (2019, 2020)). In this article, we prove the similar result on isolating the cuspidal spectrum in Beuzart-Plessis et al. (2019) for the function field case.

Eichler cohomology and zeros of polynomials associated to derivatives of L-functions

Abstract In recent years, a number of papers have been devoted to the study of zeros of period polynomials of modular forms. In the present paper, we study cohomological analogues of the Eichler–Shimura period polynomials corresponding to higher L-derivatives. We state a general conjecture about the locations of the zeros of the full and odd parts of the polynomials, in analogy with the existing literature on period polynomials, and we also give numerical evidence that similar results hold for our higher derivative “period polynomials” in the case of cusp forms. The unimodularity of the roots seems to be a very subtle property which is special to our “period polynomials”. This is suggested by numerical experiments on families of perturbed “period polynomials” (Section 5.3) suggested by Zagier. We prove a special case of our conjecture in the case of Eisenstein series. Although not much is currently known about derivatives higher than first order ones for general modular forms, celebrated recent work of Yun and Zhang established the analogues of the Gross–Zagier formula for higher L-derivatives in the function field case. A critical role in their work was played by a notion of “super-positivity”, which, as recently shown by Goldfeld and Huang, holds in infinitely many cases for classical modular forms. As will be discussed, this is similar to properties which were required by Jin, Ma, Ono, and Soundararajan in their proof of the Riemann Hypothesis for Period Polynomials, thus suggesting a connection between the analytic nature of our conjectures here and the framework of Yun and Zhang.

The Chevalley–Gras formula over global fields

Dans cet article, nous donnons une preuve adélique de la formule de Chevalley–Gras pour les corps de nombres qui, elle-même, est une généralisation de la formule du nombre de classes ambiges. L’idée est de réduire cette formule au théorème de la norme de Hasse et à des lois de réciprocité globaux. Nous donnons également une preuve adélique de la formule de Chevalley–Gras pour les groupes des classes des diviseurs de degré 0 dans le cas des corps de fonctions, qui étend un résultat de Rosen.

Local–global principles for curves over semi‐global fields

We investigate local-global principles for Galois cohomology, in the context of function fields of curves over semi-global fields. This extends work of Kato's on the case of function fields of curves over global fields.

Generic character sheaves on reductive groups over a finite ring

In this paper we propose a construction of generic character sheaves on reductive groups over finite local rings at even levels, whose characteristic functions are higher Deligne–Lusztig characters when the parameters are generic. We formulate a conjecture on the simple perversity of these complexes, and we prove it in the level two case (thus extend a result of Lusztig from the function field case). We then discuss the induction and restriction functors, as well as the Frobenius reciprocity, based on the perversity.

Witt equivalence of function fields of conics

Two fields are Witt equivalent if, roughly speaking, they have the same quadratic form theory. Formally, that is to say that their Witt rings of symmetric bilinear forms are isomorphic. This equivalence is well understood only in a few rather specific classes of fields. Two such classes, namely function fields over global fields and function fields of curves over local fields, were investigated by the authors in their earlier works [5] and [6]. In the present work, which can be viewed as a sequel to the earlier papers, we discuss the previously obtained results in the specific case of function fields of conic sections, and apply them to provide a few theorems of a somewhat quantitive flavour shedding some light on the question of numbers of Witt non-equivalent classes of such fields.

Proteomic identification of potential targets in the syncytiotrophoblast microvillous membrane for placental-specific therapeutic delivery.

A famous conjecture of Artin states that there are infinitely many prime numbers for which a fixed integer g is a primitive root, provided g≠−1 and g is not a perfect square. Thanks to work of Hooley, we know that this conjecture is true, conditional on the truth of the Generalized Riemann Hypothesis. Using a combination of Hooley's analysis and the techniques of Maynard–Tao used to prove the existence of bounded gaps between primes, Pollack has shown that (conditional on GRH) there are bounded gaps between primes with a prescribed primitive root. In the present article, we provide an unconditional proof of the analogue of Pollack's work in the function field case; namely, that given a monic polynomial g(t) which is not an vth power for any prime v dividing q−1, there are bounded gaps between monic irreducible polynomials P(t) in Fq[t] for which g(t) is a primitive root (which is to say that g(t) generates the group of units modulo P(t)). In particular, we obtain bounded gaps between primitive polynomials, corresponding to the choice g(t)=t.

Regulators of Elliptic Curves

Abstract We study the regulator of the Mordell–Weil group of elliptic curves over number fields, functions fields of characteristic 0 or function fields of characteristic $p>0$. We prove a new Northcott property for the regulator of elliptic curves of rank at least 4 defined over a number field. In the function field case, we obtain that non-trivial families of elliptic curves with bounded regulators and bounded ranks are limited families. We conclude by asking new questions.

Class numbers and p-ranks in [formula omitted]-towers

To extend Iwasawa's classical theorem from Zp-towers to Zpd-towers, Greenberg conjectured that the exponent of p in the n-th class number in a Zpd-tower of a global field K ramified at finitely many primes is given by a polynomial in pn and n of total degree at most d for sufficiently large n. This conjecture remains open for d≥2. In this paper, we prove that this conjecture is true in the function field case. Further, we propose a series of general conjectures on p-adic stability of zeta functions in a p-adic Lie tower of function fields.

Finiteness of Frobenius Traces of a Sheaf on a Flat Arithmetic Scheme

Abstract For a lisse $\ell $-adic sheaf on a scheme flat and of finite type over $\mathbb{Z}$, we consider the field generated over $ \mathbb{Q}$ by Frobenius traces of the sheaf at closed points. Assuming conjectural properties of geometric Galois representations of number fields and the Generalized Riemann Hypothesis, we prove that the field is finite over $\mathbb{Q}$ when the sheaf is de Rham at $\ell $ pointwise. This is a number field analog of Deligne’s finiteness result about Frobenius traces in the function field case.

Eventually stable rational functions

For a field [Formula: see text], rational function [Formula: see text] of degree at least two, and [Formula: see text], we study the polynomials in [Formula: see text] whose roots are given by the solutions in [Formula: see text] to [Formula: see text], where [Formula: see text] denotes the [Formula: see text]th iterate of [Formula: see text]. When the number of irreducible factors of these polynomials stabilizes as [Formula: see text] grows, the pair [Formula: see text] is called eventually stable over [Formula: see text]. We conjecture that [Formula: see text] is eventually stable over [Formula: see text] when [Formula: see text] is any global field and [Formula: see text] is any point not periodic under [Formula: see text] (an additional non-isotriviality hypothesis is necessary in the function field case). We prove the conjecture when [Formula: see text] has a discrete valuation for which (1) [Formula: see text] has good reduction and (2) [Formula: see text] acts bijectively on all finite residue extensions. As a corollary, we prove for these maps a conjecture of Sookdeo on the finiteness of [Formula: see text]-integral points in backward orbits. We also give several characterizations of eventual stability in terms of natural finiteness conditions, and survey previous work on the phenomenon.

Squarefree doubly primitive divisors in dynamical sequences

AbstractLetKbe a number field or a function field of characteristic 0, letφ∈K(z) with deg(φ) ⩾ 2, and letα∈ ℙ1(K). LetSbe a finite set of places ofKcontaining all the archimedean ones and the primes whereφhas bad reduction. After excluding all the natural counterexamples, we define a subsetA(φ,α) of ℤ⩾0× ℤ>0and show that for all but finitely many (m,n) ∈A(φ,α) there is a prime 𝔭 ∉Ssuch that ord𝔭(φm+n(α)−φm(α)) = 1 andαhas portrait (m,n) under the action ofφmodulo 𝔭. This latter condition implies ord𝔭(φu+v(α)−φu(α)) ⩽ 0 for (u,v) ∈ ℤ⩾0× ℤ>0satisfyingu<morv<n. Our proof assumes a conjecture of Vojta for ℙ1× ℙ1in the number field case and is unconditional in the function field case thanks to a deep theorem of Yamanoi. This paper extends earlier work of Ingram–Silverman, Faber–Granville and the authors.

Low-lying zeros of quadratic Dirichlet -functions: lower order terms for extended support

We study the $1$-level density of low-lying zeros of Dirichlet $L$-functions attached to real primitive characters of conductor at most $X$. Under the generalized Riemann hypothesis, we give an asymptotic expansion of this quantity in descending powers of $\log X$, which is valid when the support of the Fourier transform of the corresponding even test function $\unicode[STIX]{x1D719}$ is contained in $(-2,2)$. We uncover a phase transition when the supremum $\unicode[STIX]{x1D70E}$ of the support of $\widehat{\unicode[STIX]{x1D719}}$ reaches $1$, both in the main term and in the lower order terms. A new lower order term appearing at $\unicode[STIX]{x1D70E}=1$ involves the quantity $\widehat{\unicode[STIX]{x1D719}}(1)$, and is analogous to a lower order term which was isolated by Rudnick in the function field case.

Reflection principles for class groups

We present several new examples of reflection principles which apply to both class groups of number fields and picard groups of curves over P1/Fp. This proves a conjecture of Lemmermeyer [3] about equality of 2-rank in subfields of A4, up to a constant not depending on the discriminant in the number field case, and exactly in the function field case. More generally we prove similar relations for subfields of a Galois extension with group G for the cases when G is S3, S4, A4, D2l and Z/lZ⋊Z/rZ. The method of proof uses sheaf cohomology on 1-dimensional schemes, which reduces to Galois module computations.