Function Field Analogue Research Articles

Bounds for the stalks of perverse sheaves in characteristic p and a conjecture of Shende and Tsimerman

We prove a characteristic p analogue of a result of Massey which bounds the dimensions of the stalks of a perverse sheaf in terms of certain intersection multiplicities of the characteristic cycle of that sheaf. This uses the construction of the characteristic cycle of a perverse sheaf in characteristic p by Saito. We apply this to prove a conjecture of Shende and Tsimerman on the Betti numbers of the intersections of two translates of theta loci in a hyperelliptic Jacobian. This implies a function field analogue of the Michel–Venkatesh mixing conjecture about the equidistribution of CM points on a product of two modular curves.

A geometric approach to counting norms in cyclic extensions of function fields

In this paper we prove an explicit version of a function field analogue of a classical result of Odoni about norms in number fields in the case of a cyclic Galois extensions. In the particular case of a quadratic extension, we recover the result of Bary-Soroker, Smilanski, and Wolf which deals with finding asymptotics for a function field version on sums of two squares, improved upon by Gorodetsky , and reproved by the author in his Ph.D thesis using the method of this paper. The main tool is a twisted Grothendieck Lefschetz trace formula, inspired by the work of Church, Farb and Ellenberg on representation stability and asymptotic for point counts on varieties. Using a combinatorial description of the cohomology we obtain a precise quantitative result which works in the $q^n\rightarrow \infty$ regime, and a new type of homological stability phenomena, which arises from the computation of certain inner products of representations.

Product formulas for periods of CM abelian varieties and the function field analog

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.

On a conjecture of Furusho over function fields

In the classical theory of multiple zeta values (MZV’s), Furusho proposed a conjecture asserting that the p-adic MZV’s satisfy the same \({\mathbb {Q}}\)-linear relations that their corresponding real-valued MZV counterparts satisfy. In this paper, we verify a stronger version of a function field analogue of Furusho’s conjecture in the sense that we are able to deal with all linear relations over an algebraic closure of the given rational function field, not just the rational linear relations. To each tuple of positive integers \({\mathfrak {s}}=(s_1, \ldots , s_r)\), we construct a corresponding t-module together with a specific rational point. The fine resolution (via fiber coproduct) of this construction actually allows us to obtain nice logarithmic interpretations for both the \(\infty \)-adic MZV and v-adic MZV at \({\mathfrak {s}}\), completely generalizing the work of Anderson–Thakur (Ann Math (2) 132(1):159–191, 1990) in the case of \(r=1\). Furthermore it enables us to apply Yu’s sub-t-module theorem (Yu in Ann Math (2) 145(2):215–233, 1997), connecting any \(\infty \)-adic linear relation on MZV’s with a sub-t-module of a corresponding giant t-module. This makes it possible to arrive at the same linear relation for v-adic MZV’s.

A representation theory approach to integral moments of L-functions over function fields

We propose a new heuristic approach to integral moments of L-functions over function fields, which we demonstrate in the case of Dirichlet characters ramified at one place (the function field analogue of the moments of the Riemann zeta function, where we think of the character n^{it} as ramified at the infinite place). We represent the moment as a sum of traces of Frobenius on cohomology groups associated to irreducible representations. Conditional on a hypothesis on the vanishing of some of these cohomology groups, we calculate the moments of the L-function and they match the predictions of the Conrey-Farmer-Keating-Rubinstein-Snaith recipe. In this case, the decomposition into irreducible representations seems to separate the main term and error term, which are mixed together in the long sums obtained from the approximate functional equation, even when it is dyadically decomposed. This makes our heuristic statement relatively simple, once the geometric background is set up. We hope that this will clarify the situation in more difficult cases like the L-functions of quadratic Dirichlet characters to squarefree modulus. There is also some hope for a geometric proof of this cohomological hypothesis, which would resolve the moment problem for these L-functions in the large degree limit over function fields.

Gonality of dynatomic curves and strong uniform boundedness of preperiodic points

Fix$d\geqslant 2$and a field$k$such that$\operatorname{char}k\nmid d$. Assume that$k$contains the$d$th roots of$1$. Then the irreducible components of the curves over$k$parameterizing preperiodic points of polynomials of the form$z^{d}+c$are geometrically irreducible and have gonality tending to$\infty$. This implies the function field analogue of the strong uniform boundedness conjecture for preperiodic points of$z^{d}+c$. It also has consequences over number fields: it implies strong uniform boundedness for preperiodic points of bounded eventual period, which in turn reduces the full conjecture for preperiodic points to the conjecture for periodic points. Our proofs involve a novel argument specific to finite fields, in addition to more standard tools such as the Castelnuovo–Severi inequality.

On the Siegel-Weil formula for classical groups over function fields

We establish a Siegel-Weil formula for classical groups over a function field with odd characteristic, which asserts in many cases that the Siegel Eisenstein series is equal to an integral of a theta function. This is a function-field analogue of the classical result proved by A. Weil in his 1965 Acta Math. paper. We also give a convergence criterion for the theta integral by using Harder's reduction theory over function fields.

The fourth power mean of Dirichlet L-functions in {mathbb {F}}_q [T

We prove results on moments of L-functions in the function field setting, where the moment averages are taken over primitive characters of modulus R, where R is a polynomial in {mathbb {F}}_{q}[T]. We consider the behaviour as {{,mathrm{deg},}}R rightarrow infty and the cardinality of the finite field is fixed. Specifically, we obtain an exact formula for the second moment provided that R is square-full, an asymptotic formula for the second moment for any R, and an asymptotic formula for the fourth moment for any R. The fourth moment result is a function field analogue of Soundararajan’s result in the number field setting that improved upon a previous result by Heath-Brown. Both the second and fourth moment results extend work done by Tamam in the function field setting who focused on the case where R is prime. As a prerequisite for the fourth moment result, we obtain, for the special case of the divisor function, the function field analogue of Shiu’s generalised Brun–Titchmarsh theorem.

Limiting Properties of the Distribution of Primes in an Arbitrarily Large Number of Residue Classes

AbstractWe generalize current known distribution results on Shanks–Rényi prime number races to the case where arbitrarily many residue classes are involved. Our method handles both the classical case that goes back to Chebyshev and function field analogues developed in the recent years. More precisely, let $\unicode[STIX]{x1D70B}(x;q,a)$ be the number of primes up to $x$ that are congruent to $a$ modulo $q$. For a fixed integer $q$ and distinct invertible congruence classes $a_{0},a_{1},\ldots ,a_{D}$, assuming the generalized Riemann Hypothesis and a weak version of the linear independence hypothesis, we show that the set of real $x$ for which the inequalities $\unicode[STIX]{x1D70B}(x;q,a_{0})>\unicode[STIX]{x1D70B}(x;q,a_{1})>\cdots >\unicode[STIX]{x1D70B}(x;q,a_{D})$ are simultaneously satisfied admits a logarithmic density.

Isogenies of abelian Anderson A-modules and A-motives

As a generalization of Drinfeld modules, Greg Anderson introduced Abelian t -modules and t -motives over a perfect base field. In this article we study relative versions of these defined over base rings. We investigate isogenies among them. Our main results state that every isogeny possesses a dual isogeny in the opposite direction, and that a morphism between Abelian t -modules is an isogeny if and only if the corresponding morphism between their associated t -motives is an isogeny. We also study torsion submodules of Abelian t -modules which in general are non-reduced group schemes. They can be obtained from the associated t -motive via the finite shtuka correspondence of Drinfeld and Abrashkin. The inductive limits of torsion submodules are the function field analogs of p-divisible groups. These limits correspond to the local shtukas attached to the t -motives associated with the Abelian t -modules. In this sense the theory of Abelian t - modules is captured by the theory of t -motives.

Uniformizing the Moduli Stacks of Global G-Shtukas

Abstract This is the 2nd in a sequence of articles, in which we explore moduli stacks of global $\mathfrak{G}$-shtukas, the function field analogs for Shimura varieties. Here $\mathfrak{G}$ is a flat affine group scheme of finite type over a smooth projective curve $C$ over a finite field. Global $\mathfrak{G}$-shtukas are generalizations of Drinfeld shtukas and analogs of abelian varieties with additional structure. We prove that the moduli stacks of global $\mathfrak{G}$-shtukas are algebraic Deligne–Mumford stacks separated and locally of finite type. They generalize various moduli spaces used by different authors to prove instances of the Langlands program over function fields. In the 1st article we explained the relation between global $\mathfrak{G}$-shtukas and local ${{\mathbb{P}}}$-shtukas, which are the function field analogs of $p$-divisible groups. Here ${{\mathbb{P}}}$ is the base change of $\mathfrak{G}$ to the complete local ring at a point of $C$. When ${{\mathbb{P}}}$ is smooth with connected reductive generic fiber we proved the existence of Rapoport–Zink spaces for local ${{\mathbb{P}}}$-shtukas. In the present article we use these spaces to (partly) uniformize the moduli stacks of global $\mathfrak{G}$-shtukas for smooth $\mathfrak{G}$ with connected fibers and reductive generic fiber. This is our main result. It has applications to the analog of the Langlands–Rapoport conjecture for our moduli stacks.

Chebotarev density theorem in short intervals for extensions of 𝔽_{𝕢}(𝕋)

An old open problem in number theory is whether the Chebotarev density theorem holds in short intervals. More precisely, given a Galois extension E E of Q \mathbb {Q} with Galois group G G , a conjugacy class C C in G G , and a 1 ≥ ε > 0 1\geq \varepsilon >0 , one wants to compute the asymptotic of the number of primes x ≤ p ≤ x + x ε x\leq p\leq x+x^{\varepsilon } with Frobenius conjugacy class in E E equal to C C . The level of difficulty grows as ε \varepsilon becomes smaller. Assuming the Generalized Riemann Hypothesis, one can merely reach the regime 1 ≥ ε > 1 / 2 1\geq \varepsilon >1/2 . We establish a function field analogue of the Chebotarev theorem in short intervals for any ε > 0 \varepsilon >0 . Our result is valid in the limit when the size of the finite field tends to ∞ \infty and when the extension is tamely ramified at infinity. The methods are based on a higher dimensional explicit Chebotarev theorem and applied in a much more general setting of arithmetic functions, which we name G G -factorization arithmetic functions.

Representation of units in cyclotomic function fields

In this paper, we prove a refinement of Hilbert’s Satz 90 for the [Formula: see text]-cyclotomic function fields, where [Formula: see text] is a monic prime in [Formula: see text]. Our result can be viewed as a function field analogue of Newman’s theorem which is a refinement of Hilbert’s Satz 90 for [Formula: see text].

Differential characters of Drinfeld modules and de Rham cohomology

We introduce differential characters of Drinfeld modules. These are function-field analogues of Buium's p-adic differential characters of elliptic curves and of Manin's differential characters of elliptic curves in differential algebra, both of which have had notable Diophantine applications. We determine the structure of the group of differential characters. This shows the existence of a family of interesting differential modular functions on the moduli of Drinfeld modules. It also leads to a canonical $F$-crystal equipped with a map to the de Rham cohomology of the Drinfeld module. This $F$-crystal is of a differential-algebraic nature, and the relation to the classical cohomological realizations is presently not clear.

Angles of Gaussian primes

Fermat showed that every prime p = 1 mod 4 is a sum of two squares: p = a2 + b2. To any of the 8 possible representations (a, b) we associate an angle whose tangent is the ratio b/a. In 1919 Hecke showed that these angles are uniformly distributed as p varies, and in the 1950’s Kubilius proved uniform distribution in somewhat short arcs. We study fine scale statistics of these angles, in particular the variance of the number of such angles in a short arc. We present a conjecture for this variance, motivated both by a random matrix model, and by a function field analogue of this problem, for which we prove an asymptotic form for the corresponding variance.

The Sato-Tate distribution in thin families of elliptic curves over high degree extensions of finite fields

Over the last two decades, there has been a wave of activity establishing the Sato-Tate kind of distribution in various families of elliptic curves over prime fields. Typically the goal here is to prove this for families which are as thin as possible. We consider a function field analogue of this question, that is, for high degree extensions of a finite field where new effects allow us to study families, which are much thinner that those typically investigated over prime fields.

Nonabelian Cohen–Lenstra moments

In this article, we give a conjecture for the average number of unramified G-extensions of a quadratic field for any finite group G. The Cohen–Lenstra heuristics are the specialization of our conjecture to the case in which G is abelian of odd order. We prove a theorem toward the function field analogue of our conjecture and give additional motivations for the conjecture, including the construction of a lifting invariant for the unramified G-extensions that takes the same number of values as the predicted average and an argument using the Malle–Bhargava principle. We note that, for even |G|, corrections for the roots of unity in Q are required, which cannot be seen when G is abelian.

On Galois cohomology of connected reductive groups and Kottwitz exact sequence

We consider some function field analogs of some main cohomological results of Kottwitz theory used in his theory for stable trace formula and extend an important exact sequence of Kottwitz theory to the case of any global field and give some applications.

Irreducible Polynomials Over a Finite Field with Restricted Coefficients

AbstractWe prove a function field analogue of Maynard’s celebrated result about primes with restricted digits. That is, for certain ranges of parameters $n$ and $q$ , we prove an asymptotic formula for the number of irreducible polynomials of degree $n$ over a finite field $\mathbb{F}_{q}$ whose coefficients are restricted to lie in a given subset of $\mathbb{F}_{q}$ .

The function field Sathe--Selberg formula in arithmetic progressions and `short intervals'

We use a function field analogue of a method of Selberg to derive an asymptotic formula for the number of (square-free) monic polynomials in $\mathbb{F}_q[X]$ of degree $n$ with precisely $k$ irreducible factors, in the limit as $n$ tends to infinity. We then adapt this method to count such polynomials in arithmetic progressions and short intervals, and by making use of Weil's `Riemann hypothesis' for curves over $\mathbb{F}_q$, obtain better ranges for these formulae than are currently known for their analogues in the number field setting. Finally, we briefly discuss the regime in which $q$ tends to infinity.