It is conjectured that for fixed $A$, $r \ge 1$, and $d \ge 1$, there is a uniform bound on the size of the torsion submodule of a Drinfeld $A$-module of rank $r$ over a degree $d$ extension $L$ of the fraction field $K$ of $A$. We verify the conjecture for $r=1$, and more generally for Drinfeld modules having potential good reduction at some prime above a specified prime of $K$. Moreover, we show that within an $\Lbar$-isomorphism class, there are only finitely many Drinfeld modules up to isomorphism over $L$ which have nonzero torsion. For the case $A=\Fq[T]$, $r=1$, and $L=\Fq(T)$, we give an explicit description of the possible torsion submodules. We present three methods for proving these cases of the conjecture, and explain why they fail to prove the conjecture in general. Finally, an application of the Mordell conjecture for characteristic $p$ function fields proves the uniform boundedness for the $\pp$-primary part of the torsion for rank~2 Drinfeld $\Fq[T]$-modules over a fixed function field.
We study rational points and torsion points on Drinfeld modular curves defined over rational function fields. As a consequence we derive a conjecture of Schweizer describing completely the torsion of Drinfeld modules of rank two over $\Bbb F_2(T)$ implying Poonen's uniform boundedness conjecture in this particular case.
We derive asymptotically optimal upper bounds on the number of L-rational torsion points on a given elliptic curve or Drinfeld module defined over a finitely generated field K, as a function of the degree [L:K]. Our main tool is the adelic openness of the image of Galois representations attached to elliptic curves and Drinfeld modules, due to Serre and Pink-Ruetsche, respectively. Our approach is to prove a general result for certain Galois modules, which applies simultaneously to elliptic curves and to Drinfeld modules.
In this paper, we formulate the Drinfeld module analogue of a question raised by Lang and studied by Katz on the existence of rational points on abelian varieties over number fields. Given a maximal ideal $\fl$ of $\F_q[T]$, the question essentially asks whether, up to isogeny, a Drinfeld module $ϕ$ over $\F_q(T)$ contains a rational $\fl$-torsion point if the reduction of $ϕ$ at almost all primes of $\F_q[T]$ contains a rational $\fl$-torsion point. Similar to the case of abelian varieties, we show that the answer is positive if the rank of the Drinfeld module is $2$, but negative if the rank…
In this short note we provide a few examples of non-isomorphic arithmetically equivalent global function fields. These examples are obtained via well-known technique of adjoining the torsion points of various Drinfeld Modules to realise the $Gl_n(\mathbb F_q)$ as a Galois group of extensions of global function fields. Furthermore we afford the code of the Magma scripts to verify the results and construct more examples in similar fashion.
To each Drinfeld module over a finitely generated field with generic characteristic, one can associate a Galois representation arising from the Galois action on its torsion points. Recent work of Pink and Rütsche has described the image of this representation up to commensurability. Their theorem is qualitative, and the objective of this paper is to complement this theory with a worked out example. In particular, we give examples of Drinfeld modules of rank 2 for which the Galois action on their torsion points is as large as possible. We will follow the approach that Serre used to give explici…
Geyer and Jarden proved several results for torsion points of elliptic curves defined over the fixed field by finitely many elements in the absolute Galois group of a finitely generated field over the prime field in its algebraic closure. As an analogue of these results, this paper studies torsion points of Drinfeld modules defined over the fixed field by finitely many elements in the absolute Galois group of a finitely generated function field in its algebraic closure. We prove some results which are similar to those of Geyer and Jarden.
Let $\frak{n}$ be a square-free ideal of $\mathbb{F}_q[T]$. We study the rational torsion subgroup of the Jacobian variety $J_0(\frak{n})$ of the Drinfeld modular curve $X_0(\frak{n})$. We prove that for any prime number $\ell$ not dividing $q(q-1)$, the $\ell$-primary part of this group coincides with that of the cuspidal divisor class group. We further determine the structure of the $\ell$-primary part of the cuspidal divisor class group for any prime $\ell$ not dividing $q-1$.
We prove an equidistribution result for torsion points of Drinfeld modules of generic characteristic. We also show that similar equidistribution statements provide proofs for the Manin-Mumford and the Bogomolov conjectures for Drinfeld modules.
If M is a Drinfeld module over a local function field L, we may view M as a dynamical system, and consider its filled Julia set J. If J^0 is the connected component of the identity, relative to the Berkovich topology, we give a characterisation of the component module J/J^0 which is analogous to the Kodaira-Neron characterisation of the special fibre of a Neron model of an elliptic curve over a non-archimedean field. In particular, if L is the fraction field of a discrete valuation ring, then the component module is finite, and moreover trivial in the case of good reduction. In the context of …
Let $q$ be a power of the prime number $p$, let $K={\mathbb F}_q(t)$, and let $r\ge 2$ be an integer. For points ${\mathbf a}, {\mathbf b}\in K$ which are $\mathbb{F}_q$-linearly independent, we show that there exist positive constants $N_0$ and $c_0$ such that for each integer $\ell\ge N_0$ and for each generator $τ$ of ${\mathbb F}_{q^\ell}/{\mathbb F}_q$, we have that for all except $N_0$ values $λ\in{\overline{\mathbb{F}_q}}$, the corresponding specializations ${\mathbf a}, {\mathbf b}(τ)$ and ${\mathbf b}(τ)$ cannot have orders of degrees less than $c_0\log\log\ell$ as torsion points for …
Let $Φ^ł$ be an algebraic family of Drinfeld modules defined over a field $K$ of characteristic $p$, and let $\bfa,\bfb\in K[ł]$. Assume that neither $\bfa(ł)$ nor $\bfb(ł)$ is a torsion point for $Φ^ł$ for all $ł$. If there exist infinitely many $ł\in\Kbar$ such that both $\bfa(ł)$ and $\bfb(ł)$ are torsion points for $Φ^ł$, then we show that for each $ł\in\Kbar$, we have that $\bfa(ł)$ is torsion for $Φ^ł$ if and only if $\bfb(ł)$ is torsion for $Φ^ł$. In the case $\bfa,\bfb\in K$, then we prove in addition that $\bfa$ and $\bfb$ must be $\Fpbar$-linearly dependent.
We prove that the groups PSL(r,q^d) can be realized F_q(T)-regularly as Galois groups over the purely transcendental field F_q(T)(t_1,...,t_{r-1}) if r is even and r/2 is coprime to q^d-1. The method is to use twisted moduli schemes of Drinfeld modules (a la Shih). If one has sufficient control over modular forms, it can be made into an algorithm to find explicit equations. We do this for PSL(2,q^2) using Drinfeld modular forms.
We generalize our work on Carlitz prime power torsion extension to torsion extensions of Drinfeld modules of arbitrary rank. As in the Carlitz case, we give a description of these extensions in terms of evaluations of Anderson generating functions and their hyperderivatives at roots of unity. We also give a direct proof that the image of the Galois representation attached to the $\mathfrak{p}$-adic Tate module lies in the $\mathfrak{p}$-adic points of the motivic Galois group. This is a generalization of the corresponding result of Chang and Papanikolas for the $t$-adic case.
We consider the generalized Jacobian $\widetilde{J}$ of the modular curve $X_0(N)$ of level $N$ with respect to a reduced divisor consisting of all cusps. Supposing $N$ is square free, we explicitly determine the structure of the $\mathbb{Q}$-rational torsion points on $\widetilde{J}$ up to $6$-primary torsion. The result turns out to be very different from the case of prime power level previously studied by Yang and the second author. We also obtain an analogous result for Drinfeld modular curves. Our proof relies on similar results for classical Jacobians due to Ohta, Papikian and the first …
Let ϕ be a Drinfeld A-module of characteristic p0 over a finitely generated field K. Previous articles determined the image of the absolute Galois group of K up to commensurability in its action on all prime-to-p0 torsion points of ϕ, or equivalently, on the prime-to-p0 adelic Tate module of ϕ. In this article we consider in addition a finitely generated torsion free A-submodule M of K for the action of A through ϕ. We determine the image of the absolute Galois group of K up to commensurability in its action on the prime-to-p0 division hull of M, or equivalently, on the extended prime-to-p0 ad…
In this paper we generalize the results of \cite{sanchez} to rank one Drinfeld modules with class number one. We show that, in the present form, there does not exist a cogalois theory for Drinfeld modules of rank or class number larger than one. We also consider the torsion of the Carlitz module for the extension ${\mathbb{F}}_{q}(T)(Λ_{P^{n}}) /{\mathbb{F}}_{q}(T)(Λ_{P})$.
We prove that in the backward orbit of a non-preperiodic point under the action of a Drinfeld module of generic characteristic there exist at most finitely many points S-integral with respect to another nonpreperiodic point. This provides the answer (in positive characteristic) to a question raised by Sookdeo. We also prove that for each nontorsion point z, there exist at most finitely many torsion points which are S-integral with respect to z. This proves a question raised by Tucker and the author, and it gives the analogue of Ih's conjecture for Drinfeld modules.
In the arithmetic of function fields, Drinfeld modules play the role that elliptic curves play in the arithmetic of number fields. The aim of this paper is to study a non-existence problem of Drinfeld modules with constrains on torsion points at places with large degree. This is motivated by a conjecture of Christopher Rasmussen and Akio Tamagawa on the non-existence of abelian varieties over number fields with some arithmetic constraints. We prove the non-existence of Drinfeld modules satisfying Rasmussen-Tamagawa type conditions in the case where the inseparable degree of base fields is not …
We formulate the notion of \emph{typical boundedness} of torsion on a family of abelian varieties defined over number fields. This means that the torsion subgroups of elements in the family can be made uniformly bounded by removing from the family all abelian varieties defined over number fields of degree lying in a set of arbitrarily small density. We show that for each fixed $g$, torsion is typically bounded on the family of all $g$-dimensional CM abelian varieties. We show that torsion is \emph{not} typically bounded on the family of all elliptic curves, and we establish results -- some unc…
In this paper, we prove explicit reciprocity laws for a class of formal Drinfeld modules having stable reduction of height one, in the spirit of those existing in characteristic zero (cf. the work of Wiles). We begin by defining the Kummer pairing in the language of formal Drinfeld modules defined over local fields of positive characteristic. We then prove explicit formulas for this pairing in terms of the logarithm of the formal Drinfeld module, a certain Coleman power series, torsion points and the trace. Our results extend the explicit formulas already proved by Anglès for Carlitz modules, …
Lower Bound for the Canonical Height for Drinfeld Modules with Complex Multiplication. Let K be a fi nite extension of Fq(T), let L=K be a Galois extension with Galois group G and let E be the sub eld of L fixed by the center of G. Assume that there exists a finite place v of K such that the local degrees of E=K above v are bounded. Let $ϕ$ be a Drinfeld module with complex multiplication. We give an e fective lower bound for the canonical height of $ϕ$ on L outside the torsion points of $ϕ$ . In the number field case, this problem was solved by F. Amoroso, S. David and U. Zannier.
Let $A$ be the coordinate ring of a projective smooth curve over a finite field minus a closed point. For a nontrivial ideal $I \subset A$, Drinfeld defined the notion of structure of level $I$ on a Drinfeld module. We extend this to that of level $N$, where $N$ is a finitely generated torsion $A$-module. The case where $N=(I^{-1}/A)^d$, where $d$ is the rank of the Drinfeld module,coincides with the structure of level $I$. The moduli functor is representable by a regular affine scheme. The automorphism group $\mathrm{Aut}_{A}(N)$ acts on the moduli space. Our theorem gives a class of subgroup…
We prove that the local height of a point on a Drinfeld module can be computed by averaging the logarithm of the distance to that point over the torsion points of the module. This gives rise to a Drinfeld module analog of a weak version of Siegel's integral points theorem over number fields and to an analog of a theorem of Schinzel's regarding the order of a point modulo certain primes.
Denis associated to each Drinfeld module M over a global function function field L a canonical height function, which plays a role analogous to that of the Neron-Tate height in the context of elliptic curves. We prove that there exist constants ε>0 and C, depending only on the number of places at which M has bad reduction, such that either x in M is a torsion point of bounded order, or else the canonical height of x is bound below by εmax{h(j_M), deg(D_M)}, where j_M is a certain invariant of the isomorphism class of M, and D_M is the minimal discriminant of M. As an application, we make some …
Merel's result on the strong uniform boundedness conjecture made it meaningful to classify the torsion part of the Mordell-Weil groups of all elliptic curves defined over number fields of fixed degree $d$. In this paper, we discuss the cyclic torsion subgroup of elliptic curves over cubic number fields. For $N=49,40,25$ or $22$, we show that $\mathbb{Z}/N\mathbb{Z}$ is not a subgroup of $E(K)_{tor}$ for any elliptic curve $E$ over a cubic number field $K$.
Let $F$ be a local non-Archimedean field with ring of integers $o$. Let $\bf X$ be a one-dimensional formal $o$-module of $F$-height $n$ over the algebraic closure of the residue field of $o$. By the work of Drinfeld, the universal deformation $X$ of $\bf X$ is a formal group over a power series ring $R_0$ in $n-1$ variables over the completion of the maximal unramified extension of $o$. For $h \in \{0,...,n-1\}$ let $U_h$ be the subscheme of $\Spec(R_0)$ where the connected part of the associated divisible module of $X$ has height $h$. Using the theory of Drinfeld level structures we show tha…
We formulate and study a torsion analogue of the weight-monodromy conjecture for a proper smooth scheme over a non-archimedean local field. We prove it for proper smooth schemes over equal characteristic non-archimedean local fields, abelian varieties, surfaces, varieties uniformized by Drinfeld upper half spaces, and set-theoretic complete intersections in toric varieties. In the equal characteristic case, our methods rely on an ultraproduct variant of Weil II established by Cadoret.
The Tate conjecture for divisors on varieties over number fields is equivalent to finiteness of $\ell$-primary torsion in the Brauer group. We show that this finiteness is actually uniform in one-dimensional families for varieties that satisfy the Tate conjecture for divisors -- e.g. abelian varieties and $K3$ surfaces.
We show that for all epsilon > 0, there is a constant C(epsilon) > 0 such that for all elliptic curves E defined over a number field F with j(E) in Q we have #E(F)[tors] \leq C(epsilon)[F:Q]^{5/2+epsilon}. We pursue further bounds on the size of the torsion subgroup of an elliptic curve over a number field E/F that are polynomial in [F:Q] under restrictions on j(E). We give an unconditional result for j(E) lying in a fixed quadratic field that is not imaginary of class number one as well as two further results, one conditional on GRH and one conditional on the strong boundedness of isogenies o…
In 1983 Silverman and Tate showed that the set of points in a 1-dimensional family of abelian varieties where a section of infinite order has `small height' is finite. We conjecture a generalisation to higher-dimensional families, where we replace `finite' by `not Zariski dense'. We show that this conjecture would imply the Uniform Boundedness Conjecture for torsion points on abelian varieties. We then prove a few special cases of this new conjecture.
As a generalization of Drinfeld modules, Greg Anderson introduced abelian t-modules and t-motives over a perfect field. In this article we study relative versions of these over 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 a…
Let $k$ be a finitely generated field of characteristic $p>0$ and $X$ a smooth and proper scheme over $k$. Recent works of Cadoret, Hui and Tamagawa show that, if $X$ satisfies the $\ell$-adic Tate conjecture for divisors for every prime $\ell\neq p$, the Galois invariant subgroup $Br(X_{\overline k})[p']^{π_1(k)}$ of the prime-to-$p$ torsion of the geometric Brauer group of $X$ is finite. The main result of this note is that, for every integer $d\geq 1$, there exists a constant $C:=C(X,d)$ such that for every finite field extension $k \subseteq k'$ with $[k':k]\leq d$ and every $(\overline k/…
We study the analogy between number fields and function fields in one variable over finite fields. The main result is an isomorphism between the Hilbert class fields of class number one and a family of the function fields $\mathbf{F}_q(C)$ over a desingularized algebraic curve $C$. Our proof is based on the K-theory of the Serre $C^*$-algebras and birational geometry of the curve $C$. We apply the isomorphism to construct explicit generators of the Hilbert class fields coming from the torsion submodules of the Drinfeld module.
Let $\frak{p}$ and $\frak{q}$ be two distinct prime ideals of $\mathbb{F}_q[T]$. We use the Eisenstein ideal of the Hecke algebra of the Drinfeld modular curve $X_0(\frak{p}\frak{q})$ to compare the rational torsion subgroup of the Jacobian $J_0(\frak{p}\frak{q})$ with its subgroup generated by the cuspidal divisors, and to produce explicit examples of Jacquet-Langlands isogenies. Our results are stronger than what is currently known about the analogues of these problems over $\mathbb{Q}$.
