Adelic openness for Drinfeld modules in generic characteristic

Let φ be a Drinfeld A-module of arbitrary rank and generic characteristic over a finitely generated field K. If the endomorphism ring of φ over an algebraic closure of K is equal to A, we prove that the image of the adelic Galois representation associated to φ is open.     

Periods of orbits modulo primes

Let S be a monoid of endomorphisms of a quasiprojective variety V defined over a global field K. We prove a lower bound for the size of the reduction modulo places of K of the orbit of any point α∈V(K) under the action of the endomorphisms from S. We also prove a similar result in the context of Drinfeld modules. Our results may be considered as dynamical variants of Artin's primitive root conjecture.     

Pesenti-Szpiro inequality for optimal elliptic curves

We study Pesenti-Szpiro inequality in the case of elliptic curves over F_q(t) which occur as subvarieties of Jacobian varieties of Drinfeld modular curves. In general, we obtain an upper-bound on the degrees of minimal discriminants of such elliptic curves in terms of the degrees of their conductors and q. In the special case when the level is prime, we bound the degrees of discriminants only in terms of the degrees of conductors. As a preliminary step in the proof of this latter result we generalize a construction (due to Gekeler and Reversat) of 1-dimensional optimal quotients of Drinfeld Jacobians.     

Borne uniforme pour les homothéties dans l?image de Galois associée aux courbes elliptiques

Let K be a fixed number field and GK its absolute Galois group. We give a bound C(K), depending only on the degree, the class number and the discriminant of K, such that for any elliptic curve E defined over K and any prime number p strictly larger than C(K), the image of the representation of GK attached to the p-torsion points of E contains a subgroup of homotheties of index smaller than 12.     

Pure Anderson motives over finite fields

In the arithmetic of function fields Drinfeld modules play the role that elliptic curves take on in the arithmetic of number fields. As higher dimensional generalizations of Drinfeld modules, and as the appropriate analogues of abelian varieties, G. Anderson introduced pure t-motives. In this article we study the arithmetic of the latter. We investigate which pure t-motives are semisimple, that is, isogenous to direct sums of simple ones. We give examples for pure t-motives which are not semisimple. Over finite fields the semisimplicity is equivalent to the semisimplicity of the endomorphism algebra, but also this fails over infinite fields. Still over finite fields we study the Zeta function and the endomorphism rings of pure t-motives and criteria for the existence of isogenies. We obtain answers which are similar to Tate's famous results for abelian varieties.     

On the vanishing of Selmer groups for elliptic curves over ring class fields

Let E/Q be an elliptic curve of conductor N without complex multiplication and let K be an imaginary quadratic field of discriminant D prime to N. Assume that the number of primes dividing N and inert in K is odd, and let Hc be the ring class field of K of conductor c prime to ND with Galois group Gc over K. Fix a complex character χ of Gc. Our main result is that if LK(E,χ,1)≠0 then Selp(E/Hc)χ_⊗W=0 for all but finitely many primes p, where Selp(E/Hc) is the p-Selmer group of E over Hc and W is a suitable finite extension of Zp containing the values of χ. Our work extends results of Bertolini and Darmon to almost all non-ordinary primes p and also offers alternative proofs of a χ-twisted version of the Birch and Swinnerton-Dyer conjecture for E over Hc (Bertolini and Darmon) and of the vanishing of Selp(E/K) for almost all p (Kolyvagin) in the case of analytic rank zero.     

A Weil-Barsotti formula for Drinfeld modules

We study the group of extensions in the category of Drinfeld modules and Anderson's t-modules, and we show in certain cases that this group can itself be given the structure of a t-module. Our main result is a Drinfeld module analogue of the Weil-Barsotti formula for abelian varieties. Extensions of general t-modules are also considered, in particular extensions of tensor powers of the Carlitz module. We motivate these results from various directions and compare to the situation of elliptic curves.     

The Galois representations associated to a Drinfeld module in special characteristic—II: Openness

Let φ be a Drinfeld A-module in special characteristic p₀ over a finitely generated field K. For any finite set P of primes p≠p₀ of A let Γ_P denote the image of Gal(K^sep/K) in its representation on the product of the p-adic Tate modules of φ for all p∈P. We determine Γ_P up to commensurability.     

Generalized theta functions, Drinfeld modules and some arithmetic consequences

In a paper from 1994, G.W. Anderson studies the relation between theta functions and rank-one Drinfeld modules. Here, we study generalized theta functions in relation to rank-n Drinfeld modules, explicitly obtaining Plucker coordinates for Drinfeld modules.     

Bernoulli-Hurwitz numbers, Wieferich primes and Galois representations

Let K be a quadratic imaginary number field with discriminant DK≠−3,−4 and class number one. Fix a prime p?7 which is unramified in K. Given an elliptic curve A/Q with complex multiplication by K, let be the representation which arises from the action of Galois on the Tate module. Herein it is shown that, for all but finitely many inert primes p, the image of a certain deformation of is “as large as possible”, that is, it is the full inverse image of a Cartan subgroup of SL(2,Zp). If p splits in K, then the same result holds as long as a certain Bernoulli-Hurwitz number is a p-adic unit which, in turn, is equivalent to a prime ideal not being a Wieferich place. The proof rests on the theory of elliptic units of Robert and Kubert-Lang, and on the two-variable main conjecture of Iwasawa theory for quadratic imaginary fields.     

Root numbers and ranks in positive characteristic

For a global field K and an elliptic curve E_η over K(T), Silverman's specialization theorem implies rank(E_η(K(T)))?rank(E_t(K)) for all but finitely many t∈P¹(K). If this inequality is strict for all but finitely many t, the elliptic curve E_η is said to have elevated rank. All known examples of elevated rank for K=Q rest on the parity conjecture for elliptic curves over Q, and the examples are all isotrivial.Some additional standard conjectures over Q imply that there does not exist a non-isotrivial elliptic curve over Q(T) with elevated rank. In positive characteristic, an analogue of one of these additional conjectures is false. Inspired by this, for the rational function field K=κ(u) over any finite field κ with characteristic ≠2, we construct an explicit 2-parameter family E_c,d of non-isotrivial elliptic curves over K(T) (depending on arbitrary c,d∈κ^×) such that, under the parity conjecture, each E_c,d has elevated rank.     

Stable anti-Yetter–Drinfeld modules

We define and study a class of entwined modules (stable anti-Yetter–Drinfeld modules) that serve as coefficients for the Hopf-cyclic homology and cohomology. In particular, we explain their relationship with Yetter–Drinfeld modules and Drinfeld doubles. Among sources of examples of stable anti-Yetter–Drinfeld modules, we find Hopf–Galois extensions with a flipped version of the Miyashita–Ulbrich action. To cite this article: P.M. Hajac et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

On the variation of Tate-Shafarevich groups of elliptic curves over hyperelliptic curves

Let E be an elliptic curve over F=F_q(t) having conductor (p)·∞, where (p) is a prime ideal in F_q[t]. Let d∈F_q[t] be an irreducible polynomial of odd degree, and let . Assume (p) remains prime in K. We prove the analogue of the formula of Gross for the special value L(E⊗_FK,1). As a consequence, we obtain a formula for the order of the Tate-Shafarevich group Ш(E/K) when L(E⊗_FK,1)≠0.     

On elliptic units and p-adic Galois representations attached to elliptic curves

Let K be a quadratic imaginary number field with discriminant D_K≠-3,-4 and class number one. Fix a prime p?7 which is not ramified in K and write h_p for the class number of the ray class field of K of conductor p. Given an elliptic curve A/K with complex multiplication by K, let be the representation which arises from the action of Galois on the Tate module. Herein it is shown that if then the image of a certain deformation of is “as big as possible”, that is, it is the full inverse image of a Cartan subgroup of SL(2,Z_p). The proof rests on the theory of Siegel functions and elliptic units as developed by Kubert, Lang and Robert.     

Weil Pairing for Drinfeld Modules

As is well-known, there exists a Weil pairing for elliptic curves which is a perfect bilinear form from the m-torsion of the elliptic curve E to the m-th roots of unity. In this paper we will show how Andersons paper [1] gives rise to an analogue of this pairing for Drinfeld modules.The author was supported by NWO Grant 613.007.040. The author would like to thank G. Böckle and S. J. Edixhoven for their comments.     

Elliptic units in K2

We present certain norm-compatible systems in K₂ of function fields of some CM elliptic curves. We demonstrate that these systems have some properties similar to elliptic units.     

Superspecial abelian varieties and the Eichler Basis Problem for Hilbert modular forms

Let p be an unramified prime in a totally real field L such that h⁺(L)=1. Our main result shows that Hilbert modular newforms of parallel weight two for Γ₀(p) can be constructed naturally, via classical theta series, from modules of isogenies of superspecial abelian varieties with real multiplication on a Hilbert moduli space. This may be viewed as a geometric reinterpretation of the Eichler Basis Problem for Hilbert modular forms.     

K3 surfaces, rational curves, and rational points

We prove that for any of a wide class of elliptic surfaces X defined over a number field k, if there is an algebraic point on X that lies on only finitely many rational curves, then there is an algebraic point on X that lies on no rational curves. In particular, our theorem applies to a large class of elliptic K3 surfaces, which relates to a question posed by Bogomolov in 1981.     

Image of the group ring of the Galois representation associated to Drinfeld modules

Let φ be a Drinfeld A-module of arbitrary rank and arbitrary characteristic over a finitely generated field K, and set GK=Gal(Ksep/K). Let E=EndK(φ). We show that for almost all primes p of A the image of the group ring A[GK] in EndA(Tp(φ)) is the commutant of E. In the special case E=A it follows that the representation of GK on the p-torsion points φ[p](Ksep) of φ is absolutely irreducible for almost all p.     

Torsion points on elliptic curves over function fields and a theorem of Igusa

If F   is a global function field of characteristic p>3$p>3$, we employ Tate's theory of analytic uniformization to give an alternative proof of a theorem of Igusa describing the image of the natural Galois representation on torsion points of non-isotrivial elliptic curves defined over F. Along the way, using basic properties of Faltings heights of elliptic curves, we offer a detailed proof of the function field analogue of a classical theorem of Shafarevich according to which there are only finitely many F-isomorphism classes of admissible elliptic curves defined over F with good reduction outside a fixed finite set of places of F. We end the paper with an application to torsion points rational over abelian extensions of F.