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.     

The Galois representations associated to a Drinfeld module in special characteristic—I: Zariski density

Let ? be a Drinfeld A-module of rank r over a finitely generated field K. Assume that ? has special characteristic p₀ and consider any prime p≠p₀ of A. If End_K^sep(?)=A, we prove that the image of Gal(K^sep/K) in its representation on the p-adic Tate module of ? is Zariski dense in GL_r.     

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.     

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.     

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.     

The Galois group of the Eisenstein polynomial X 5 + aX + a

Let p be a rational prime and let a be an integer which is divisible by p exactly to the first power. Then the Galois group of the Eisenstein polynomial f = X ^p + aX + a is known to be either the full symmetric group S ^p or the affine group A(1, p), and it is conjectured that always G = S _p . In this note we settle this conjecture for p = 5 and, answering a question by J.-P. Serre, we show that this does not carry over when replacing the integer a by some rational number with 5-adic valuation equal to 1. Received: 6 June 2007     

Galois module structure for dihedral extensions of degree 8: Realizable classes over the group ring

Let k be a number field with ring of integers O_k, and let Γ be the dihedral group of order 8. For each tame Galois extension N/k with group isomorphic to Γ, the ring of integers O_N of N determines a class in the locally free class group Cl(O_k[Γ]). We show that the set of classes in Cl(O_k[Γ]) realized in this way is the kernel of the augmentation homomorphism from Cl(O_k[Γ]) to the ideal class group Cl(O_k), provided that the ray class group of O_k for the modulus 4O_k has odd order. This refines a result of the second-named author (J. Algebra 223 (2000) 367-378) on Galois module structure over a maximal order in k[Γ].     

On the independence of Heegner points in the function field case

Let ∞ be a fixed place of a global function field k. Let E be an elliptic curve defined over k which has split multiplicative reduction at ∞ and fix a modular parametrization ΦE:X₀(N)→E. Let be Heegner points associated to the rings of integers of distinct quadratic “imaginary” fields K₁,…,Kr over (k,∞). We prove that if the “prime-to-2p” part of the ideal class numbers of ring of integers of K₁,…,Kr are larger than a constant C=C(E,ΦE) depending only on E and ΦE, then the points P₁,…,Pr are independent in . Moreover, when k is rational, we show that there are infinitely many imaginary quadratic fields for which the prime-to-2p part of the class numbers are larger than C.     

Equidistribution for Torsion Points of a Drinfeld Module

We prove an equidistribution result for torsion points of Drinfeld modules of generic characteristic. We also show that similar equidistribution statements would provide proofs for the Manin–Mumford and the Bogomolov conjectures for Drinfeld modules.     

The modular automorphisms of the Drinfeld modular curve $$ X_{1}(\frak n) $$

In this work we determine the group of modular automorphisms of the Drinfeld modular curve for any nonconstant polynomial .Received: 19 September 2002     

The Rubin-Stark conjecture for a special class of function field extensions

We prove a strong form of the Brumer-Stark Conjecture and, as a consequence, a strong form of Rubin's integral refinement of the abelian Stark Conjecture, for a large class of abelian extensions of an arbitrary characteristic p global field k. This class includes all the abelian extensions K/k contained in the compositum k_p∞?k_p·k_∞ of the maximal pro-p abelian extension k_p/k and the maximal constant field extension k_∞/k of k, which happens to sit inside the maximal abelian extension k^ab of k with a quasi-finite index. This way, we extend the results obtained by the present author in (Comp. Math. 116 (1999) 321-367).     

Drinfeld Modules of Rank 1 and Algebraic Curves with Many Rational Points

 We construct algebraic curves C defined over a finite prime field such that the number of -rational points of C is large relative to the genus of C. The methods of construction are based on the relationship between algebraic curves and their function fields, as well as on narrow ray class extensions obtained from Drinfeld modules of rank 1. Received 21 July 1997; in revised form 5 February 1998     

Arithmetic equivalence for function fields, the Goss zeta function and a generalisation

A theorem of Tate and Turner says that global function fields have the same zeta function if and only if the Jacobians of the corresponding curves are isogenous. In this note, we investigate what happens if we replace the usual (characteristic zero) zeta function by the positive characteristic zeta function introduced by Goss. We prove that for function fields whose characteristic exceeds their degree, equality of the Goss zeta function is the same as Gaßmann equivalence (a purely group theoretical property), but this statement can fail if the degree exceeds the characteristic. We introduce a ‘Teichmüller lift’ of the Goss zeta function and show that equality of such is always the same as Gaßmann equivalence.     

Integral and modular representations attached to the Selmer group of an elliptic curve with complex multiplication

Let E be an elliptic curve with complex multiplication over the ring of integers of an imaginary quadratic field K. Denote by p an odd prime that splits into in and by the unique -extension of K totally ramified above . It is well-known that the Selmer group attached to any finite extension of is analogous to the minus part of the p-class group of divisors of the cyclotomic - extensions of CM number fields. One of the most striking examples of this analogy is the existence of a translation formula à la Kida for the codimension of the Selmer group at the top of the tower. In this article we carry on the analogy with the presentations of results similar to those proven by Gold and Madan in the cyclotomic case (see [8]), which were the continuation of Kida's work. More precisely, we describe the -structure of the Selmer group when G is a cyclic group of order p or . In addition, we study the modular representation of G on the subgroup of points of order p of the Selmer group, when G is cyclic of order . Received December 3, 1997     

Additive Hilbert's Theorem 90 in the ring of algebraic integers

For a number field K, we give a complete characterization of algebraic numbers which can be expressed by a difference of two K-conjugate algebraic integers. These turn out to be the algebraic integers whose Galois group contains an element, acting as a cycle on some collection of conjugates which sum to zero. Hence there are no algebraic integers which can be written as a difference of two conjugate algebraic numbers but cannot be written as a difference of two conjugate algebraic integers. A generalization of the construction to a commutative ring is also given. Furthermore, we show that for n ?_ 3 there exist algebraic integers which can be written as a linear form in n K-conjugate algebraic numbers but cannot be written by the same linear form in K-conjugate algebraic integers.     

The distribution of class groups of function fields

For any sufficiently general family of curves over a finite field F_q and any elementary abelian ?-group H with ? relatively prime to q, we give an explicit formula for the proportion of curves C for which Jac(C)[?](F_q)≅H. In doing so, we prove a conjecture of Friedman and Washington.     

Constructing function fields¶with many rational places via the Carlitz module

We consider certain decomposition fields in extensions of ?q (Z) by the Carlitz module and give formulas for their genera and numbers of rational places, suitable for automatic computations. By extensive calculations we found some function fields which have more rational places than the known examples of the respective genus. Received: 7 September 2001/ Revised version: 19 October 2001     

Galois modular representation of associated Jacobians in the tamely ramified cyclic case

Let L/K be an ℓ-cyclic extension with Galois group G of algebraic function fields over an algebraically closed field k of characteristic p ≠  ℓ. In this paper, the -module structure of the ℓ-torsion of the Jacobian associated to L is explicitly determined.     

On the zeta function associated with module classes of a number field

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The goal of this note is to generalize a formula of Datskovsky and Wright on the zeta function associated with integral binary cubic forms. We show that for a fixed number field K of degree d, the zeta function associated with decomposable forms belonging to K in d−1 variables can be factored into a product of Riemann and Dedekind zeta functions in a similar fashion. We establish a one-to-one correspondence between the pure module classes of rank d−1 of K and the integral ideals of width <d−1. This reduces the problem to counting integral ideals of a special type, which can be solved using a tailored Moebius inversion argument. As a by-product, we obtain a characterization of the conductor ideals for orders of number fields.

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=RePyaF8vDnE.     

On the group of modular units and the ideal class group

J.-M. Kim, S. Bae and I.-S. Lee showed that there exists an isomorphism between the p-primary part of the ideal class group and p-primary part of the unit group modulo cyclotomic unit group in Q⁺(ζpn) for all sufficiently large n under some conditions. In the present paper, we shall give an analogue of their result for modular units.