On the Proﬁnite Regular Inverse Galois Problem

Let k be a field finitely generated over Q and p a prime. The torsion conjecture (resp. p-primary torsion conjecture) for abelian varieties over k predicts that the k-rational torsion (resp. the p-primary k-rational torsion) of a d-dimensional abelian variety A over k should be bounded only in terms of k and d. These conjectures are only known for d = 1. The p-primary case was proved by Y. Manin, in 1969; the general case was completed by L. Merel, in 1996, after a series of contributions by B. Mazur, S. Kamienny and others. Due to the fact that moduli of elliptic curves are 1-dimensional, the d = 1 case of the torsion conjecture (resp. p-primary torsion conjecture) is closely related to the following. For any k-curve S and elliptic scheme E → S, the k-rational torsion (resp. the p-primary k-rational torsion) is uniformly bounded in the fibres E s , s ∈ S(k). In this paper, we extend this result in the p-primary case to arbitrary abelian schemes over curves.More precisely, we prove the following. Denote by k the absolute Galois group of k. For an abelian variety A over k and a character χ : k → Z * p , define A[p ∞ ](χ ) to be the module of p-primary torsion of A(k) on which k acts as χ -multiplication. Assume that χ does not appear as a subrepresentation of the p-adic representation associated with an abelian variety over k. Then A[p ∞ ](χ ) is always finite, but the exponent of A[p ∞ ](χ ) may de-

show abstract

“…Since all the construction is natural and K -equivariant, the second assertion follows from Lemma 5.12 (2), (3).…”

Section: N B = (ẽ F ) Ab /(ẽ F ) Ab [B] ⊃ G/g[b] (G/g[e])/(g/g[e])[b]mentioning

confidence: 90%

“…When k is finitely generated over its prime field, the trivial character [20] and the p-adic cyclotomic character [3,Lemma 2.6] are typical examples of non-Tate characters.…”

Section: Non-tate Charactersmentioning

confidence: 99%

Uniform boundedness of p-primary torsion of abelian schemes

Cadoret

Tamagawa

2011

Invent. math.

Self Cite

View full text Add to dashboard Cite

show abstract

“…The non-existence of projective systems of K-rational points on a modular tower first appeared in the Bailey-Fried paper [BF02]; the result was then refined and extended to more general situations by Kimura [Kim05] and the first author [Cad04] [Cad07c]. The case r = 4 considered in statement (c) has been thoroughly studied by Fried [BF02], [Fri06].…”

Section: Modular Tower Diophantine Conjecturementioning

confidence: 96%

Abelian constraints in inverse Galois theory

Cadoret

Dèbes

2008

manuscripta math.

Self Cite

View full text Add to dashboard Cite

Abstract. We show that if a finite group G is the Galois group of a Galois cover of P 1 over Q, then the orders p n of the abelianization of its p-Sylow subgroups are bounded in terms of their index m, of the branch point number r and the smallest prime | |G| of good reduction of the branch divisor. This is a new constraint for the regular inverse Galois problem: if p n is suitably large compared to r and m, the branch points must coalesce modulo small primes. We further conjecture that p n should be bounded only in terms of r and m. We use a connection with some rationality question on the torsion of abelian varieties. For example, our conjecture follows from the so-called torsion conjectures. Our approach also yields a new viewpoint on Fried's Modular Tower program.

show abstract

On the Proﬁnite Regular Inverse Galois Problem

Cited by 2 publications

References 12 publications

Uniform boundedness of p-primary torsion of abelian schemes

Uniform boundedness of p-primary torsion of abelian schemes

Abelian constraints in inverse Galois theory

Contact Info

Product

Resources

About