A finiteness conjecture on abelian varieties with constrained prime power torsion

Abstract

The pro-Galois representation attached to the arithmetic fundamental group of a curve influences heavily the arithmetic of its branched ' -covers.' In many cases, the -power torsion on the Jacobian of such a cover is fixed by the kernel of this representation, giving explicit information about this kernel. Motivated by the relative scarcity of interesting examples for -covers of the projective line minus three points, the authors formulate a conjecture to quantify this scarcity. A proof for certain genus one cases is given, and an exact set of curves satisfying the required arithmetic conditions in the base case is determined.

On the uniform Rasmussen–Tamagawa conjecture in the CM case

“…Proof. It is clear by definition that we must have C(1, 2) ≥ C(1, 1) ≥ 163, where the optimal value of C(1, 1) = 163 is taken from [14] (see also [2]). Consider now an abelian surface A/Q admitting potential complex multiplication, and suppose first that A Q is isogenous to the product of two elliptic curves.…”

“…Motivated by previous work of Anderson and Ihara [1], in [14] and [15] Rasmussen and Tamagawa have formulated (and partially proven) a series of finiteness conjectures for abelian varieties A over number fields K such that the extension K(A[ℓ ∞ ])/K(µ ℓ ∞ ) is both pro-ℓ and unramified away from ℓ. The strongest form of their conjecture, as stated in [15,Conj.…”

A uniform version of a finiteness conjecture for CM elliptic curves

“…Proof. A version of this appears as Lemma 3 in [17], and a more general version appears in [16]. Note that although the result in [16] is stated for abelian varieties A/F where F (A[ℓ ∞ ]) is both a pro-ℓ extension of F (µ ℓ ) and unramified away from ℓ, the ramification requirement is not used in the proof.…”

“…This implies that, for a fixed F and g, there exists a constant C for which A (F, g, ℓ) = ∅ when ℓ > C. One may ask whether stronger behavior should be expected for the bound C, and indeed the following uniform (meaning uniform in the degree of F/Q) conjecture appears in [16,Conj. 2]: Conjecture 2.…”

On Congruences of Galois Representations of Number Fields

“…

We give a criterion for two ℓ-adic Galois representations of an algebraic number field to be isomorphic when restricted to a decomposition group, in terms of the global representations mod ℓ. This is applied to prove a generalization of a conjecture of Rasmussen-Tamagawa [14] under a semistablity condition, extending some results [12] of one of the authors. It is also applied to prove a congruence result on the Fourier coefficients of modular forms.

…”