Generalization of a theorem of Siegel

Comparison of Picard groups in dimension 1

“…Our proofs are constructive and the results generalize a theorem of J. Sands [11].MSC 2000: 13C15, 13C20, 03F65, 13F45 …”

“…In [11] J. Sands generalizes a theorem of Siegel [13]. He explains the link between the Picard group of the integer ring of a number field and that of an order of this number field.…”

Factoring abelian groups into uniquely complemented subsets

“…is a normalized factorization of G. Here B is not periodic and none of the factors C i; j is a subgroup of G. This contradicts [5,Theorem 5]. r…”

Extreme Values of Geodesic Periods on Arithmetic Hyperbolic Surfaces

“…Here, the diophantine problem consists of counting η ∈ R(n) that are at distance ≪ √ n from 0 and are close to where L is a lift of an arbitrary ℓ ∈ Λ dF (the volume is independent of the choice), B dF (a) is defined similarly to B L (a) (see Proposition 7.2) and B dF (a) ≥ B L (a). We conclude by choosing the amplifier as in Proposition 7.5, taking C > 0 large enough and comparing (29) to the standard pre-trace formula. (In the latter, we don't bother to extract the contribution of even φ j , since we can discard the contribution of odd ones using positivity, in the endgame.)…”