Abstract. Assuming Lang's conjectured lower bound on the heights of non-torsion points on an elliptic curve, we show that there exists an absolute constant C such that for any elliptic curve E/Q and non-torsion point P ∈ E(Q), there is at most one integral multiple [n]P such that n > C. The proof is a modification of a proof of Ingram giving an unconditional but not uniform bound. The new ingredient is a collection of explicit formulae for the sequence v(Ψ n ) of valuations of the division polynomials. For P of non-singular reduction, such sequences are already well described in most cases, but for P of singular reduction, we are led to define a new class of sequences called elliptic troublemaker sequences, which measure the failure of the Néron local height to be quadratic. As a corollary in the spirit of a conjecture of Lang and Hall, we obtain a uniform upper bound on h(P)/h(E) for integer points having two large integral multiples.
The ring and polynomial learning with errors problems (Ring-LWE and Poly-LWE) have been proposed as hard problems to form the basis for cryptosystems, and various security reductions to hard lattice problems have been presented. So far these problems have been stated for general (number) rings but have only been closely examined for cyclotomic number rings. In this paper, we state and examine the Ring-LWE problem for general number rings and demonstrate provably weak instances of the Decision Ring-LWE problem. We construct an explicit family of number fields for which we have an efficient attack. We demonstrate the attack in both theory and practice, providing code and running times for the attack. The attack runs in time linear in q, where q is the modulus.Our attack is based on the attack on Poly-LWE which was presented in [EHL]. We extend the EHL-attack to apply to a larger class of number fields, and show how it applies to attack Ring-LWE for a heuristically large class of fields. Certain Ring-LWE instances can be transformed into Poly-LWE instances without distorting the error too much, and thus provide the first weak instances of the Ring-LWE problem. We also provide additional examples of fields which are vulnerable to our attacks on Poly-LWE, including power-of-2 cyclotomic fields, presented using the minimal polynomial of ζ 2 n ±1.
We study the orbit of R under the Bianchi group PSL2(OK ), where K is an imaginary quadratic field. The orbit, called a Schmidt arrangement SK , is a geometric realisation, as an intricate circle packing, of the arithmetic of K. This paper presents several examples of this phenomenon. First, we show that the curvatures of the circles are integer multiples of √ −∆ and describe the curvatures of tangent circles in terms of the norm form of OK . Second, we show that the circles themselves are in bijection with certain ideal classes in orders of OK , the conductor being a certain multiple of the curvature. This allows us to count circles with class numbers. Third, we show that the arrangement of circles is connected if and only if OK is Euclidean. These results are meant as foundational for a study of a new class of thin groups generalising Apollonian groups, in a companion paper.
Abstract. An amicable pair for an elliptic curve E/Q is a pair of primes (p, q) of good reduction for E satisfying #Ẽ p (F p ) = q and #Ẽ q (F q ) = p. In this paper we study elliptic amicable pairs and analogously defined longer elliptic aliquot cycles. We show that there exist elliptic curves with arbitrarily long aliqout cycles, but that CM elliptic curves (with j = 0) have no aliqout cycles of length greater than two. We give conjectural formulas for the frequency of amicable pairs. For CM curves, the derivation of precise conjectural formulas involves a detailed analysis of the values of the Grössencharacter evaluated at primes p in End(E) having the property that #Ẽ p (F p ) is prime. This is especially intricate for the family of curves with j = 0.
We study the orbit of R under the Möbius action of the Bianchi group PSL 2 (O K ) on C, where O K is the ring of integers of an imaginary quadratic field K. The orbit S K , called a Schmidt arrangement, is a geometric realisation, as an intricate circle packing, of the arithmetic of K. We give a simple geometric characterisation of certain subsets of S K generalizing Apollonian circle packings, and show that S K , considered with orientations, is a disjoint union of all primitive integral such K-Apollonian packings. These packings are described by a new class of thin groups of arithmetic interest called K-Apollonian groups. We make a conjecture on the curvatures of these packings, generalizing the local-to-global conjecture for Apollonian circle packings.
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