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Abstract: In 1987, Dan Gordon defined an elliptic curve analogue to Carmichael numbers known as elliptic Carmichael numbers. In this paper, we prove that there are infinitely many elliptic Carmichael numbers. In doing so, we resolve in the affirmative the question of whether there exist infinitely many Lucas–Carmichael numbers (that is, squarefree, composite integers n such that for every prime p that divides n, p + 1 | n + 1) . [ABSTRACT FROM AUTHOR]

Let L be a number field and let E/L be an elliptic curve with complex multiplication by the ring of integers OK of an imaginary quadratic field K. We use class field theory and results of Skorobogatov and Zarhin to compute the transcendental part of the Brauer group of the abelian surface E × E. The results for the odd-order torsion also apply to the Brauer group of the K3 surface Kum(E × E). We describe explicitly the elliptic curves E/Q with complex multiplication by OK such that the Brauer group of E × E contains a transcendental element of odd order. We show that such an element gives rise to a Brauer-Manin obstruction to weak approximation on Kum(E × E), while there is no obstruction coming from the algebraic part of the Brauer group. [ABSTRACT FROM AUTHOR]

We prove that the (logarithmic) Mahler measure m(P) of P(x, y) = x + 1/x + y + 1/y + 3 is equal to the L-value 2L'(E, 0) attached to the elliptic curve E : P(x, y) = 0 of conductor 21. In order to do this, we investigate the measure of a more general Laurent polynomial Pa,b,c(x, y) = a(x + 1/x)+ b(y + 1/y)+ c and show that the wanted quantity m(P) is related to a 'half-Mahler' measure of Pữ(x, y) = P√7,1,3(x, y). In the finale, we use the modular parametrization of the elliptic curve Pữ(x, y) = 0, again of conductor 21, due to Ramanujan and the Mellit-Brunault formula for the regulator of modular units. [ABSTRACT FROM AUTHOR]

The goal of this article was to study the Iwasawa theory of an abelian variety A that has complex multiplication by a complex multiplication (CM) field F that contains the reflex field of A, which has supersingular reduction at every prime above p. To do so, we make use of the signed Coleman maps constructed in our companion article [Kâzım Büyükboduk and Antonio Lei, 'Integral Iwasawa theory of motives for non-ordinary primes', 2014, in preparation, draft available upon request] to introduce signed Selmer groups as well as a signed p-adic L-function via a reciprocity conjecture that we formulate for the (conjectural) Rubin-Stark elements (which is a natural extension of the reciprocity conjecture for elliptic units). We then prove a signed main conjecture relating these two objects. To achieve this, we develop along the way a theory of Coleman-adapted rank-g Euler-Kolyvagin systems to be applied with Rubin-Stark elements and deduce the main conjecture for the maximal Zp-power extension of F for the primes failing the ordinary hypothesis of Katz. [ABSTRACT FROM AUTHOR]

In this note, we consider the bifurcation of a singular homoclinic orbit to periodic ones in a 4-dimensional slow-fast system of ordinary differential equations, having a 2-dimensional normally elliptic slow manifold, originally studied by Fečkan and Rothos. Assuming an extra degree of differentiability on the system, we can refine their perturbation scheme, in particular the choice of approximate solution, and obtain improved estimates. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We combine the method of Faltings (Arakelov, Paršin, Szpiro) with the Shimura-Taniyama conjecture to prove effective finiteness results for integral points on moduli schemes of elliptic curves. For several fundamental Diophantine problems, such as for example [ABSTRACT FROM AUTHOR]

We construct liftings of reduction maps from complex multiplication (CM) points to supersingular points for general quaternion algebras and use these liftings to establish a precise correspondence between CM points on indefinite quaternion algebras with a given conductor and CM points on certain corresponding totally definite quaternion algebras. [ABSTRACT FROM PUBLISHER]

In this paper we define and study Hopf C$^*$-algebras. Roughly speaking, a Hopf C$^*$-algebra is a C$^*$-algebra $A$ with a comultiplication $\phi : A \rightarrow M(A \otimes A)$ such that the maps $a \otimes b \mapsto \phi(a)(1 \otimes b)$ and $a \otimes b \mapsto (a \otimes 1)\phi(b)$ have their range in $A \otimes A$ and are injective after being extended to a larger natural domain, the Haagerup tensor product $A \otimes_h A$. In a purely algebraic setting, these conditions on $\phi$ are closely related to the existence of a counit and antipode. In this topological context, things turn out to be much more subtle, but nevertheless one can show the existence of a suitable counit and antipode under these conditions.The basic example is the C$^*$-algebra $C_0(G)$ of continuous complex functions tending to zero at infinity on a locally compact group where the comultiplication is obtained by dualizing the group multiplication. But also the reduced group C$^*$-algebra $C^*_r(G)$ of a locally compact group with the well-known comultiplication falls in this category. In fact all locally compact quantum groups in the sense of Kustermans and the first author (such as the compact and discrete ones) as well as most of the known examples are included.This theory differs from other similar approaches in that there is no Haar measure assumed.2000 Mathematics Subject Classification: 46L65, 46L07, 46L89. [ABSTRACT FROM AUTHOR]

The j-invariants of the quadratic Q-curves without complex multiplication are studied. Some properties of the norms of these invariants are shown and a relationship between the field Q(j) and the degree of an isogeny of the Q-curve to its Galois conjugate is found. In the case when the degree of the isogeny is a prime p, some properties of the primes of potentially multiplicative reduction for the Q-curve and of the reduction of j modulo a prime 5 in Q(j) over p when the Q-curve has potentially good reduction at 5 are found. [ABSTRACT FROM AUTHOR]

In this Letter we propose complex modular multiplication of two N-bit negabinary complex numbers by adopting digital partitioning and weighting-shifting techniques. The module (-2)N - 1 is introduced for its simple arithmetic. Both complex numbers are treated as two N-bit negabinary real numbers. All four real numbers are decomposed into several short bytes; a corresponding complex product modular (-2)N - 1 is achieved by cyclic convolution. The subtraction in complex multiplication is converted to addition by weighting shifting. The algorithms can be implemented in a fully parallel manner, especially for optics. Both pre- and postprocessing are simple. The proof-of-principle experiment is performed by an incoherent optical correlator, and results are given. [ABSTRACT FROM AUTHOR]

We show that any abelian surface with multiplication by the quaternion ℚ-algebra of discriminant 6, with field of moduli ℚ and which is a Jacobian in characteristic 2 and 3, has infinitely many primes of superspecial reduction. This is done by examining complex multiplication points in characteristic 0 and p and the values of a certain j-function on the associated moduli space at these points. [ABSTRACT FROM AUTHOR]

The following two problems are open. Do two sets of positive integers [script A] and ℬ exist, with at least two elements each, such that [script A]+ℬ coincides with the set of primes ℘ for sufficiently large elements? Let [script A]={6, 12, 18}. Is there an infinite set ℬ of positive integers such that [script A]ℬ+1⊂℘? A positive answer would imply that there are infinitely many Carmichael numbers with three prime factors. In this paper we prove the multiplicative analogue of the first problem, namely that there are no two sets [script A] and ℬ, with at least two elements each, such that the product [script A]ℬ coincides with any additively shifted copy ℘+c of the set of primes for sufficiently large elements. We also prove that shifted copies of sets of integers that are generated by certain subsets of the primes cannot be multiplicatively decomposed. [ABSTRACT FROM AUTHOR]