Compatible families of elliptic type

Compatible Families of Elliptic Type

In axiomatizing their study of Frobenius distributions [5], Lang and Trotter introduce the notion of an adelic Galois representation of elliptic type, and they ask in passing whether every such representation arises from an elliptic curve (see pp. 5 and 19 of [5]). Formulated in the language of `-adic representations [7], their question is as follows. Put G = Gal(Q/Q), let p denote a prime, and...

Complete characterization of the Mordell-Weil group of some families of elliptic curves

 The Mordell-Weil theorem states that the group of rational points‎ ‎on an elliptic curve over the rational numbers is a finitely‎ ‎generated abelian group‎. ‎In our previous paper, H‎. ‎Daghigh‎, ‎and S‎. ‎Didari‎, On the elliptic curves of the form $ y^2=x^3-3px$‎, ‎‎Bull‎. ‎Iranian Math‎. ‎Soc‎.‎‎ 40 (2014)‎, no‎. ‎5‎, ‎1119--1133‎.‎, ‎using Selmer groups‎, ‎we have shown that for a prime $p...

Commutative Families of the Elliptic Macdonald Operator

In the paper [J. Math. Phys. 50 (2009), 095215, 42 pages], Feigin, Hashizume, Hoshino, Shiraishi, and Yanagida constructed two families of commuting operators which contain the Macdonald operator (commutative families of the Macdonald operator). They used the Ding–Iohara–Miki algebra and the trigonometric Feigin–Odesskii algebra. In the previous paper [arXiv:1301.4912], the present author const...

Constructing Families of Pairing-Friendly Elliptic Curves

We present a general method for constructing families of elliptic curves with prescribed embedding degree and prime order. We demonstrate this method by constructing curves of embedding degree k = 10, a value which has not previously appeared in the literature, and we show that our method applies to existing constructions for k = 3, 4, 6, and 12. We give evidence that our method is unlikely to ...

Lattice structure of the families of compatible frames of discernment