Towers Of Number Fields Research Articles

Growth of in towers for isogenous curves

We study the growth of$\unicode[STIX]{x0428}$and$p^{\infty }$-Selmer groups for isogenous abelian varieties in towers of number fields, with an emphasis on elliptic curves. The growth types are usually exponential, as in the ‘positive${\it\mu}$-invariant’ setting in the Iwasawa theory of elliptic curves. The towers we consider are$p$-adic and$l$-adic Lie extensions for$l\neq p$, in particular cyclotomic and other$\mathbb{Z}_{l}$-extensions.

Growth of Selmer Groups of CM Abelian Varieties

AbstractLet p be an odd prime. We study the variation of the p-rank of the Selmer groups of Abelian varieties with complex multiplication in certain towers of number fields.

INFINITE HILBERT CLASS FIELD TOWERS FROM GALOIS REPRESENTATIONS

We investigate class field towers of number fields obtained as fixed fields of modular representations of the absolute Galois group of the rational numbers. First, for each k ∈ {12, 16, 18, 20, 22, 26}, we give explicit rational primes ℓ such that the fixed field of the mod-ℓ representation attached to the unique normalized cusp eigenform of weight k on SL2(ℤ) has an infinite class field tower. Further, under a conjecture of Hardy and Littlewood, we prove the existence of infinitely many cyclotomic fields of prime conductor, providing infinitely many such primes ℓ for each k in the list. Finally, given a non-CM curve E/ℚ, we show that there exists an integer ME such that the fixed field of the representation attached to the n-division points of E has an infinite class field tower for a set of integers n of density one among integers coprime to ME.

Systematic growth of Mordell-Weil groups of abelian varieties in towers of number fields

Let A be an abelian variety over the number field K. One knows, by the Mordell-Weil Theorem, that the Mordell-Weil group A(K) of K-rational points of A is finitely generated. Unfortunately, one knows little more than that; in particular, one can rarely find conceptual means for exhibiting points of infinite order in A(K). A few years ago, however, Kurchanov [Ku 1] and Mazur [-M2] succeeded in doing just that, when (1) A is an elliptic curve over Q, admitting a surjective map Xo(N )--, A, where Xo(N ) is the modular curve corresponding to the group

Hasse invariants and anomalous primes for elliptic curves with complex multiplication

Let C be an elliptic curve defined over Q. Let p be a prime where C has good reduction. By definition, p is anomalous for C if the Hasse invariant at p is congruent to 1 modulo p. The phenomenon of anomalous primes has been shown by Mazur to be of great interest in the study of rational points in towers of number fields. This paper is devoted to discussing the Hasse invariants and the anomalous primes of elliptic curves admitting complex multiplication. The two special cases Y 2 = X 3 + a 4 X and Y 2 = X 3 + a 6 are studied at considerable length. As corollaries, some results in elementary number theory concerning the residue classes of the binomial coefficients ( n 2n) (Resp. ( n 3n)) modulo a prime p = 4 n + 1 (resp. p = 6 n + 1) are obtained. It is shown that certain classes of elliptic curves admitting complex multiplication do not have any anomalous primes and that others admit only very few anomalous primes.

Rational points of abelian varieties with values in towers of number fields

Rational points of abelian varieties with values in towers of number fields