Explicit Hyperelliptic Curves With Real Multiplication and Permutation Polynomials

Abstract

The aim of this paper is to present a very explicit construction of one parameter families of hyperelliptic curves C of genus (p−1 )/ 2, for any odd prime number p, with the property that the endomorphism algebra of the jacobian of C contains the real subfield Q(2 cos(2π/p)) of the cyclotomic field Q(e2π i/p).Two proofs of the fact that the constructed curves have this property will be given. One is by providing a double cover with the pth roots of unity in its automorphism group. The other is by explicitly writing down equations of a correspondence in C x C which defines multiplication by 2cos(2π/ p) on the jacobian of C. As a byproduct we obtain polynomials which define bijective maps Fℓ → Fℓ for all prime numbers in certain congruence classes.

Sato-Tate distribution for abelian varieties with real multiplication over function fields

“…Proof. This is the main theorem of [11], except for the assertion that Rosati acts trivially on the image of O F , which is not explicitly mentioned in the references. This can be seen as follows.…”

“…By explicit calculation, we have Proof. This is implicit in [11] and [2], Prop 2.2. We give the details here.…”

“…[6]. We apply this technique to the one parameter family of curves considered in [11], whose Jacobians have real multiplication by the totally real field F := Q(cos 2π r ), with r a prime (see section 4). The condition that the family has real multiplication puts a constraint on the monodromy group.…”

On the index of degeneracy of a CM abelian variety

“…These are CM Jacobian varieties treated in [8]. For these abelian varieties, we can determine the dimensions of the Mumford-Tate groups.…”

Galois invariants of dessins d’enfants

“…has real multiplication by the subfield of index 2 in Q(ζ p ) [24,5], possibly after some finite extension of the baseQ(t). Let A ∈ Sp 2g (Ẑ) be the image of the action of End(C t × Q(t)) on H 1 (C t × Q(t),Ẑ).…”