Endomorphism algebras of Jacobians of certain superelliptic curves

Endomorphism Algebras of Abelian Varieties with Special Reference to Superelliptic Jacobians

“…It is known [39,44,42,25] that J (f,q) is positive-dimensional and J(C f,q ) is Kisogenous to a product r j=1 J (f,p j ) . E.g., if q = p (i.e, r = 1) then J(C f,p ) = J (f,p) .…”

“…In addition, if k − 1 is odd (i.e., k is even) then n J (f,q) ,i = 1. (iii) If n ≥ 5, p divides n but q does not divide n then n J (f,q) ,i = 1 [25,Prop. 2.2 and Remark 2.3].…”

“…Now the desired result follows from Theorem 8.3. without any additional conditions on n and q. The case when either p does not divide n or q | n was done in [39], the case when p | n but q does not divide n was done in [25]. The proofs in [39] are based on the notion of a very simple representation that was introduced in [28], see also [40].…”

“…(A) The groupG λ,J (f,q) ,K is isomorphic to Gal(f ). (B) If p does not divide n and Gal(f ) is doubly transitive then EndG λ,J (f,q) If q divides n and either Gal(f ) is 3-transitive orEnd Gal(f ) ((F R f p ) 00 ) = F p then EndG λ,J (f,q) ,K (J (f,q) λ ) = F p .4 J. Xue[25] assumed that char(K) = 0. However, all his arguments related to the computation of dim J (f,q) and J(f,q) λ work under a weaker assumption that char(K) = p. (D) Suppose that p divides n but q does not divide n. Assume also that Gal(f ) is transitive (i.e., f (x) is irreducible over K) and the Gal(f )-module (F R f p ) 00 is absolutely simple.…”

Endomorphism algebras of factors of certain hypergeometric Jacobians

“…and h is given by(2.30). Then h is primitive if one of the following condition holds:• gcd(n, N ) = 1, (by Proposition 5.3); • n = 3, and N = 3 r ≥ 9, (by[29, Lemma 4.2]).…”