Endomorphism algebras of factors of certain hypergeometric Jacobians

Abstract

We classify the endomorphism algebras of factors of the Jacobian of certain hypergeometric curves over a field of characteristic zero. Other than a few exceptional cases, the endomorphism algebras turn out to be either a cyclotomic field E = Q(ζq), or a quadratic extension of E, or E⊕E. This result may be viewed as a generalization of the well known results of the classification of endomorphism algebras of elliptic curves over C.

Fixed points and homology of superelliptic Jacobians

“…We refer to [17,Subsection 2.11] for the following proposition. Any point x = [D] ∈ ker β M is fixed by (δ N ) M .…”

“…By [17,Lemma 4.1], a divisor of degree zero of the form D = n i=1 a i Q i is linear equivalent to zero if and only if there exists c ∈ Z such that a i ≡ ce i (mod N ) for all 1 ≤ i ≤ n. For this c ∈ Z, we have 0 =…”

“…In this case J old q = J q/p . By [17,Subsection 2.11], J new q ∩ J q/p = J q/p [p]. It follows that…”

“…In particular, if p > 7, then End 0 (J new q ) is either Q(ζ q ), a quadratic field extension of Q(ζ q ), or Q(ζ q ) ⊕ Q(ζ q ). The generic case was treated in [22] by Zarhin and the classification was given in [17] jointly by the second and third named authors. Now suppose K ⊆ C, deg f (x) ≥ 3, and End 0K (J new q ) = Q(ζ q ).…”

“…). The generic case was treated in [22] by Zarhin and the classification was given in [17] jointly by the second and third named authors. Now suppose K ⊆ C, deg f (x) ≥ 3, and End 0 K (J new q ) = Q(ζ q ).…”