Journal of the Ramanujan Mathematical Society

Arsen Elkin ¹, Yuri G. Zarhin ²

Affiliations
1 Department of Mathematics, Colorado State University, Fort Collins, CO 80523, US
2 Department of Mathematics, Pennsylvania State University, University Park, PA 16802, US

Abstract

We prove that the jacobian of a hyperelliptic curve y²=f(x) is either absolutely simple or isogenous to a self-product of a CM elliptic curve if deg (f)=q+1 where q is a power prime congruent to 7 modulo 8, the polynomial f(x) is irreducible over the ground field of characteristic zero and its Galois group is L₂(q).

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Arsen Elkin ¹, Yuri G. Zarhin ²

Affiliations
1 Institute of Mathematics, Hebrew University of Jerusalem, Givat Ram, Jerusalem-91904, IL
2 Department of Mathematics, Pennsylvania State University, University Park, PA-16802, US

Abstract

Let K be a field with char (K)≠2. Let us fix an algebraic closure K_a of K. Let us put Gal(K):=Aut(K_a/K). If X is an abelian variety of positive dimension over K_a then we write End(X) for the ring of all its K_a-endomorphisms and End⁰(X) for the corresponding (semisimple finite-dimensional) ℚ-algebra End(X)⊗ℚ. We write End _K(X) for the ring of all K-endomorphisms of X and End⁰_K(X) for the corresponding (semisimple finite-dimensional) ℚ-algebra End_K(X)⊗ℚ.

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