Computational aspects of the endomorphism ring of the Jacobian of a curve of genus two

William David Butske, Purdue University


Building on a method of Zarhin, we determine the tensor of the endomorphism ring of the Jacobian over the field of two elements in the case of a genus two curve defined as a double cover over the projective line by an irreducible polynomial of degree five over the rational numbers. In particular, we show that there are three possibilities, the field of two elements, the field of four elements and the field of sixteen elements. Furthermore, we give examples that realize all three. Under an assumption about two-torsion, we show as a corollary that quaternionic multiplication cannot occur in such a case. Zarhin shows that if the Galois group of this polynomial is either the symmetric group or the alternating group then the endomorphism ring is trivial. As a further corollary to our work, we show that such a relation between the Galois group of the polynomial defining the curve and the endomorphism ring of the Jacobian does not extend to other Galois groups.




Matsuki, Purdue University.

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