Generalized Legendre curves and quaternionic multiplication
By using results by \textit{J. Wolfart} [Invent. Math. 92, No. 1, 187--216 (1988; Zbl 0649.10022)], \textit{N. Archinard} [Can. J. Math. 55, No. 5, 897--932 (2003; Zbl 1038.11036)] and \textit{G. Wüstholz} [in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 476--483 (1987; Zbl 0679.10023)] on periods and decompositions of abelian varieties, or results by \textit{K. Yamamoto} [J. Comb. Theory 1, 476--489 (1966; Zbl 0173.03901)] on Gauss sums, the authors consider abelian varieties from generalized Legendre curves, particularly their corresponding Galois representations, periods and endomorphism algebras. For certain one parameter families of two-dimensional abelian varieties, the question whether the endomorphism algebra of each fiber defined over the algebraic closure of \(\mathbb{Q}\) contains a quaternion algebra, is answered.
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