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 Citació: David, C.; Jimenez, J. Square-free discriminants of Frobenius rings. "International journal of number theory", Setembre 2010, vol. 6, núm. 6, p. 1391-1412. Títol: Square-free discriminants of Frobenius rings Autor: David, Chantal; Jiménez Urroz, Jorge Data: set-2010 Tipus de document: Article Resum: Let E be an elliptic curve over Q. It is well known that the ring of endomorphisms of $E_p$, the reduction of E modulo a prime p of ordinary reduction, is an order of the quadratic imaginary field $Q(\pi_p)$ generated by the Frobenius element $\pi_p$. When the curve has complex multiplication (CM), this is always a fixed field as the prime varies. However, when the curve has no CM, very little is known, not only about the order, but about the fields that might appear as algebra of endomorphisms varying the prime. The ring of endomorphisms is obviously related with the arithmetic of $a^2_p$−4p, the discriminant of the characteristic polynomial of the Frobenius element. In this paper, we are interested in the function $\pi^{sf}_{E,r,h}(\chi)$ counting the number of primes p up to x such that $a^2_p$ is square-free and in the congruence class r modulo h. We give in this paper the precise asymptotic for $\pi^{sf}_{E,r,h}(\chi)$ when averaging over elliptic curves defined over the rationals, and we discuss the relation of this result with the Lang-Trotter conjecture, and with some other problems related to the curve modulo p. ISSN: 1793-0421 URI: http://hdl.handle.net/2117/10419 DOI: 10.1142/S1793042110003599 Versió de l'editor: http://www.mathstat.concordia.ca/faculty/cdavid/PAPERS/SFdiscriminants-2008.pdf Apareix a les col·leccions: Altres. Enviament des de DRACMAK - Matemàtica Aplicada a la Criptografia. Articles de revistaDepartaments de Matemàtica Aplicada. Articles de revista Comparteix: