SQUARE-FREE DISCRIMINANTS OF FROBENIUS RINGS
| Autor: | Chantal David, Jorge Jiménez Urroz |
|---|---|
| Přispěvatelé: | Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada IV, Universitat Politècnica de Catalunya. MAK - Matemàtica Aplicada a la Criptografia |
| Rok vydání: | 2010 |
| Předmět: |
Discrete mathematics
Ring (mathematics) Pure mathematics Endomorphism rings Algebra and Number Theory Mathematics - Number Theory Corbes el·líptiques Mathematics::Number Theory Sato–Tate conjecture Complex multiplication Àlgebra abstracta Twists of curves Supersingular elliptic curve Curves Elliptic Anells (Àlgebra) Matemàtiques i estadística::Àlgebra::Ordres reticles estructures algebraiques ordenades [Àrees temàtiques de la UPC] Modular elliptic curve Algebra Abstract FOS: Mathematics Counting points on elliptic curves Number Theory (math.NT) Schoof's algorithm Mathematics |
| Zdroj: | UPCommons. Portal del coneixement obert de la UPC Universitat Politècnica de Catalunya (UPC) Recercat. Dipósit de la Recerca de Catalunya instname |
| ISSN: | 1793-7310 1793-0421 |
| DOI: | 10.1142/s1793042110003599 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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