Centeleghe, Tommaso Giorgio Integral Tate modules and splitting of primes in torsion fields of elliptic curves. (English) Zbl 1408.11055 Int. J. Number Theory 12, No. 1, 237-248 (2016). Summary: Let \(E\) be an elliptic curve over a finite field \(k\), and \(\ell\) a prime number different from the characteristic of \(k\). In this paper, we consider the problem of finding the structure of the Tate module \(T_{\ell}(E)\) as an integral Galois representations of \(k\). We indicate an explicit procedure to solve this problem starting from the characteristic polynomial \(f_E(x)\) and the \(j\)-invariant \(j_E\) of \(E\). Hilbert class polynomials of imaginary quadratic orders play an important role here. We give a global application to the study of prime-splitting in torsion fields of elliptic curves over number fields. Cited in 10 Documents MSC: 11G05 Elliptic curves over global fields 11G07 Elliptic curves over local fields 11G20 Curves over finite and local fields Keywords:elliptic curve; integral Tate module; reciprocity law Software:IntegralFrobenius.m PDFBibTeX XMLCite \textit{T. G. Centeleghe}, Int. J. Number Theory 12, No. 1, 237--248 (2016; Zbl 1408.11055) Full Text: DOI arXiv References: [1] 1. T. Centeleghe and P. Tsaknias, Integral Frobenius, Magma package available at: http://math.uni.lu/ tsaknias/sfware.html. [2] 2. D. Cox, Primes of the Form x2 + ny2. Fermat, Class Field Theory, and Complex Multiplication (Wiley, New York, 1989). [3] 3. W. Duke and Á. Tóth, The splitting of primes in division fields of elliptic curves, Experiment. Math.11(4) (2002) 555-565. genRefLink(16, ’S1793042116500147BIB003’, ’10.1080 [4] 4. N. Elkies, The existence of infinitely many supersingular primes for every elliptic curve over Q , Invent. Math.89 (1987) 561-567. genRefLink(16, ’S1793042116500147BIB004’, ’10.1007 [5] 5. S. Lang, Elliptic Functions, 2nd edn., Graduate Texts in Mathematics, Vol. 112 (Springer, New York, 1987). genRefLink(16, ’S1793042116500147BIB005’, ’10.1007 [6] 6. D. Mumford, Abelian Varieties, 2nd edn. (Tata Institute of Fundamental Research, Bombay, 1974). [7] 7. J.-P. Serre, Propriétés galoisiennes des point d’ordre fini des courbes elliptiques, Invent. Math.15 (1972) 259-331. genRefLink(16, ’S1793042116500147BIB007’, ’10.1007 [8] 8. J.-P. Serre and J. Tate, Good reduction of abelian varieties, Ann. Math.88(3) (1968) 492-517. genRefLink(16, ’S1793042116500147BIB008’, ’10.2307 [9] 9. G. Shimura, A reciprocity law in non-solvable extensions, J. Reine Angew. Math.221 (1966) 209-220. genRefLink(128, ’S1793042116500147BIB009’, ’A19667384100015’); · Zbl 0144.04204 [10] 10. J. H. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, Vol. 106 (Springer, New York, 1986). genRefLink(16, ’S1793042116500147BIB010’, ’10.1007 · Zbl 0585.14026 [11] 11. J. Tate, Endomorphisms of abelian varieties over finite fields, Invent. Math.2 (1966) 134-144. genRefLink(16, ’S1793042116500147BIB011’, ’10.1007 · Zbl 0147.20303 [12] 12. J. Tate, Classes d’isogénie des variétés abéliennes sur un corps fini, Sém. Bourbaki21 (1968/69) Exposé 352. [13] 13. W. C. Waterhouse, Abelian varieties over finite fields, Ann. Sci. École. Norm. Sup. (4)2 (1969) 521-560. · Zbl 0188.53001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.