Perfect squares representing the number of rational points on elliptic curves over finite field extensions |
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Affiliation: | 1. Institute of Analysis and Number Theory, Graz University of Technology, Kopernikusgasse 24/II, A-8010 Graz, Austria;2. School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland;3. School of Mathematics, Wits University, Johannesburg, South Africa;4. Research Group in Algebraic Structures and Applications, King Abdulaziz University, Jeddah, Saudi Arabia;5. Centro de Ciencias Matemáticas, UNAM, Morelia, Mexico;1. Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, Via Vanvitelli 1, 06123 Perugia, Italy;2. Department of Geometry and MTA–ELTE Geometric and Algebraic Combinatorics Research Group, ELTE Eötvös Loránd University, Budapest, 1117 Budapest, Pázmány Péter sétány 1/C, Hungary;3. FAMNIT, University of Primorska, 6000 Koper, Glagoljaška 8, Slovenia;1. Department of Mathematics, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Prague 8, Czech Republic;2. Institut de Mathématiques de Toulon, campus La Garde, 83041 Toulon, France;1. Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Krijgslaan 281, Building S8, 9000 Gent, Flanders, Belgium;2. Department of Mathematics and Data Science, University of Brussels (VUB), Pleinlaan 2, Building G, 1050 Elsene, Brussels, Belgium;3. Department of Mathematics, University of Rijeka, Radmile Matejčić 2, 51000 Rijeka, Croatia;1. Department of Applied Mathematics and Computer Science, Technical University of Denmark, Matematiktorvet 303B, 2800 Kgs. Lyngby, Denmark;2. Department of Mathematics and Statistics, University of Tromsø, Hansine Hansens veg 18, 9019, Norway;1. Department of Mathematics and Computer Science, Perugia University, Perugia 06123, Italy;2. Institute for Information Transmission Problems (Kharkevich Institute), Russian Academy of Sciences, Moscow 127051, Russian Federation |
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Abstract: | Let q be a perfect power of a prime number p and be an elliptic curve over given by the equation . For a positive integer n we denote by the number of rational points on E (including infinity) over the extension . Under a mild technical condition, we show that the sequence contains at most 10200 perfect squares. If the mild condition is not satisfied, then is a perfect square for infinitely many n including all the multiples of 12. Our proof uses a quantitative version of the Subspace Theorem. We also find all the perfect squares for all such sequences in the range and . |
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Keywords: | Elliptic curves Subspace theorem Recurrence sequence |
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