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SEMINAR:On the number of rational points of curves over a surface in P3

Speaker: Dr. Elena Berardini, Eindhoven University of Technology

Title:On the number of rational points of curves over a surface in P3

Date/Time:Nov 16, 2022 / 13:40 – 14:30 (Online only)

Zoom Link:https://sabanciuniv.zoom.us/j/99963877622?pwd=UklXR3FUMDlQcGI0bm1oL0pkamRxQT09

Meeting ID: 999 6387 7622

Passcode: algebra

Abstract:In this talk, we will show that the number of rational points of an irreducible curve of degree δ defined over a finite field Fq lying on a surface S in P3 of degree d is, under certain conditions, bounded by δ(d + q − 1)/2. Within a certain range of δ and q, this result improves all other known bounds in the context of space curves. The method we used is inspired by techniques developed by St¨ohr and Voloch [2]. In their seminal work of 1986, they introduced the Frobenius orders of a projective curve and used them to give an upper bound on the number of rational points of the curve. After recalling some general results on the theory of orders of a space curve, we will study the arithmetic properties of curves lying on a surface in P3 , to finally prove the bound. The talk is based on the preprint [1], accepted for publication at Acta Arithmetica.

Keywords: algebraic curves, embedded surfaces, rational points, finite fields

MSC: 11G20, 14G05, 14H50, 14J70

References

[1] Elena Berardini and Jade Nardi. Curves on Frobenius classical surfaces in P 3 over finite fields, arXiv preprint arXiv:2111.09578, 2021.

[2] Karl-Otto St¨ohr and Jos´e Felipe Voloch. Weierstrass Points and Curves Over Finite Fields, Proceedings of The London Mathematical Society,1, 1–19,1986.