These notes give an account of a series of lectures at the University of Lecce as well as two at the University of Bari, all during Apri1 1986.
§§1-15 are based on the thesis [18], of J.-F.Voloch, a part from some background remarks and classical interpolations. They deal with the number of points on an algebraic curve over a finite field. The main results of the thesis are also contained in [14], §16 records some classical results on elliptic curves and §17, following Voloch [19], proves the existence of complete k-arcs form any values of k by taking half the points on anelliptic curve. §§18-19 discusses the values of n(2,q), the size of the smallest k- art in PG(2,q), and m'(2,q), the size of the second largest complete k- art in PC(2,q), the main result of §19 follows a proof of Segre using an improved bound for the number of points on a curve from §§ll and 14. Finally, §20 summarizes the best, known estjmates for $m_2(d,q)$, the irrgest size of k- cap in PG(d,q).
Table Of Contents
The maximum number of points on an algebraic curve |
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1-1 |
The deduction of Serre's and Ihara's results from the Riemann hypothesis |
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1-4 |
The essential idea in a particular case |
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4-7 |
Weierstrass points in characteristic zero |
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7-8 |
Fundamental definitions in algebraic geometry |
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8-11 |
The canonical series |
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10-11 |
The osculating hyperplane of a curve |
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11-14 |
The generalized Wronskian |
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15-17 |
Construction of some linear systems |
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18-19 |
The essential construction |
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19-24 |
Elliptic curves |
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24-25 |
Hyperelliptic curves |
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25-26 |
Plane curves |
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26-28 |
The maximum number of points on an algebraic curve |
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28-29 |
Elliptic curves: fundamental aspects |
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30-35 |
k-arcs on elliptic curves |
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k-arcs in |
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42-43 |
k-caps in |
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J. W. P. Hirschfeld |
46-47 |
Bibliography |
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48-49 |
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