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2009 The DLP on Elliptic and Hyperelliptic Curves with the same endomorphism ring (in English, 39 slides)
We ask whether the discrete logarithm problem (DLP) in the divisor class group has the same complexity for all curves over finite fields with Jacobian varieties having the same ring of endomorphisms. We present a result of Jao, Miller and Venkatesan who proved that the answer to our question is positive for elliptic curves. We try to use the same methods to extend the result to the genus 2 case in the case that the Jacobian is of CM type and we present the work we have done so far. Finally, we explain which phenomena can occur for curves of genus 3.
2008 Iwasawa's theorem (in English, 15 pages)
We prove the Iwasava's Theorem, which describes the behaviour of the class number in an extension of a finite field.
[dvi] [ps] [pdf]Arithmetic of Quaternion Algebras: Orders and Ideals (in English, 17 pages)
The basics on the arithmetic on quaternion algebras is introduced: (maximal) orders, (principal) ideals, (reduced) norm/discriminant, ideal classes, etc.
[dvi] [ps] [pdf]The DLP on Elliptic Curves with the same order (in English, 20 pages)
We ask whether the discrete logarithm problem (DLP) has the same difficulty for all curves with the same order over a finite field. We present the result of Jao, Miller and Venkatesan who proved that the answer to our question is positive if you limit ourselves to curves with the same endomorphism ring.
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2007 The Tensor Product Theorem (in English, 11 pages)
The Tensor Product Theorem from Flath asserts that if A is the adele ring of a global field F and G is a reductive algebraic group over F, then G(A) decomposes into a "restricted tensor product" of representations of the groups G(Fυ). We give a proof of the theorem.
[dvi] [ps] [pdf]
2006 Modular forms of weight 1 (in English, 22 pages)
We study modular forms and Galois representations over finite and fields and over the complex numbers. We give the proof of an important theorem from Serre and Deligne that in every modular form of weight 1 we can attach a linear representation. This representation is unique up to isomorphism.
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2004 Primes of the form x2 + ny2 (in English, 24 pages)
We study ring class fields of orders in imaginary quadratic fields to determine which primes are of the form x2 + ny2, where x, y integers, for arbitrary n. We give certain examples how our result works in practice.
[dvi] [ps] [pdf]Optimal linear codes over GF(4) (in Greek, 18 pages)
A central problem in coding theory is that of finding the smallest length for which there exists a linear code of dimension k and minimum distance d, over a filed of q elements. We consider here the problem for quaternary codes (q = 4), solving the problem for k < 5 for all values of d.
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2003 Primality test (Algorithms and Complexity) (in Greek, 26 pages)
We consider the primality problem, to decide whether a number is prime or composite. In this survey we show that PRIMES is in coNP and in NP. Then we try a probabilistic approach and we show that PRIMES is in coRP and in ZPP. Finally we present one of the most significant results of the last years: that PRIMES is in P.
Last update: Aug 31, 2005
[doc] [mdi] [pdf]Smooth numbers and the quadratic sieve (in Greek, 12 pages)
With the help of Analytic Number Theory we consider the problem of optimizing the bound used in the quadratic sieve to factorise numbers.
[doc] [ps] [pdf]The main linear coding theory problem (in Greek, 27 pages)
Central problem in coding theory is that of constructing optimal codes for a variable (length, dimension, minimum distance), over a field of q elements, while keeping the other two constant. Here we present one version of the problem, with the help of Finite Geometries, and all the known results until now.
Last update: Jan 14, 2004
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2002 A lecture on Groebner bases (in Greek, 12 pages)
We study the methods of the Groebner bases in order to solve problems concerning polynomial ideals with algorithmic or computable methods.
[doc] [ps] [pdf] [html]This page is not updated anymore. Please update your links to:
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