# Od kongruencijskih brojeva do eliptičkih krivulja

Cicvarić, Borna (2015) Od kongruencijskih brojeva do eliptičkih krivulja. Diploma thesis, Faculty of Science > Department of Mathematics.

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Language: Croatian

To sumarize, we began with the congruent number problem and related it to the eliptic curve $y^2=x^3-n^2x$. Next, we defined eliptic curves and eliptic functions. We showed that every eliptic function can be written as a rational function of the Weierstrass $\wp$ function, and used it to construct a bijection between $\mathbb{C} / L$ and points of an eliptic curve in $\mathbb{P}_{\mathbb{C}}^2$. Using that bijection, we defined addition of points of an eliptic curve and explained its geometrical interpretation. Now the points of an eliptic curve have the properties of a group, so we examined the points of finite order, which form a subgroup. In the end we focus on the curve $y^2 = x^3-n^2x$, (which we will denote $E_n$) over $\mathbb{Q}$. Using reduction modulo $p$, where $p \in \mathbb{N}$ is a prime, we prove that the torsion subgroup of $E_n(\mathbb{Q})$ consists of only 4 elementa, and all the other elements of $E_n(\mathbb{Q})$ have infinite order. At the end we use our results and go back to the congruent number problem to prove the following theorem: \textbf{Teorem. } \textit{$n \in \mathbb{N}$ is a congruent number if and only if $E_n(\mathbb{Q})$ contains a point of infinite order (in other words, if there exist $P, Q \in E_n(\mathbb{Q})$, with $P \neq 0$ such that $P = 2Q$).}