La distribution des zéros des fonctions L
The Katz and Sarnak philosophy states that the distribution laws of zeros of $L$-functions follow the distribution laws of eigenvalues of random matrices. The zeros near the central point would reveal the symmetry type of our family of $L$-functions. Once the symmetry has been identified, it is conjectured that many statistics associated to the zeros would be predicted by the eigenvalues of the corresponding group of random matrices. This thesis will study the low-lying zeros of the family of elliptic curves over $\mathbb{Q}[i]$. Brumer computed the symmetry type of the family of elliptic curves over $\mathbb{Q}$ in 1992. New challenges arising from this generalisation over number fields of his work will be revealed in this thesis.
L-functions Low-lying zeros Katz and Sarnak philosophy Random matrix theory Elliptic curves