The fifth day of CIMPA SCHOOL 2025 began with a lecture by Dr. Francesco Campagna, who continued his discussion on Elliptic Curves over Finite Fields. He introduced the notion of the boundedness of the number of solutions of elliptic curves and stated the Hasse–Weil Theorem. The lecture then moved on to Frobenius isogeny and the degree of isogenies, followed by a clear proof demonstrating that (E) possesses a ring structure.

The next lecture was delivered by Prof. Michel Waldschmidt, who began his first lecture on Lattices. He introduced the topology on subgroups of the real numbers and the classification of closed subgroups of ℝⁿ. Using foundational theory on vector spaces E, he presented two equivalent definitions of lattices within E. The lecture continued with the study of key properties when E = ℝⁿ, including the concepts of the fundamental domain, covolume, density, and the dual lattice and its relationship with the original lattice. Prof. Waldschmidt concluded the lecture by providing a detailed example on ℝ².

Exercise Session

Participants continued their exercises on modular forms in a productive environment, actively interacting with the speaker team for guidance and discussion.

Extra Activity

At the end of the day, there were two short presentations where participants shared their research interests, allowing them to introduce their work and explore possible collaborations with fellow participants.