The fourth day of CIMPA SCHOOL 2025 began with a lecture by Prof. Fabien Pazuki on Algebraic Number Theory, continuing from his previous session. He presented a theorem on the boundedness of the norm of elements in non-zero ideals of the ring of integers, using this theorem to show that there are only finitely many ideal classes in the ring of integers. He then discussed Dirichlet’s theorem on the finite rank of the unit group of the ring of integers. Prof. Pazuki concluded his lecture with a valuable remark:
“To study algebraic numbers, you study ideals, and it is computable.”

The next session was led by Dr. Pee Choon Toh, who continued his lecture on Modular Forms. He discussed the Valence Formula and additional properties of modular forms, providing further theoretical grounding for participants.

Following this, Dr. Samuele Anni continued the lecture series, beginning with the definition of the Eisenstein Series of weight 2 (G_2) and explaining why G_2 fails to transform like a modular form of weight 2. With motivating examples and properties, he moved on to the differentiation of modular forms and its failure of modularity. Using Serre derivation, Dr. Anni demonstrated that the Serre derivation of a modular form f in M_k lies in M_{k+2}. He concluded his session with an engaging discussion on Ramanujan’s Tau Function and Congruences.

Exercise Session

The day concluded with a productive exercise session where participants continued working on problems in algebraic number theory. In the final segment, participants began exercises on modular forms, engaging in active discussions and receiving guidance from the speaker team in a collaborative environment.