LA PORTA, Lorenzo (2017) Geometric aspects of p-adic modular forms. [Magistrali biennali]
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The first definition that one usually encounters of modular forms (automorphic functions of the complex Poincaré half-plane) might not appear self-explanatory at first, and the understanding of such definition is made much easier by the geometric interpretation of the modular forms as k-differential forms on the Riemann surfaces associated to each congruence subgroup of SL2(Z), that are one first instance of modular curves. This interpretation of the modular forms can be used the free oneself from the limitation of considering modular forms over the complex numbers, and give a definition in a much more abstract context. Similarly, one can proceed from the usual definition over the complex numbers and consider some congruences of the q-expansions of modular forms with integral or p-integral coefficients, and through elementary techniques this leads to the definition of the p-adic modular forms. These, exactly as the classical ones, admit a geometric description as global sections of a sheaf on a modular curve. This sheaf’s existence is linked to some p-adic representations, and our goal is trace out its construction in the second Chapter, while keeping in mind the analogy with the classical situation, and the elementary definition of p-adic modular forms that is detailed as well in the first Chapter.
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