CIMPA research school in Ho Chi Minh city in 2016

CIMPA research school on

Lattices and applications to cryptography and coding theory
Saigon University, Ho Chi Minh City, Vietnam.
August 1^st - 12^th, 2016

Lattices and Modular Forms
Lecturers: René Schoof (Tor Vergata) - Valerio Talamanca (Roma Tre)

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Program

Brief review of holomorphic and meromorphc functions. Lattices in the complex plane and elliptic curves. Action of SL_2(Z) on the set of lattice in the complex plane. Moduli space for complex elliptic curves.

Classsical theta functions (Jacobi e Riemann) and their properties. Modular forms for SL_2(Z), cusp forms. The space M_k of weight 2k modular forms. Invariants of elliptic curves as modular functions. Eisenstein series and the explicit determination of a basis of M_k.

Theta series of an arbitrary lattices. Poisson summation formulas. Even Unimodular lattices. The Theta series of even unimodular lattices is a modular forms. Explicit examples.