Conjecture on theta-blocks of order one. Part 2: root system A_4 and Borcherds products

Theta-blocks are special automorphic products which are holomorphic Jacobi forms. These objects are important in number theory, the theory of automorphic forms, Lie algebras, algebraic geometry and string theory. Conjecture on theta-blocks of order one was formulated by Gritsenko, Poor and Yuen in 2013. In the first talk we gave a general review of the construction of theta-blocks in the context of Siegel modular forms. In the second talk we propose an explanation of the existence of the theta-block of weight 2 and prove the theta-block conjecture for them. The main ingredients in our construction are the affine root system A_4 and the Borcherds products for the dual modular lattice A*_4(5)with determinant 125. This is my new result with H. Wang (Lille).