A Siegel paramodular form of level N is a Siegel modular form of degree two

with respect to the paramodular group of level N. The Fourier-Jacobi

coefficients of such modular form are holomorphic Jacobi forms in the sense

of Eichler-Zagier. Conversely, there are two ways to construct paramodular

forms from Jacobi forms, namely additive Jacobi lifts (a generalization of

Maass lifts) and Borcherds products. In 2013, Gritsenko, Poor and Yuen

proposed the theta-block conjecture which characterizes paramodular forms

which are simultaneously Borcherds products and additive Jacobi lifts.

Theta-block is a great way to construct holomorphic Jacobi forms of small

weight proposed by Gritsenko-Skoruppa-Zagier. They are defined as the

product of odd Jacobi theta-series divided by Dedekind eta-function. In

this talk, I will prove the theta-block conjecture for two infinite series

of theta blocks of minimal weight 2 related to root systems A_4 and

A_1+B_3, here the infinite series mean that the level of paramodular forms

goes to infinite. In some sense, the additive lift is like an infinite sum

and the Borcherds product has an infinite product expansion. Thus our

results give generalized Euler type identities of the form "infinite sum =

infinite product".

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