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We use Borcherds products and their restrictions to Hirzebruch-Zagier curvesto determine generators and relations for the graded rings of Hilbert modularforms for the fields $\mathbb{Q}(\sqrt{29})$ and $\mathbb{Q}(\sqrt{37})$. Theseseem to be the first cases where the graded ring can be computed despiteobstructions to the existence of Borcherds products with arbitrary divisors.

The rings of Hilbert modular forms for $\mathbb{Q}(\sqrt{29})$ and $\mathbb{Q}(\sqrt{37})$