Rings of modular forms and a splitting of TMF0(7)

Among topological modular forms with level structure, {{\,\mathrm{TMF}\,}}_0(7) at the prime 3 is the first example that had not been understood yet. We provide a splitting of {{\,\mathrm{TMF}\,}}_0(7) at the prime 3 as {{\,\mathrm{TMF}\,}}-module into two shifted copies of {{\,\mathrm{TMF}\,}} and two shifted copies of {{\,\mathrm{TMF}\,}}_1(2). This gives evidence to a much more general splitting conjecture. Along the way, we develop several new results on the algebraic side. For example, we show the normality of rings of modular forms of level n and introduce cubical versions of moduli stacks of elliptic curves with level structure.