# modular forms – Why is theory of newforms for half-integral weight difficult? Why does the \$U\$-operator show up in def of oldforms?

Integer weight case: Modular forms in $$S_{2k}(N) := S_{2k}(Gamma_{0}(N))$$ can come from lower levels, and we’d like to know when that happens. This is where we call a modular form $$f$$ “old” if in fact it belongs to $$S_{2k}(M)$$ for some $$M mid N$$ and $$M < N$$. How do we characterize such forms? We notice that if $$g in S_{2k}(M)$$ then $$g mid V(d) in S_{2k}(dM)$$. One can then define the space of “oldforms” by

$$S_{2k}^{old}(N) := bigoplus_{substack{dM mid N \ M neq N}} S_{2k}(M) mid V(d).$$

The space $$S_{2k}^{new}(N)$$ of “newforms” will then be the orthogonal complement in $$S_{2k}(N)$$ of the oldform space with respect to the Petersson inner product. A “newform” is then a normalized form in $$S_{2k}^{new}(N)$$ that is an eigenform for all Hecke operators and for all of the Atkin-Lehner involutions $$W_{p}$$ for $$p mid N$$ and $$W(N)$$.

Half-weight case: Winfried Kohnen has built a theory of newforms for the half-weight subspaces $$S_{k+1/2}^{+}(4N) := {f in S_{k+1/2}(4N) : a_{f}(n) = 0 text{ if } (-1)^{k}n equiv 2, 3 mod 4}$$ where $$N$$ is odd and square-free. Why must he hone in on the plus space to develop this theory? Others such as Ueda and Manickam further develop theories of newforms on the more general space $$S_{k+1/2}(4N)$$, and they even go further to $$S_{k+1/2}(2^{alpha}N)$$ for $$2 leq alpha leq 5$$. But setting up the oldform space looks odd to me. For instance, for level $$4$$ we have

$$S_{k+1/2}^{old}(4) = S_{k+1/2}^{+}(4) oplus S_{k+1/2}(4) mid U(4).$$

In general, people like Manickam seem to like to define the oldform spaces like

$$S_{k+1/2}^{old}(4N) = sum_{substack{d mid 2N \ d < 2N}} left(S_{k+1/2}^{+}(4d) + S_{k+1/2}^{+}(4d)mid U(4N^{2}/d^{2})right) + sum_{substack{d mid 2N \ d text{ odd}}} left(S_{k+1/2}(4d) + S_{k+1/2}(4d) mid U(4N^{2}/d^{2}) right)$$

(there is an easier way to write this by defining $$S_{k+1/2}(N)$$ to be either plus space of level $$4N$$ when $$N$$ odd or non-plus space of level $$2N$$ when $$N$$ even).

Why are $$U$$-operators used in this definition? Why can we not just define oldforms just as in the integer-weight case? Why is it difficult to get a “theory of newforms” for these half-weight spaces? What makes higher orders of $$2^{alpha}$$ dividing $$N$$ so difficult?

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