S2(K(587))+
Proving the existence of a nonlift.
-
There are no nonlift Borcherds product in S2(K(587))+
- There are three [nonlift] Borcherds products Borch(ψj) in S2(K(1174))−
for the following ψj, each given in the form of
(φ|V2)/φ + (a weight 12 cusp form divided by Δ):
- Each Borch(ψi)|AL2 − Borch(ψi)
lies in S2(K(1174))−.
We trace down Borch(ψ2)|AL2 − Borch(ψ2) to S2(K(587))+,
and show that the result is not in the subspace of Gritsenko lifts.
Thus
there is a nonlift in S2(K(587))+.
This procedure would also have succeeded with Borch(ψ3)|AL2 − Borch(ψ3).
- Once the existence of a nonlift in S2(K(587))+ is proven using the above arguments,
the previously conjectured formula for the nonlift eigenform f in S2(K(587))+
as a quotient f = Q/L of a weight 4 paramodular form with a weight 2 paramodular form becomes a theorem.
See
here for the formula
f = Q/L.
(The formula was part of
this project
)