L4-norms and sign changes of Maass forms

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Haseo Ki, Yonsei University

Unconditionally, we prove the Iwaniec-Sarnak conjecture for L4-norms of the Hecke-Maass cusp forms. From this result, we can justify that for even Maass cusp form ϕ with the eigenvalue λϕ=14+t2ϕ, for a>0, a sufficiently large h>0 and for any 0<ϵ1<ϵ/107 (ϵ>0) , for almost all 1k<t1ϵϕ, we are able to find βk={Xk+yi:a<y<a+h} with 12+k1t1ϵϕXk12+kt1ϵϕ such that the number of sign changes of ϕ along the segment βk  is ϵt1ϵ1ϕ as tϕ. Also, we obtain the similar result for horizontal lines. On the other hand, we conditionally prove that for a sufficiently large segment β on R(z)=0 and I(z)>0, the number of sign changes of ϕ along β is ϵt1ϵϕ and consequently, the number of inert nodal domains meeting any compact vertical segment on the imaginary axis is ϵt1ϵϕ as tϕ.