Bounds for twisted symmetric square L-functions via half-integral weight periods

TY - JOUR

T1 - Bounds for twisted symmetric square L-functions via half-integral weight periods

AU - Nelson, Paul D.

PY - 2020/11

Y1 - 2020/11

N2 - We establish the first moment bound [formula presented] for triple product L-functions, where ψ is a fixed Hecke-Maass form on SL2 and ψ runs over the Hecke-Maass newforms on Γ0(p) of bounded eigenvalue. The proof is via the theta correspondence and analysis of periods of half-integral weight modular forms. This estimate is not expected to be optimal, but the exponent 5/4 is the strongest obtained to date for a moment problem of this shape. We show that the expected upper bound follows if one assumes the Ramanujan conjecture in both the integral and half-integral weight cases. Under the triple product formula, our result may be understood as a strong level aspect form of quantum ergodicity: for a large prime p, all but very few Hecke-Maass newforms on Γ0(p) of bounded eigenvalue have very uniformly distributed mass after pushforward to SL2(Z)\H. Our main result turns out to be closely related to estimates such as [formula presented] where the sum is over those n for which np is a fundamental discriminant and χnp denotes the corresponding quadratic character. Such estimates improve upon bounds of Duke-Iwaniec.

AB - We establish the first moment bound [formula presented] for triple product L-functions, where ψ is a fixed Hecke-Maass form on SL2 and ψ runs over the Hecke-Maass newforms on Γ0(p) of bounded eigenvalue. The proof is via the theta correspondence and analysis of periods of half-integral weight modular forms. This estimate is not expected to be optimal, but the exponent 5/4 is the strongest obtained to date for a moment problem of this shape. We show that the expected upper bound follows if one assumes the Ramanujan conjecture in both the integral and half-integral weight cases. Under the triple product formula, our result may be understood as a strong level aspect form of quantum ergodicity: for a large prime p, all but very few Hecke-Maass newforms on Γ0(p) of bounded eigenvalue have very uniformly distributed mass after pushforward to SL2(Z)\H. Our main result turns out to be closely related to estimates such as [formula presented] where the sum is over those n for which np is a fundamental discriminant and χnp denotes the corresponding quadratic character. Such estimates improve upon bounds of Duke-Iwaniec.

UR - http://www.scopus.com/inward/record.url?scp=85096641077&partnerID=8YFLogxK

U2 - 10.1017/fms.2020.33

DO - 10.1017/fms.2020.33

M3 - Journal article

SN - 2050-5094

VL - 8

JO - Forum of Mathematics, Sigma

JF - Forum of Mathematics, Sigma

M1 - e44

ER -