Holomorphic Hoffstein-Lockhart

Question

In the article Hoffstein, Jeffrey; Lockhart, Paul "Coefficients of Maass forms and the Siegel zero." Ann. of Math. (2) 140 (1994), no. 1, 161–181, it is stablished a good bound for the Petersson norm of a Hecke Mass newform. As they indicate themselves, their method also works for classical holomorphic newforms of arbitrary level and weight.

Is this version explicitly written somewhere?

If $N$ is the level and $k$ is the weight, the bound is a function of $N$ and $k$, which is NOT exactly the same as in the Maass case (even if we replace $k$ by the corresponding eigenvalue). It would be useful to see the explicit form of the bound neatly written.

I know it follows from the Rankin-Selberg method, but the important point is to control a special value of an automorphic form on $GL_3$ and I am not familiar with such forms.

Answer 1

You can find a detailed treatment for all cuspidal representations of $GL(2)$ over a number field in Péter Maga's thesis, see his Proposition 3.2 on Page 20. Note that this is really what you need, because the residue appearing in the proposition equals, up to an explicit constant depending on the level and the number field (which is essentially the volume of the fundamental domain), the ratio of the squared norm of a newvector and the squared norm of the corresponding vector in the Whittaker model, cf. (3.3) and (3.4) in the mentioned work. This is the adelic analogue of (0.5) in Hoffstein-Lockhart who normalize $f$ to have $\|f\|=1$.

Added. I add more details responding to the OP's comment. There is no need to be familiar with the general theory to use the above results. Let $$f(z)=\sum_{n=1}^\infty a(n)e(nz)$$ be a holomorphic newform of weight $k$, level $N$, and arbitrary nebentypus. Then $a(n)$ can be expressed as $a(1)\lambda(n)n^{\frac{k-1}{2}}$, where $\lambda(n)$ is the $n$-th Hecke eigenvalue normalized so that the Ramanujan conjecture (proved by Deligne for this setting) reads $|\lambda(p)|\leq 2$ for $p$ prime. Using the Rankin-Selberg unfolding technique (see Section 1.6 in Bump: Automorphic forms and representations) we readily get $$\frac{3}{\pi}({\rm SL}_{2}(\mathbb{Z}) : \Gamma_{0}(N))^{-1}\int_{\Gamma_0(N)\backslash\mathcal{H}} y^k|f(x+iy)|^2 \frac{dxdy}{y^2}\\=(4\pi)^{-k}\Gamma(k)\cdot\mathrm{res}_{s=1}\sum_{n=1}^\infty\frac{|a(n)|^2}{n^{s+k-1}},$$ that is, $$\mathrm{res}_{s=1}\sum_{n=1}^\infty\frac{|\lambda(n)|^2}{n^{s}}=\frac{3}{\pi}({\rm SL}_{2}(\mathbb{Z}) : \Gamma_{0}(N))^{-1}\cdot\frac{(4\pi)^k}{\Gamma(k)}\cdot\frac{\|f\|^2}{|a(1)|^2}.$$ The Dirichlet series on the left hand side agrees with $L(s,\pi\otimes\tilde\pi)$ apart from having different Euler factors at the ramified primes $p\mid N$, which are easy to estimate. Here I denoted by $\pi$ the cuspidal representation generated by $f(z)$. Hence the general bound mentioned in my original post above yields $$ (Nk)^{-\epsilon}\ll \frac{1}{N}\cdot\frac{(4\pi)^k}{\Gamma(k)}\cdot\frac{\|f\|^2}{|a(1)|^2}\ll (Nk)^\epsilon,$$ where the implied constant depends only on $\epsilon$. Of course this is a standard result that can be used in a research paper without any further comment. If $f(z)$ is a non CM form, the bounds can be improved further, see e.g. the Appendix by Goldfeld-Hoffstein-Lieman to the paper by Hoffstein-Lockhart.

Answer 2

The reference pointed out by GH gives an excellent general write-up. For those interested in completely explicit bounds in the classical case of $GL(2)$ over $\mathbb{Q}$, I've worked these out in a few cases, though none in complete generality. For example if $f$ is a newform in $S_{2}(\Gamma_{0}(D), \chi_{D})$ that does not have complex multiplication and $D$ is a fundamental discriminant, then Proposition 11 of this paper shows that $$ \frac{3}{\pi [{\rm SL}_{2}(\mathbb{Z}) : \Gamma_{0}(D)]} \iint_{\mathbb{H} / \Gamma_{0}(D)} |f(x+iy)|^{2} y^{2} \, \frac{dx \, dy}{y^{2}} > \frac{3}{208 \pi^{4} \log(D)} \prod_{p | D} \left(\frac{p}{p+1}\right). $$ The appendix to Hoffstein and Lockhart's paper explains how one gets a zero-free regions for $L(f \otimes \overline{f}, s)$ near $s = 1$, and a good reference for the classical problem of turning a zero-free region into a lower bound is Hoffstein's paper "On the Siegel-Tatuzawa theorem" (in Acta Arithmetica, 1980/1981).