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Let $L(s,\chi)$ be the $L$-function of a non-trivial Hecke character of a general number field $K$, so that $L(s,\chi)$ which has no pole or zero at $s=1$.

I am looking for a reference for upper bounds for $\alpha$ in the convexity estimate $$L(s,\chi)\ll_{\epsilon,K,\chi}|\Im(s)|^{\alpha+\epsilon}, \qquad\frac{1}{2}<\Re(s)\leq 1,$$ for all $\epsilon>0$. I care only about the dependence on $s$ but not the dependence on other parameters (e.g.the conductor). I am aware that equation (5.20) of Iwaniec-Kowalski gives $$\alpha=[K:\mathbb{Q}]\frac{1-\Re(s)}{2},$$ but I was wondering if there is a better bound.

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    $\begingroup$ I added a paragraph about Wu's thesis. Please give me a few minutes to polish this. $\endgroup$
    – GH from MO
    Commented May 13, 2015 at 17:44
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    $\begingroup$ I finished editing the added paragraph about Wu's thesis, which gives an explicit $\lambda<1/2$ and speculates about the scope of Wu's methods (which are based on the ideas of Michel and Venkatesh). $\endgroup$
    – GH from MO
    Commented May 13, 2015 at 17:51

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You have to be careful: the implied constant also depends on $\chi$ (not just on $\epsilon$), in particular it depends on the number field $K$. (I edited your post to reflect this.)

There is a better (subconvex) bound available, namely there is an absolute constant $\lambda<1/2$ such that $$ L(\sigma+it,\chi)\ll_{\epsilon,K} C(\chi|\cdot|^{it})^{\lambda(1-\sigma+\epsilon)},\qquad 1/2\leq\sigma\leq 1, $$ where $C(\chi|\cdot|^{it})$ is the analytic conductor of the Hecke character $\chi|\cdot|^{it}$. Here $|\cdot|$ stands for the norm of ideals or ideles, depending on how you think of Hecke characters (Grössencharacters or idele class characters). This follows from the Phragmén-Lindelöf convexity principle combined with Theorem 5.1 in this paper of Michel and Venkatesh. In particular, $$\alpha=\lambda[K:\mathbb{Q}](1-\Re(s))$$ is admissible in your bound with the same absolute constant $\lambda<1/2$.

Added 1. The state-of-the art seems to be in Han Wu's thesis, see Theorem 0.3.4 there. According to this result, $\lambda=(5+2\theta)/12$ is available, where $\theta=7/64$ is the current record towards the Ramanujan-Selberg conjecture on $GL(2)$ over $K$. Perhaps the Burgess-like exponent $\lambda=(3+2\theta)/8$ is also within reach, especially in the light of the other main results of the thesis (for twists of cusp forms on $GL(2)$ over $K$), which have appeared in GAFA recently. See here.

Added 2. Han Wu recently established $\lambda=(3+2\theta)/8$ in this preprint.

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  • $\begingroup$ So improving on $\lambda$ would that give any (other) applications? $\endgroup$
    – Dr. Pi
    Commented May 13, 2015 at 18:25
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    $\begingroup$ @CaptainDarling: I don't know of a specific application of improving $\lambda$ here, i.e. improving the $s$-aspect. On the other hand, improving on Wu's global exponent has applications to arithmetic equidistribution problems. Namely, a better exponent yields better equidistribution, e.g. in the analogue of Duke's theorem on the equidistribution of closed geodesics in $SL(2,\mathbb{Z})\backslash H$ over number fields. $\endgroup$
    – GH from MO
    Commented May 13, 2015 at 19:38

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