Skip to main content
added 134 characters in body
Source Link
GH from MO
  • 105.2k
  • 8
  • 292
  • 398

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.

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. 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.

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.

added 453 characters in body
Source Link
GH from MO
  • 105.2k
  • 8
  • 292
  • 398

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. 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.

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. The state-of-the art seems to be in Han Wu's thesis, see Theorem 0.3.4 there.

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. 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.

deleted 4 characters in body
Source Link
GH from MO
  • 105.2k
  • 8
  • 292
  • 398

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,Here $|\cdot|$ stands for the norm if we think in terms of Grössencharactersideals or ideles, and for the adelic norm if wedepending on how you think in terms 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. The state-of-the art seems to be in Han Wu's thesis, see Theorem 0.3.4 there.

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

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 if we think in terms of Grössencharacters, and for the adelic norm if we think in terms of 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 $\lambda<1/2$.

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. The state-of-the art seems to be in Han Wu's thesis, see Theorem 0.3.4 there.

added 2 characters in body
Source Link
GH from MO
  • 105.2k
  • 8
  • 292
  • 398
Loading
Source Link
GH from MO
  • 105.2k
  • 8
  • 292
  • 398
Loading