Timeline for Modern Algebraic Geometry and Analytic Number Theory
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Feb 26, 2019 at 16:21 | vote | accept | lulu2612 | ||
| Feb 26, 2019 at 16:21 | vote | accept | lulu2612 | ||
| Feb 26, 2019 at 16:21 | |||||
| Feb 26, 2019 at 13:02 | comment | added | KConrad | Kumar Murty showed ("On the Sato-Tate conjecture", pp. 195-205 in Number Theory related to Fermat’s Last Theorem, 1982) that analytic continuation to ${\rm Re}(s) = 1$ of the $m$-th symmetric power $L$-functions for all $m \geq 1$ is enough, as their nonvanishing can be deduced from the analytic continuation. (The paper also has a result about Sato-Tate in the function field case, and unfortunately the MathSciNet review of this paper --- see MR0685296 -- mentions only the function field result.) | |
| Feb 26, 2019 at 12:20 | comment | added | KConrad | The Tauberian theorem used in the book is the Wiener-Ikehara one, but they could also have used Newman's Tauberian theorem, which gets the same conclusion (an asymptotic relation) by a less technical argument at the cost of an added hypothesis that is often easy to verify in practice. | |
| Feb 26, 2019 at 12:18 | comment | added | KConrad | @reuns the deduction of Sato-Tate from analytic continuation and nonvanishing of symmetric-power $L$-functions on the line ${\rm Re}(s) = 1$ (which is the boundary of the half-plane of convergence of their Euler products) is discussed in the Murty-Murty book "Non-vanishing of $L$-functions and Applications." See Section 7 of Chapter IV, where the argument relies on an equidistribution theorem (Theorem 3.1 on p. 68), whose proof relies on Weyl's equidistribution theorem in a compact group (Cor. 2.2 on p. 67) and a Tauberian theorem (Thm. 1.1 on p. 7) | |
| Feb 25, 2019 at 22:08 | comment | added | reuns | Could you tell where heavy estimates are needed ? One looks at the symmetric power L-functions $L(s,\text{sym}^mE)=\exp(\sum_{p^k} \frac{p^{-sk}}{k}\frac{\sin t_p k(m+1)}{\sin t_p k})$ where $p+1−\#E(\mathbf{F}_p)=2\cos(t_p)p^{1/2}$ having no pole at $s=1$ iff E has no CM, from the PNT-like asymptotic $\sum_{p \le x} \frac{\sin t_p m}{\sin t_p }=o(\pi(x))$ and the orthonormal basis for $(u,v)=\frac1\pi\int_0^\pi u(t)v(t)\sin^2(t)dt$ one finds $\sum_{p \le x}f(t_p) \sim (f,1)\pi(x)$ for any $f$ continuous, which implies Sato-Tate | |
| S Feb 25, 2019 at 11:00 | history | answered | KConrad | CC BY-SA 4.0 | |
| S Feb 25, 2019 at 11:00 | history | made wiki | Post Made Community Wiki by KConrad |