Timeline for Hecke character and CM elliptic curve
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 4, 2016 at 1:48 | comment | added | GH from MO | @paulgarrett: I think the symbol $L(s,\pi)$ should be reserved for the finite $L$-function $\prod_p L_p(s,\pi)$ with functional equation $s\leftrightarrow 1-s$ and $\pi\leftrightarrow\tilde\pi$. Anything else, e.g. $L(s,\pi)$ with the gamma factors attached, or $L(s,\pi)$ shifted by some constant or rescaled in any way, should be denoted with a different symbol such as $\Lambda(s,\pi)$ or $D(s,\pi)$. This could be carried out in our universe. | |
| Jan 3, 2016 at 22:53 | comment | added | paul garrett | Somewhat related to @DenisChaperondeLauzières's comment, although I myself do like the $s\leftrightarrow 1-s$ convention, it is true that Deligne's conjectures, e.g., proven in some cases by Shimura and others for holomorphic elliptic or Hilbert modular forms' L-functions, about special values, do seem to "want" a wider critical strip, etc. And, yes, these results are highly anomalous. Once again, I think there's no single best convention... so one always must think about context. In a different universe things might have been otherwise... | |
| Jan 3, 2016 at 19:27 | comment | added | GH from MO | @DenisChaperondeLauzières: No doubt it makes sense, but I don't like it, especially that most automorphic $L$-functions seem highly transcendental. Algebraicity is a rare feature. | |
| Jan 3, 2016 at 19:05 | comment | added | Denis Chaperon de Lauzières | It makes sense to use the other normalization for algebraicity issues; for instance, the field generated over $\mathbf{Q}$ by the coefficients of the Hecke-normalized $L$-function of an elliptic curve (i.e., functional equation between $s$ and $1-s$) is of infinite degree over $\mathbf{Q}$, and depends on the elliptic curve (through the set of supersingular primes, for instance). | |
| Jan 3, 2016 at 17:41 | comment | added | paul garrett | Indeed, specifically, for elliptic curves there is a "competing" tradition of making the critical strip $0\le \sigma\le 2$, and critical line $\sigma=1$... | |
| Jan 3, 2016 at 17:16 | history | answered | GH from MO | CC BY-SA 3.0 |