Timeline for Special case of Eichler–Shimura
Current License: CC BY-SA 4.0
18 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 16, 2023 at 19:51 | comment | added | Dendrit | @FrançoisBrunault ok, thanks, i will try to study | |
| Oct 16, 2023 at 19:38 | comment | added | François Brunault | @Dendrit This is explained in the notes linked to by Chris Wuthrich. | |
| Oct 16, 2023 at 18:57 | comment | added | Dendrit | @FrançoisBrunault thanks! But what is $X_0(11)$ and $X_1(11)$? | |
| Oct 16, 2023 at 18:54 | comment | added | Dendrit | @ChrisWuthrich thank you for link! But modularity means only that such product exists, am I right? Or it is possible to somehow deduce from modularity explicit formula for $F(q)$? | |
| Oct 16, 2023 at 14:11 | comment | added | François Brunault | @Dendrit Eichler-Shimura tells you that the $L$-function of the modular form $F(q)$ is equal to the $L$-function of the elliptic curve $X_0(11)$. This elliptic curve is isogenous to $X_1(11)$, which is itself isomorphic to $C$ (on a historical note, an equation of $X_0(11)$ has been determined by Fricke around 1920). This gives you the result. | |
| Oct 15, 2023 at 22:28 | comment | added | Chris Wuthrich | I believe the modularity of this elliptic curve is well explain in these notes. The curve in question here is isomorphic to $y^2+y=x^3-x$, which is shown to be the equation of $X_1(11)$. You will see this is really a bit more difficult than the level of Silverman-Tate, yet the notes give a good introduction into the subject. | |
| Oct 15, 2023 at 17:58 | comment | added | GH from MO | It is hard to tell what is needed for a proof before finding the proof. You will see more clearly after you have read the chapter I recommended. | |
| Oct 15, 2023 at 17:54 | comment | added | Dendrit | @GHfromMO thank you! So Hecke characters are necessary to obtain relation between $M_p$ and $N_p$? | |
| Oct 15, 2023 at 17:47 | comment | added | GH from MO | Read Iwaniec's "topics" book (bookstore.ams.org/gsm-17), especially Chapter 8. Note also that L-functions of modular forms are more fundamental than L-functions of elliptic curves (although many colleagues would disagree). | |
| Oct 15, 2023 at 17:43 | comment | added | Dendrit | @GHfromMO what is $L(s, F)?$ I know only about L-functions for elliptic curves | |
| Oct 15, 2023 at 17:27 | comment | added | Dendrit | @FrançoisBrunault so maybe you know where i can read proof of this theorem? thanks! | |
| Oct 15, 2023 at 17:22 | history | edited | LSpice | CC BY-SA 4.0 |
Link to article; removed "thanks"
|
| Oct 15, 2023 at 17:18 | history | edited | GH from MO |
edited tags
|
|
| Oct 15, 2023 at 17:18 | comment | added | GH from MO | The exercise is equivalent to the statement that the $L(s,C)=L(s,F)$. On a smaller note, please use a high-level tag like "nt.number-theory". I added this tag now. | |
| Oct 15, 2023 at 17:16 | comment | added | François Brunault | This question is double-starred in the exercise. This is of the same difficulty as Eichler-Shimura. | |
| Oct 15, 2023 at 16:42 | history | edited | Dendrit | CC BY-SA 4.0 |
edited title
|
| S Oct 15, 2023 at 16:41 | review | First questions | |||
| Oct 15, 2023 at 17:44 | |||||
| S Oct 15, 2023 at 16:41 | history | asked | Dendrit | CC BY-SA 4.0 |