Timeline for Are there 'analytic' $p$-adic modular forms.
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 26, 2011 at 8:46 | vote | accept | wood | ||
| May 24, 2011 at 20:58 | comment | added | S. Carnahan♦ | $E_4$ converges nowhere ,so it is completely boring. Proof: for any fixed $\tau$, choose $n$ a big enough power of $p$ so that $n$ times anything near $\tau$ is small. You get infinitely many summands with $m$ congruent to $1$ mod $p$. $\Delta$ converges in the $p$-adic unit disc, since it is the $q$-expansion of a modular form. | |
| May 24, 2011 at 20:22 | comment | added | wood | .... As the answers so far suggest, these probably will not have to do a lot with $p$-adic modular forms ala Serre or Katz. But do these functions play a role anywhere. As in the early days, the modular transformation properties of $\eta$ where used by Rademacher to find asymptotics for the partition function. I guess this is not the way modular forms are seen nowadays. But do the $p$-adic analytic properties reveal anything interesting? | |
| May 24, 2011 at 20:19 | comment | added | wood | @all: All answers and comments so far have been focusing on the relation between modular forms and elliptic curves and how this translates into the $p$-adic world. Maybe the question above is not posed as I wanted it to be understood. Take the $E_4$ and $\Delta$ as defined above ($\tau,q$ in suitable some subset of $\mathbb{C}_p$). Do these functions converge anywhere and if yes do they have interesting properties. At least $E_4$ will have transformation properties.... | |
| May 24, 2011 at 20:05 | comment | added | Kevin Buzzard | ...you get a function on $\tau$'s obeying the usual relations -- but in the $p$-adic world you need more than this so the theory is perhaps a bit deeper. | |
| May 24, 2011 at 20:04 | comment | added | Kevin Buzzard | Here's a more sensible and I hope comprehensible way of thinking about it. A modular form isn't really a function on the upper half plane! It's a function on elliptic curves, or pairs consisting of an elliptic curve and a differential. This definition works well $p$-adically and over the complexes. Now over the complexes every elliptic curve is isomorphic to $\mathbf{C}/\langle 1,\tau\rangle$ for some $\tau$ in the upper half plane. But over the $p$-adics this is not true; the curves that can be uniformised are those with bad reduction. So for a classical modular form... | |
| May 24, 2011 at 19:57 | answer | added | S. Carnahan♦ | timeline score: 2 | |
| May 24, 2011 at 19:32 | comment | added | Kevin Buzzard | ..."over the p-adics most elliptic curves don't; only the ones with good reduction do". So $q$ is not the full story (although of course by analytic continuation properties, the power series determines the form). | |
| May 24, 2011 at 19:31 | comment | added | Kevin Buzzard | Because of convergence issues, I don't think one can make sense of the notion that a $p$-adic modular form is a function on $\mathbf{C}_p$. But here's something one can do: a $p$-adic modular form still has a $q$-expansion which converges for $|q|<1$, because $|q|<1$ is a (rather small) region on the modular curve where these things are defined, and so you could think of a $p$-adic modular form as a function on this disc defined by a power series. But the hard part is saying what "transformation law" it has to satisfy, because "every elliptic curve over the complexes has a $q$, but..." | |
| May 24, 2011 at 15:43 | history | edited | wood | CC BY-SA 3.0 |
added 5 characters in body
|
| May 24, 2011 at 14:38 | answer | added | David Loeffler | timeline score: 6 | |
| May 24, 2011 at 14:25 | answer | added | Ramsey | timeline score: 11 | |
| May 24, 2011 at 13:51 | history | asked | wood | CC BY-SA 3.0 |