Timeline for Ordinary vs Non-ordinary for GL(2)-type Abelian Surfaces over Q
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 15, 2013 at 8:42 | comment | added | N. Kumar | Do we know that there exists at least one prime p for which $A_f$ mod p is ordinary? | |
| Nov 17, 2012 at 20:45 | comment | added | Barinder Banwait | Section 7 of Pink's Crelle paper "l-adic algebraic monodromy groups, cocharacters, and the Mumford-Tate conjecture" states a conjecture which predicts that, for any abelian variety over a number field, the set of primes where the variety has ordinary reduction has density 1 "in the potential sense". As Felipe observed, this is a theorem in dimensions 1 and 2. The extension of the base is also conjectured to be the smallest field over which the monodromy groups are connected. | |
| Nov 17, 2012 at 20:43 | comment | added | Rob Harron | ... Given an abelian variety as in this question, do you expect there to be any supersingular primes at all? If so, how big is the smallest one? And do you expect infinitely many? | |
| Nov 17, 2012 at 20:42 | comment | added | Rob Harron | @Felipe: It's the potential-ness matched with the requirement that the primes under consideration have degree 1. For instance, (unless I'm confusing myself), if E is an elliptic curve with CM by an imag. quad. field K, then after base changing to K you can say that E is ordinary at all unramified primes of degree 1 (despite the fact that we know that E is supersingular at half of the unramified primes of Q). While he's still telling you that E is ordinary at infinitely many places, it's the quantity of places of supersingular reduction that I would find more interesting. (cont'd) | |
| Nov 17, 2012 at 19:50 | comment | added | Felipe Voloch | @Rob: I looked it up (I was going by memory, which is not as good as it used to be) and that's about it (2.7,2.8 and 2.9). I am not sure why he needs potential, as ordinariness should be invariant under ground field extension. His result certainly implies infinitely many places of ordinary reduction. | |
| Nov 17, 2012 at 19:07 | comment | added | Rob Harron | @Felipe: Can you be more specific about the location in Ogus' paper? Skimming through it I only found results about how often an abelian variety is ordinary in a potential sense at degree one primes (this is statement 2.7.1 of the article). | |
| Nov 17, 2012 at 15:06 | comment | added | Tommaso Centeleghe | @ Felipe and Filippo: thanks for your references! | |
| Nov 17, 2012 at 15:04 | comment | added | Filippo Alberto Edoardo | Of course, you know Elkies' result about infinite supersingular primes for elliptic curves over the rationals. In 2008 Baba and Granath generalized his techniques and proved that there are infinitely many supersingular (or superspecial) primes for certain abelian surfaces with QM by the quarternion algebra of discriminant $6$ (plus some condition), see their paper in Bull. London Math. Society (40). They also quote some reference at the end for other special cases. | |
| Nov 17, 2012 at 14:58 | comment | added | Felipe Voloch | The paper by Ogus in "Hodge cycles, motives, and Shimura varieties", Springer Lecture Notes in Mathematics 900 (1982), has results on this and states what is expected. I don't know if that is still the state of the art. | |
| Nov 17, 2012 at 14:51 | history | asked | Tommaso Centeleghe | CC BY-SA 3.0 |