# Ordinary vs Non-ordinary for GL(2)-type Abelian Surfaces over Q

Let $A_f$ be an abelian surface over $\mathbf{Q}$ of $\mathbf{GL}_2$-type arising from a weight $2$ cuspidal eigenform $f\in S_2(\Gamma_0(N))$. What is known (or expected to be true) for the size of the set of primes $p$ such that $A_f$ mod $p$ is ordinary (resp. non-ordinary)? Do you know of a reference where this is discussed?

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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. – Felipe Voloch Nov 17 '12 at 14:58
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. – Filippo Alberto Edoardo Nov 17 '12 at 15:04
@ Felipe and Filippo: thanks for your references! – Tommaso Centeleghe Nov 17 '12 at 15:06
@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). – Rob Harron Nov 17 '12 at 19:07
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. – Barinder Banwait Nov 17 '12 at 20:45