What integer value can be the conductor of a $g$-dimensional abelian variety over $\mathbb Q$? Fix a positive integer $g$. What positive integer $N$ can  be the conductor of a $g$-dimensional abelian variety over $\mathbb Q$ ?
For example, as there is no abelian varieties over $\mathbb Z$, $N$ can not be $1$. And for elliptic curves, $N$ must be no less than $11$.
 A: You seem to be asking about which small integers cannot occur as the conductor $N$ of an abelian variety over $\mathbb Q$ of dimension $g$, but possibly you'll also be interested in other constraints on the value of $N$. It's relatively easy to see that for primes $p>2g+1$, one has $\operatorname{ord}_p(N)\le2g$. For $p\le2g+1$, there are also upper bounds for $\operatorname{ord}_p(N)$, but they're more complicated. See for example


*

*Lockhart-Rosen-Silverman, An upper bound for the conductor of an
abelian variety. J. Algebraic Geom. 2 (1993), no. 4, 569–601
(MR1227469). 

*Brumer--Kramer, The conductor of an abelian variety.
Compositio Math. 92 (1994), no. 2, 227–248 (MR1283229).

A: There is work by Brumer and Kramer, Paramodular abelian varieties of odd conductor, on the possible conductors. They show for example that if $A$ is a semistable abelian surface over $\mathbb{Q}$ of odd non-square conductor $N$, then $N \geq 249$ (and there is an explicit abelian surface with this conductor). They also give tables of possible odd conductors $N \leq 1000$.
The Langlands philosophy predicts that abelian varieties of dimension $g$ over $\mathbb{Q}$ should (roughly) correspond to automorphic forms on $\mathrm{GSp}_{2g}/\mathbb{Q}$. There is even a precise conjecture, see Section 8 in the article of Brumer and Kramer. If you assume this conjecture, then you are lead to investigate the space of such automorphic forms, and for what levels the space is non-trivial. This gives only a necessary condition, because the field of Hecke eigenvalues may be larger than $\mathbb{Q}$. There has been work by Poor and Yuen, Paramodular cusp forms where they classify such automorphic forms. Their results support the above conjecture.
