TITLE: What is exceptional about the prime numbers 2 and 3?
QUESTION [31 upvotes]: Admittedly this question is  vague. But I hope to convey my point. Feel free to downvote this.
Permit me to define prime number  the following way:
A number $n>1$ is a prime if all integers $d$ with $1< d \leq \sqrt{n}$ give non-zero remainders  while dividing $n$.   This will expose the primes 2 and 3: they qualify as primes  as this condition is vacuous (there are simply no integers in that interval to check divisibility).
Lot of exceptions to  some theory (quadratic forms, elliptic curves, representation theory)  happen at primes 2 and 3. Is it because 2 and 3 entered The Prime Club by this backdoor?  It is my wild guess. Perhaps experts might be able to say if I am wrong. Or possibly there are more conceptual reasons. Kindly
enlighten me.

REPLY [6 votes]: The quote from Serre-Tate in jmc's answer indicates that for an abelian variety $A$ of dimension $d$, the primes that have the most complicated behavior are those satisfying $p\le 2d+1$, but it doesn't explain WHY those primes appear. The answer has to do with the representation of $\hbox{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}$ on the Tate module, or more simply on the $\ell$-torsion $A[\ell]$. Things become more complicated, for example the computation of the conductor is quite complicated, if this representation is wildly ramified. And it is not that hard to prove (as is done in [ST]), by looking at the order of $\hbox{GL}_{2d}(\mathbb{Z}/\ell\mathbb{Z})$, that the representation can only be ramified at primes $p\le 2d+1$. For example, the gcd over all (sufficiently large) $\ell$ of $\#\hbox{GL}_{2}(\mathbb{Z}/\ell\mathbb{Z})$ is divisible only by 2 and 3, which is why they are the primes for elliptic curves that may be wildly ramified.