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AVERAGE OF THE FIRST INVARIANT FACTOR OF THE REDUCTIONS OF ABELIAN VARIETIES OF CM TYPE
 Author(s): Kim, Sungjin;
 Advisor(s): Duke, William;
 et al.
Abstract
Let $E$ be an elliptic curve defined over $\Q$ and with complex multiplication by $\mO_K$, the ring of integers in an imaginary quadratic field $K$. Let $p$ be a prime of good reduction for $E$. It is known that $E(\F_p)$ has a structure
\begin{equation}
E(\F_p)\simeq \Z/d_p\Z \oplus \Z/e_p\Z
\end{equation}
with uniquely determined $d_pe_p$.
We give an asymptotic formula for the average order of $e_p$ over primes $p\leq x$ of good reduction, with improved error term $O(x^2/\log^A x)$ for any positive number $A$, which previously $O(x^2/\log^{1/8} x)$ by ~\cite{Wu}. Further, we obtain an upper bound estimate for the average of $d_p$, and a lower bound estimate conditionally on nonexistence of Siegelzeros for Hecke Lfunctions.
Then we extend the methods to abelian varieties of CM type. For a field of definition $k$ of an abelian variety $\Av$ and prime ideal $\ip$ of $k$ which is of a good reduction for $\Av$, the structure of $\Av(\F_{\ip})$ as abelian group is:
\begin{equation}
\Av(\F_{\ip})\simeq \Z/d_1(\ip)\Z\oplus\cdots\oplus\Z/d_g(\ip)\Z\oplus\Z/e_1(\ip)\Z\oplus\cdots\oplus\Z/e_g(\ip)\Z,
\end{equation}
where $d_i(\ip)d_{i+1}(\ip)$, $d_g(\ip)e_1(\ip)$, and $e_i(\ip)e_{i+1}(\ip)$ for $1\leq i
We use the class field theory and the main theorem of complex multiplication to obtain the average behaviors of $d_1(\ip)$ when averaged over primes $\ip$ in $k$ with $N\ip
Finally, for elliptic curves $E$ over $\Q$, the asymptotic density $C_{E,j}$ of primes $p\leq x$ with $d_p=j$ which is given by ~\cite{C2}:
\begin{equation}
C_{E,j}=\sum_{k=1}^{\infty} \frac{\mu(k)}{[\Q(E[jk]):\Q]}.
\end{equation}
We prove under an appropriate conditions that these constants are positive.
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