core files come from a core dump. A core dump can happen for lots of reasons, but in shared hosting it generally occurs if:

1. Your site software is not compatible with the latest server software versions.

2. The server software itself was/is having issues.

I'd verify your cart is fully compatible with the PHP version your server runs as well as MySQL 4.1. It would not hurt to check and see about compatibility for the coming months about MySQL 5.x which is planned soon and possibly PHP 5 as well.