Ajax not working for themed form

Question

I have a form for which I want to theme with a template file, and I also need to use ajax in it. But I get this error messages in the log and ajax doesn't work :

• message 1

  Invalid form POST data.

• message 2

  Notice: Undefined index: form_build_id in ajax_get_form() (line 320 of /Applications/MAMP/htdocs/ta/includes/ajax.inc).

ajax_test.module :

function ajax_test_menu() {
    return array(
        'ajax_test' => array(
            'title' => t('Ajax test'),
            'page callback' => 'drupal_get_form',
            'page arguments' => array('_ajax_test_form'),
            'access arguments' => array('access content'),
        )
    );
}

function ajax_test_theme() {
    return array(
        '_ajax_test_form' => array(
            'template' => 'ajax_test-form',
            'render element' => 'form',
        ),
    );

}

function _ajax_test_form() {
    return array(
        'name' => array(
            '#type' => 'select',
            '#id' => "select-id",
            '#options' => range(1, 40),
            '#ajax' => array(
                'callback' => '_ajax_test_callback',
                'wrapper' => "more-options",
            ),
        ),
    );
}

function _ajax_test_callback($form, $form_state) {
    return '<h1>Hello world!</h1>';
}

ajax_test-form.tpl.php :

<div>
    <? echo drupal_render($form['name']); ?>
</div>
<div style="height: 20px; border: 2px dashed blue" id="more-options"></div>