What is Tachanun, anyway?
After the sin of the eigel hazahav, Moshe Rabbeinu davened to HaShem in 3 positions: sitting, standing, and falling (onto the floor). [And since this is found in Devorim 9:9, 10:10, and 9:18, and since they are all found in this week’s sedrah, Eikev (you knew that already, right?), you can stop wondering why I have chosen to talk about this today.]
We therefore follow this precedent: birchos shema (sitting), shemoneh essrei (standing), and tachanun (falling upon our faces).
The minhag is for women to omit Tachanun*.
The Zohar and other sources state that Tachanun is an especially intense Tefillah, and therefore its full intensity—which essentially involves falling upon one’s face–is only done according to most minhagim when there is a sefer Torah there (representing the Shechinah).
This should teach us about the added kedushah in a Shul or Beis HaMedrash where we do whatever we do in front of an Aron Kodesh containing many sifrei Torah.
*I am conducting an experiment to see how many people care enough to ask why.