The Amidah describes Hashem as being "Avinu" (our Father) twice. Once in the fifth beracha and once in the sixth beracha. The fifth beracha says, "Hashiveinu Avinu L'sorasecha" (Bring us back, our Father, to Your Torah). The sixth beracha states, "Selach Lanu Avinu Ki Chatanu" (Forgive us, our Father for we have sinned).
The Tur (Rabbi Ya'akov ben Rabbi Asher, b. 1269 Germany, d. 1343 Toledo, Spain, in Orach Chaim, chap. 115) explains why we describe Hashem as "Avinu" specifically in these two blessings. In the fifth blessing, we mention the Torah. Therefore, we also mention that Hashem is "Avinu" because the Gemara (Kiddushin, chap. 1, "Ha-isha Niknis", pg. 29a) says that a father has an obligation to teach his son Torah (based on Parshas Eikev, 11:19).
By mentioning Hashem as our Father, we are basically saying that we are His children. As such, He has an obligation to teach us Torah, as any father would have.
In the sixth beracha, we are asking Hashem to pardon us for our sins. In order to be forgiven for past crimes, we must rely on Hashem's compassion like that of a father's compassion towards his children (Tehillim, 103:13). Therefore, we mention "Avinu" in order to evoke that compassion.
Getting back to the fifth beracha and God's obligation to teach His children Torah, we must add the Midrash (Vayikra Rabba, Parshas Bechukosai, 35:3, based on Parshas Emor, 22:9) which says that Hashem chooses to abide by Torah law. Since Hashem imposes Torah law onto Himself, He must teach us the Torah because He is our Father and we are His children, and a father must teach his children Torah.
However, Hashem is not only referred to as "Avinu" in this beracha. He is also referred to as "Malkeinu" (our King), as it says, Vikarveinu Malkeinu La'avodasecha" (And bring us near, our King, to Your service). Why would the beracha begin describing Hashem as "Avinu" and conclude by describing Hashem as "Malkeinu?" This is an apparent lack of consistency.
The Shvilei Pinchas says that the answer is based on the Zohar (Parshas Behar, pg. 111b) which says that one difference between an eved (servant) and ben (son) is that an eved must do the will of his superior. But, an eved has no permission to search out his majesty's secrets and hidden treasures of his household. For example, an eved has no right to sneak into the king's bedroom, open the drawer of his night table, take out his majesty's diary, and start reading it. Such an eved has crossed a red line.
However, although a ben must also fulfil every command of his king, he is allowed to search out his secrets and hidden treasures because, after all, he is the king's son. As heir to the throne, he must become familiar with the inner workings of the palace.
Every single one of us must try to become more and more of a ben with respect to Hashem and search out the secrets and hidden treasures of our Father, the King. We do this every time we read our King's diary, the Torah.
This explains why we describe Hashem as "Avinu" and as "Malkeinu" within the same blessing. It comes to show that we want to be like an eved who fulfills every single command of His Majesty. Although we are His children, we do not want to take advantage by not abiding to His laws.
On the other hand, we also want to hold on to the title of ben, exercising the right of a prince to delve into the secrets of our Father, the King. We accomplish this by digging into the Torah.
Having said that, we must wonder why we are even asking Hashem to teach us Torah (Hashiveinu Avinu L'sorasecha) in a beracha whose thematic idea is teshuva (repentance)?
One approach to understanding the connection between Torah and teshuva can be seen through a controversial Midrash.
The Tanchuma (Parshas Ha'azinu, 4, based on Parshas Naso, 6:26) says that teshuva only works for Jews. However, the nations of the world cannot do teshuva. This can be extremely disturbing to many people. Are we playing favorites?
The Chida (Rabbi Chaim Yosef Dovid Azulai, 1724-1806, Jerusalem; in Birkei Yosef, Orach Chaim, 115:2; and in Rosh Dovid, Parshas Emor) addresses this by pointing out that the verse says about the Jews, "Banim Atem La'Hashem Elokeichem" (you are children to Hashem your God; Parashas Re'eh, 14:1). If we are God's children, then, Hashem is like our Father. This is why we can do teshuva. It is because the Gemara (Kiddushin, chap. 1, "Ha-isha Niknis", pg. 32a, Rav Masna in the name of Rav Chisda) says that a father who chooses to forgo his honor may do so.
For example, imagine a child is very disrespectful to his father. The seething father is deciding what form of punishment to implement. Then, right before punishing the child, the child grabs his father's leg, cries, and says, "I'm sorry dad, please forgive me." A father has the prerogative to forgive his child and "let it slide" completely dismissing the punishment.
This is how it works with us and God. We anger Him with our sins. He contemplates our punishment. Then, suddenly, we cry, apologize and ask for His forgiveness. As our Father, Hashem has the right to pardon without demanding that any penalty be implemented.
However, the nations of the world are only considered to be God's servants. There is no verse which refers to them as God's children. Verses only speak about the nations as Hashem's "avadim" (servants; see Tehillim, 47:9; Yirmiya, 10:7). The reason for this is as follows.
Before Mount Sinai, all people had the status of Hashem's servants. However, at Mount Sinai something changed. The Jewish people said, "Na'aseh V'nishma" (we will do and obey; Parshas Mishpatim, 24:7). This pledge meant that we wanted to enter into a deeper relationship with God. We no longer wanted to only be His servants. We wanted to be His children also.
The nations of the world rejected this proposal. As such, they remained with the old status as servants. However, the doors are not closed. Any person from amongst the nations of the world who wishes to become God's child, can say, "Na'aseh V'nishma" and convert to Judaism just like the rest of us did at Sinai.
Therefore, when the nations sin against God, they cannot do teshuva. This is because the Gemara (Kiddushin, chap. 1, "Ha-isha Niknis", pg. 32b, Rav Ashi; based on Parshas Shoftim, 17:15) says that if a king wishes to forgo his honor, he may not do so. A king must uphold the law.
We don't have to live in a world of kings and queens to understand this. Just think about our judicial system. Imagine that a crime was committed. Imagine that the judge ruled that the defendant was guilty as charged and sentenced to forty years imprisonment. Imagine that the defendant breaks down, begins to cry, and says, "Your honor, I'm so sorry, please forgive me and let it slide." Would any judge dismiss the penalty? No! As judge, he must uphold the law. The judge would probably say, "You should have thought of that before you committed the crime. Now, you have to pay the price."
This is how the system works for the nations of the world. When God finds them guilty of sin, they must be penalized. There is no such thing as teshuva which gets them off the hook without any sort of consequence. The only way for the nations to be cleansed of the sin is to suffer through the penalty. They must pay the price.
This explains why we refer to Hashem as "Avinu" in the beracha about teshuva. In order for our teshuva to be accepted, it is imperative that we establish that God is our Father. Only a Father can wave the punishment and pardon without us suffering any sort of consequence (Chida).
This leads right into the connection between "Avinu" and Torah. The Midrash (Bereishis Rabba, 1:1) says that the Torah said, "I was God's vessel. He looked into me and created the world."
The Shvilei Pinchas says that since the Torah referred to itself as a "vessel" of God, it is safe to refer to the Torah as the "scepter" of God. Just as a king of flesh and blood uses his scepter to command others to keep the law, similarly, Hashem uses His scepter, the Torah, to command us to keep the law.
Obviously, an eved has no jurisdiction to touch, let alone handle his majesty's scepter. Were he to do so, he would be executed immediately.
However, imagine the king's son jumping up onto the royal throne, sitting down, grabbing the royal scepter, and playing with it. The king begins to smile. He thinks to himself, "Good. He must become comfortable with this seat. He must become familiar with the scepter. After all, he is heir to the throne."
This explains why the Gemara (Sanhedrin, chap. 7, "Arba Misos", pg. 59a) says that if the nations of the world study Torah, they deserve the death penalty. It is because the Torah is His Majesty's royal scepter. For a servant to use the royal scepter is crossing a red line. It is so audacious that it demands the death penalty.
However, Jews may, and are even obligated to study the Torah. There is no problem with us "playing" with His Majesty's royal scepter because we are not just servants, we are His children. This is our right.
Therefore, when we study Torah, we demonstrate that we are Hashem's children. Subsequently, our teshuva must be accepted by God because, as children, our Father has the prerogative to forgive us without making us pay the price.
But, it's even deeper than that. The Belzer Rebbe (Rabbi Yissachar Dov Rokeach, 1854-1926, West Ukraine) proves from the Rambam (Sefer Hamitzvos, Mitzva 11, based on Parshas Vaeschanan, 6:7) that learning Torah and teaching Torah to one's child is considered to be one mitzva, not two. Therefore, if a father learns Torah, but does not teach it to his son, it's only considered half a mitzva.
Now, the Midrash (Bereishis Rabba, 8:2; based on Mishlei 8:30) says that Hashem fulfills the mitzva of learning Torah. But, so long as Hashem does not teach us the Torah, His own mitzva of Talmud Torah is lacking. This is one reason why Hashem was so anxious that we accept the Torah. It was so that He would be able to fulfil His own mitzva of Torah study in a complete and total way.
This explains why the Tanna D'bei Eliyahu Rabba (chap. 18, based on Eicha, 2:19) says that when we sit down to study Torah, Hashem sits down opposite us and learns with us. This means that Hashem sits down with us to teach us Torah. One reason for this is so that He can fulfil His own mitzva of Talmud Torah in totality.
Finally, we can understand the famous Gemara (Shabbos, chap. 9, "Amar Rebbi Akiva", pg. 88b) which tells us that when Moshe went on High to receive the Torah, the angels asked Hashem what this earthling was doing in Heaven. God told them that he came to bring the Torah down to Earth. The angels complained that people would sin against the Torah. They requested that the Torah remain in Heaven where it could be guarded properly (Tehillim, 8:5,10, and 2).
Hashem told Moshe, "Hechzer Lahem Teshuva" (give them an answer). Moshe said to Hashem that he was concerned about engaging in a conversation with the angels, lest the fire that emanates from their mouths burn him to a crisp. Hashem said not to worry, just grab hold of My Throne of Glory and "Chazor Lahem Teshuva" (give them an answer).
The Shela Hakadosh (Shavuos, Torah Ohr, 23) says that Hashem did not just tell Moshe to give them an answer, rather, Hashem told Moshe what it was specifically that he was supposed to answer them with. First Hashem said, "Hechzer Lahem" that means, "answer them." Then Hashem added what that answer should be. The next word is "Teshuva." Meaning, tell the angels about the fact that the Jews can do Teshuva for their sins.
In other words, tell the angels that the Jews are My children. Tell them that I am their Father. Tell them that as their Father, I can pardon them freely. Then go on to tell them that, as their Father, I have an obligation to teach them Torah. Ask the angels if they still want to withhold the Torah from them which will result in making God's own Torah study half baked. Is there any angel who would want to prevent Hashem from fulfilling His own mitzva in a total and complete way? Of course not. The angels were silent and the Torah was brought down to the Jews.
Practically speaking, whenever we say the "Hashiveinu" beracha, let us keep in mind that we are God's children and that Hashem is our father. As such, He has an obligation to teach us Torah.
Let us also keep in mind that as a result of this, we can do Teshuva because He is our Father and a Father can always wave the penalty.
More importantly, let us try to become even more God-like by trying to teach Torah to our children a little bit more and a little bit better.
Even people without children can teach another person Torah. When they do, the student becomes that person's spiritual child.
Either way, the Torah we teach should be filled with love, it should be sweet, empowering, comforting, and confidence building.
For, when we do so, we will have completed our own Torah learning.
So, may we children of God be blessed to be taught the scepter of Torah by Hashem with so much love that we are forgiven for all of our sins which will usher in the Messianic era.