How is shmita related to Mt. Sinai?

B"H

In Pirkei Avot (Ethics of the Fathers) Ch. 5 it reads:
גלות באה לעולם על עבודה זרה, ועל גילוי עריות, ועל שפיכות דמים, ועל שמיטת הארץ.

Loosely translated:
Exile manifests in the world due to idol-worship, immoral relations, murder and not obeying shmita laws (=laws pertaining to the sabbatical year we must give the land on a 7-year cycle.)

What immediately jumps out are two questions. But first some background:
1) The prohibition of idol-worship is at the very foundation (and indeed fabric) of Jewish belief and practice.
2) Murder is a very serious crime which warrens an equivalently serious punishment. Practically every civilized society has laws pertaining to murder.
3) Immoral relations are also very serious crime in Judaism to the point that any product of such relations is forever branded a mamzer. These relations are also widely outlawed in most civilized societies.
Indeed all of the above three appear as prohibitions in the ten commandments as well as the 7 Noachide laws, outlawing them for all of humanity.

In light of this, we can see why the exile be brought-upon by transgressing these three prohibitions, as they are very serious indeed.

So, the first question here is: what does transgressing shmita have to do with exile and why does it appear at the same place and said with one breath along with idol-worship, murder and immoral relations? The question becomes even stronger when we consider that the first three are prohibited to all of humanity, whereas shmita only applies to the Jewish people. Not only that, but not even all Jews, just those living in the Land of Israel. And amongst those, even, it only applies to Jews who work the land.

The second question is: if the exile is a product, by divine decree, for transgressing the laws of shmita, then while the Jewish people are being punished in exile, they certainly cannot keep the laws of shmita (which only apply in the Land of Israel, as previously stated.) In other words, how does the punishment fit the crime, since by the very nature of the punishment (=exile) the laws of shmita cannot be kept? Indeed, during the time of the Babylonian exile, which lasted 70 years, the Jewish people have "missed" 10 potential shmitas that they could have kept, had they not been exiled for not keeping shmita. (See Rashi on Vayikra (Leviticus) Ch. 25:18)

A possible answer:
Rashi begins his commentary on the portion Behar with the following idea:
Why are the laws of shmita listed in great detail at the beginning of Behar? To teach us that just as the laws of shmita were given in detail and at great length, right there at mount Sinai, so too were all other Torah laws given in great detail on the same occasion.

On this very Rashi, a child might pose the question: Why specifically all the laws of shmita given? The Torah could have chosen any other law and listed its details at great length -- and Rashi could have had the very same commentary with a slight change of wording to account for the specific law chosen. We're forced to conclude, therefore, that there's a special connection between the laws of shmita and the revelation at mount Sinai during the giving of the Torah. In other words, somehow the laws of shmita, specifically, are representative of all other Torah laws, more so than any other law. But How?

To answer, we need to delve into some Jewish philosophy:
According to Jewish philosophy there is one level of emunah (=trust and belief in the Almighty) that a Jewish farmer may posses where he/she believes that the Almighty created nature and instilled in it everlasting laws. And because we trust the Almighty, we also trust that his laws are eternal. The farmer's emunah then is that if he/she plows, plants, waters and takes care of the field, and all the other conditions are just right (e.g., the temperature, humidity and wind are just right) then according to the laws of nature, which the Almighty himself created, there will be a good yield of crop.

Above that there's a higher level of emunah: the Jewish farmer believes that the Almighty is actively involved in the laws of nature. In other words, this emunah is at that the Almighty is pro-active in the development of the crop, not merely by orchestrating nature as a whole, via some laws, but rather in detail.

Both these levels of emunah are confined to nature, since they are solely within the framework of nature: either at the macro level (the first level of emunah) or at a micro level (the second level of emunah).

However, there's a higher level of emunah yet, as we can see in Vayikra Ch. 25:3-4 which read:
שש שנים תזרע שדך ושש שנים תזמר כרמך ואספת את תבואתה
ובשנה השביעת שבת שבתון יהיה לארץ שבת להשם שדך לא תזרע וכרמך לא תזמר

Loosely translated:
For six years you shall seed your wine and you shall harvest it, and the seventh year shall be a sabbatical to the land (i.e., shmita), a sabbatical to G-d, your field you shall not seed and your vine you shall not prune.

At this is the level of emunah that the farmer believes that really everything is in the hands of the Almighty. On the seventh year the Jewish farmer is not required to plow, plant, water or otherwise take care of the field. As a matter of fact, he/she is prohibited to do so. Yet the land will continue to produce. This teaches us that really, it's the Almighty in control -- not us, for if it was up to us, inaction of the seventh year would mean no food. See the article about Superrational Trust for further clarification on this point.

Whereas the first two levels of emunah are limited by the framework of nature, the third level is not. At the third and higher level of emunah the Jew (farmer or otherwise) believes that although the Almighty acts within nature (making the crop give good yield, for instance) however the flux of abundance comes from a level higher than nature. In other words, Hashem acts within nature but from beyond nature.

What better mitzvah (=commandment), from the set of 613, is there to demonstrate that the flux of abundance originates beyond nature? What better mitzvah is there to demonstrate that the Almighty's ways are incomparable to nature?

Even the mitzvah of the Red Heifer, which is the most irrational mitzvah (as testified by Shlomo HaMelech, King Salomon, the wisest of all people) lacks in this respect. Whereas the Red Heifer defies logic to demonstrate the Hashem is beyond logic and understanding, more so than any other mitzvah, by purifying the tainted while tainting the pure in the process, there is no visible difference to the human eye. An impure person looks no different than a perfectly pure one. We know that there's a difference, but we cannot perceive it. This is precisely where the laws of shmita excel: there is nothing more tangible to a person than food on the table. There's nothing that announces that "The Almighty manipulates nature from beyond the laws of nature" than when the Jewish farmer abstains from working the land on shmita years, and yet there's food in the field, on the table and in the tummy. The farmer made no effort on his-/herown behalf, and yet all his essential needs were taken care of by Hashem.

This is then the connection between Behar and shmita: Rashi's comment can now be understood to also mean that just as the laws of shmita were given in their entirety (that is, in a way that would announce loud and clear that Hashem works within nature but from beyond nature, i.e, just as the laws of shmita affect this lowly world, but are rooted in higher realms) so too all the other mitzvahs. This includes those that we can logically explain -- although there's a logical explanation, you should know that at their essence these mitzvahs are rooted in higher reals, they're all supernal and super-rational. To come to this realization, we specifically need the most tangible instance of such a mitzvah -- shmita laws.

We can now undertand better Pirkei Avot: The first question was asked why shmita was mentioned in the same breath as the prohibition of idol worship, murder and improper relation, in lieu of these three being fundamental laws. The explanation is, then, that shmita is most fundamental from a philosophical point of view as it reflect the supernal source of all Torah laws.

The second question was what's the connection between shmita and the exile, especially in lieu of the fact that shmita laws can certainly not be kept while in exile. The explanation to that is already given at length in the Superrational Trust article. See the part about Hashem defying logic due to the promise/blessing that on the sixth year the land gives the greatest yield.

Good shabbos with lots of emunah and yield both physically and spiritually.