By Liona Fan-Chiang
It was seen earlier that any complex number can be either a residue of a modulus, or itself be the defining measure against which all other numbers are measured. What if we were now to ask what the relation is between a particular number's role as the one, and its role as the many, were those roles to be played simultaneously? This question leads us to Gauss's main focus in treating moduli throughout his life: the question of reciprocity.
In the image shown above, -1+2i is congruent to zero in its own modulus, as well as with respect to its associates, but is congruent to -i in the mirrored modulus.
What is the relationship between the two numbers above, which are both represented as -1+2i, ?
Let's begin with a simple investigation: an inquiry into left and right handedness (chirality).
Notice that -2+i is a biquadratic residue when considered in the context of the modulus defined by its chiral partner -2-i.
But, reciprocally, when redefining all the complex numbers in accordance with the measure -2+i, we find -2-i, as a residue of the modulus defined by -2+i, is also a biquadratic residue!
Moreover, in a similar way, the other 3 pairs of associates are not left out of this dance; they too participate in mutual reciprocity.
Does this reciprocity simply result from the overlay of two moduli which correspond to the same norm, 5? What would happen if modulus 5 and modulus 13 were tiled?
modulus 5 (purple) and modulus 13 (green)
Does reciprocity still hold?
Remember that 5 and 13 were both distinguished as asymmetric moduli whose norms could be expressed as p=8n+5, which, as a category, had some very unique characteristics. One of which is that every associate set contains a biquadratic residue, a quadratic residue, and two non-residues, in the same configuration. Therefore, in consideration of two moduli, both of the form p=8n+5, the two associate sets will always have four corresponding pairs of reciprocal residues!
Here are few other examples of this case:
modulus 13 (teal) and the right and left hand of modulus 29 (maroon)
modulus 5 with modulus 37 (left) and modulus 29 (right)
As you see in these cases, since each set of associates contains a residue belonging to each class, A, B, C, D, there will always be a direct correspondence of the two sets. However, if simply given the two sets of associates, how do we know which of the four associates will be the biquadratic residue? For example, in the tiling of 5 and 13, the biquadratic pairs were -2+i and 2-3i.
The question becomes more dire when we go on to consider tilings of two different types of moduli, for example, modulus 5 and modulus 17, one of which is a p=8n+5 and the other, a p=8n+1 modulus.
modulus 5 and modulus 17
Now, the four associates which correspond to modulus 17 are asymmetric within the modulus of 5, again consisting of a biquadratic, a quadratic, and two quadratic non-residues. However, the associates of -1-2i are all quadratic residues with respect to modulus 17! Thus, only one pair between the two set will be able to participate in reciprocity, but which one?
At this point, Gauss introduces the concept of the primary associate.
Since the connection between every four associated complex numbers is analogous to the connection between two opposite real numbers (i.e. between two numbers which are considered as absolutely equal, but with opposite signs) Of these, the positive number generally tends to be correctly considered as a sort of primary number, and so the question arises, whether a similar distinction can be established between every set of four associated complex numbers and must be considered advantageous....we will consider as the primary number among four associated uneven complex numbers, that which will be congruent to unity with respect to modulus 2 + 2i. -Second Treatise on Biquadratic Residues, §36
Here are the associates of 3+2i mapped onto the modulus of 2+2i.
Notice that only one associate satisfies the condition of being º 1 mod 2+2i, while each of the other associates are congruent to i, -1, and -i. Try this out for another set of associates. Must this always be the case?
From this standpoint, look again at the tiling of modulus 5 and 13, this time, the two mapped onto modulus 2+2i.
Shown here are only the two sets of associates of 5 and 13, and their respective classes relative to each other's moduli.
In this case, the primaries are both quadratic non-residues. What of the biquadratic residues? -2+i º -i mod 2+2i, while 2-3i.....is also º -i mod 2+2i. In fact, reciprocity seems to hold, not only for the primaries, but with any two which are congruent to the same residue with respect to modulus 2+2i!
Consider again the case of an asymmetric modulus in the context of a symmetric modulus. Does this restriction resolve the ambiguity encountered when 5 and 17 considered each other?
Modulus 5 and modulus 17 with respect to modulus 2+2i.
There is still one more case of tiling which has not been considered: the case of p=8n+1 in relation to 8n+1.
Here are some examples:
Modulus 17 with modulus 41
Modulus 17 with modulus 73
Modulus 41 with modulus 89
Again, does reciprocity hold?
Gauss, what are you implying?
How does reciprocity help Gauss solve the inverse question? Take a question like "which moduli are -1+2i a biquadratic residue of?" We now know that any primary number which is congruent to a biquadratic residue of the modulus -1+2i, will serve as a modulus for which -1+2i is reciprocally a biquadratic residue! For -1+2i, since it only has one biquadratic residue, the solutions are reduced to those primary numbers, which are º 1 mod -1+2i.
Shown in this diagram, are the primary associates of all prime numbers of the form 4n+1, up to 100. The grid displayed is modulus 2+2i. The primary associate, as shown above, is always the associate which is º 1 mod 2+2i. Remember, every modulus has a right and left hand representation, therefore, there are two sets of associates related to the complex modulus which represent each modulus, and thus two primary associates for every prime number modulus of the form p=4n+1.
Shown here are all the primary associates for all 4n+1 primes up to 100, viewed from the standpoint of -1+2i. Which primaries are quadratic or biquadratic residues of modulus -1+2i? According to the law of biquadratic reciprocity, there are two cases of of reciprocity:
1) If the prime is of the form 8n+1, then the reciprocity is direct. That is, if a primary is a biquadratic residue of modulus -1+2i (i.e. modulus 5), then -1+2i is a biquadratic residue of that prime.Thus, from the diagram above, the search for which moduli -1+2i is a biquadratic residue of, is soon ended.
Therefore, the beautiful theory of biquadratic reciprocity reduces the inverse problem to a direct problem, as if to say, the reply to the question, "How does the universe shape the individual?" is "How does the individual shape the universe?"
File translated from TEX by TTH, version 3.80.
On 06 Mar 2008, 21:28.