Birch's hypothesis - Swinnerton-Dyer

This remarkable hypothesis connects the behavior of the function L where it is currently unknown whether it is defined and the order of the group W , about which it is not known whether it is finite!
JTTate, The arithmetic of elliptic curves, Inventiones mathematicae 23 (1974)

(A brief note on the relevance of a 40-year-old quotation: after Wiles and Co., it became known that the function L can be defined on the entire complex plane. The finiteness of group III in the general case remains unknown.)

It remains to discuss the possibility of error. As a precaution against internal computer errors, you can run all calculations twice or make checks inside the program. Moreover, computers, unlike people, are designed in such a way that their mistakes are usually too large to be overlooked. We are sure that there are no such errors in our results. On the other hand, when coding an intricate calculation scheme into a computer program, programmer errors are inevitable. Most of them are detected before the main launches, due to the fact that the program hangs or gives ridiculous results. But the program, which is considered to be working, can still contain logical errors, manifested in rare coincidences of circumstances: indeed, most computers are subject to anomalies, because of which they sometimes behave differently than they should according to specifications. In essence, our program for stage (ii) turned out to be inaccurate and missed a very small number of equivalences that it had to find.

For these reasons, we believe that you should not automatically trust the results obtained on the computer. In some cases, they can be verified through properties that were not essentially involved in the calculations and which would hardly have survived a possible error. (For example, a table of values ​​of a smooth function obtained without using interpolation can be checked by calculating differences in neighboring values.) But if such checks are not available, you should not completely trust the results until they have been independently confirmed by another programmer on another computer. We do not think that this sets an excessive standard at a time when computers are becoming so widely available; and we are sure that low standards have already led to publication and belief in incorrect results.

BJBirch and HPFSwinnerton-Dyer, Notes on elliptic curves. I, Journal für die reine und angewandte Mathematik 212 (1963)

Under the cut there will be no formulation of the hypothesis; knowledgeable expressions like “Euler product” and “holomorphic continuation” (both in the sense of language and in the sense of denoted concepts) can read the five-page pdf from the Clay Institute website. Under the cut - some attempt to clarify in which direction of development of mathematical thought is the Birch – Swinnerton-Dyer conjecture in general. And also - how can you count up to large numbers like those shown on CDRV in less than a second.

It will be about finding rational solutions of equations with two variables.

Linear and quadratic equations

The simplest case is linear equations: a x + b y + c = 0 (where a , b , c are rational). Here the solution is simple: if we exclude the degenerate cases with a = b = 0, one of the variables can take any rational value, and the other is uniquely calculated from the first.

The next case is quadratic equations. Here there are more different cases, but if we cross out those where everything is clear ( y - x ² = 0, y ²- x ² = 0), the remaining linear change of variables reduces to a x ² + b y ² + c = 0, where a , b , c are nonzero rational. Take three characteristic examples:

Obviously, the first example has no rational solutions, because the left-hand side is always at least one and cannot be zero. It can be said differently: the first example does not even have real solutions, and rational numbers are a subset of real ones, so there are no rational solutions.

It is less obvious that the second example has no rational solutions. We give x and y to the lowest common denominator, let x = k / n , y = m / n , where k , m , n are integers and mutually simple in the aggregate (that is, there is no number greater than one that would divide all three at the same time) . Then the equation is converted to .
Consider it modulo 3. The square of an integer modulo 3 can be either 0 or 1; the sum of two squares modulo 3 can be zero only if both numbers are divisible by 3. Hence, k and m must be divisible by 3.
Take now the module 9 = 3². The left side is divisible by 9, hence 3 n ² should be divisible by 9 and n should be divisible by 3. Hence, there are no solutions where k , m , n are integers and are mutually simple, and the original equation does not have rational solutions like and it was promised.


The third equation has several solutions to indicate easily: for example, x = 1, y = 0. When one solution of a quadratic equation is known, it is not difficult to find everything by a method that goes back to Diofant: a straight line passing through a point (1,0) intersects the circle at exactly one point, and this point will be rational if and only if direct coefficient is rational. Specifically for a circle, if the angular coefficient is denoted by - t , the straight line has the form y = (1 - x ) t , and the second point of intersection with the circle x ² + y ² = 1 has coordinates
- this is the general solution of the third example (with the exception of the point (1,0) itself, which in a sense corresponds to ).

Hasse principle

The idea of ​​presenting equations with integer or rational coefficients modulo any prime number, as well as powers of the same number, is extremely fruitful. The special construction “let's say that we are only interested in what happens modulo p , p ² and all other powers of p at once, and ignore other arithmetic properties” is called “ p-adic numbers ” (there could be a link to Wikipedia, but there sometimes come across scary words); The p-adic numbers, like the real ones, contain rational numbers and add many others (for example, the square root of 2 is extracted modulo 7 and all the powers of the seven, so that it is among the 7-adic numbers; on the other hand, not extracted neither modulo 4 nor modulo 3, so it is not among the 2- and 3-adic numbers). The reasoning about the second example is literally carried over to 3-adic numbers, and one can say: the second example does not even have 3-adic solutions, and rational numbers are a subset of 3-adic, so there are no rational solutions.
Working with one prime number is often easy and pleasant. In the end, all options modulo a prime number can be sorted. On the other hand, working with more than one prime number tends to be much more difficult - the problem of Goldbach and the problem of simple twins is vivid evidence of that.

The transition to real and p-adic numbers well helps to prove that there are no solutions. And vice versa, the Hasse principle says that if there are real and p-adic solutions for all p, then there are necessarily rational solutions. (Of course, there are infinitely many primes, and testing all can be delayed. But you can prove that if p does not divide either numerators or denominators a , b , c and more than 2, then there are always p-adic solutions, and for abc divisors you can effectively verify the existence of solutions using the quadratic reciprocity law.)

Unfortunately, the Hasse principle does not hold for equations of a higher degree. For example, one can prove that the equation 3 x ³ +4 y ³ + 5 = 0 has real and p-adic solutions for all p, but has no rational solutions.

Elliptic curves

The third degree equation, if, again, crossing out different degenerate cases like y ( x ²-1) = x ³ -1 and y ² = x ³ + x ², sets a curve of genus 1 (this is not a definition, there are other curves of the genus one). There may be no points on the curve (solutions of the equation). If there are points and if you select one of them, you get an elliptic curve ; in the case of an elliptic curve, you can always change the variables to send the selected point to infinity and get the equation of the standard form y ² = x ³ + a x + b , a , b are rational (and even integers, you can always get rid of the denominators by another change of variables), we will do it: only elliptic curves (over rational numbers) will continue.

Informally speaking, the extent to which the Hasse principle is violated for an individual elliptic curve and related entities is characterized by the Tate-Shafarevich group of the curve, with the Tate filing traditionally denoted by Cyrillic III, even in articles where most of the letters are English. It is assumed that it is finite. The Birch – Swinnerton-Dyer conjecture involves the order of this group.

Unlike quadratic equations, after finding a single point (and sending it to infinity) everything is just beginning. It is well known that points on an elliptic curve can be added. If you take one point and start folding it with itself ( P , P + P = 2 P , P + P + P = 3 P , ...), then two options are possible: or after some number of steps you will get infinitely distant a point (after which the next step will give P again and the process will loop), or all the obtained points will be different (and then it makes sense to take more - P , - 2P and so on). In the first case, a point is called a torsion point ; for a single curve, there can be from 1 to 12, except for 11 (the always existing infinitely distant point is also a torsion point). The Mordell – Weil theorem says that one can always find a finite number (possibly, 0) points of the second type P ₁ , ..., P _r such that any point of the curve is uniquely written as n ₁ P ₁ + ... + n _r P _r + Q , where Q is some kind of torsion point, and n _i are integers. The number r is called the rank of the curve. For example, the curve y ² = x ³ +877 x , drawn on the CCDT, has rank 1 and two torsion points; Any (rational) point of the curve is either n P or n P + (0,0), where the coordinates of P are signed in the picture.

All the torsion points are relatively easy to find. For example, in the case of integer coefficients a and b, all the torsion points (excluding the infinitely remote) themselves have integer coordinates, and the y coordinate is either zero or y ² divides 4 a ³ +27 b ². Calculating the rank and finding the generating points P _{i is} much more difficult.

Time to calculate something

You can search for points on a curve simply by going through the numerator and denominator of the x- coordinates in ascending order and checking whether a rational y is obtained. It is easy to figure out that in the case of integer coefficients of a curve, the denominator x must be an exact square, and the denominator y must be a cube of the same number (the denominators on KDPV are 78841535860683900210 squared and cubed), which makes life a little easier. However, the curve on the KDPV is specially chosen so that a look at it stops such thoughts in the bud ( P is the point with the lowest denominator, not counting (0,0)).

In principle, there is a general procedure for n- descent ( n is a natural number not less than 2), which allows you to calculate r and find r independent points if the Tate-Shafarevich group does not have elements of order n , and get the upper and lower estimates on r and find some independent points in the general case. (In the latter case, you can repeat the descent, choosing the growing degree of some prime number as n ; if the Tate-Shafarevich group is finite, then the process will converge in a finite time.) But in practice it is extremely inconvenient to perform. Birch and Swinnerton-Dyer in their first of two articles proposed a method for 2-descent, not going beyond real arithmetic. Those who want an exact description can look at the source code of mwrank , and specifically in mwrank1.cc , here will be some results for y ² = x ³ +877 x .

At the first stage, the method looks for quartic curves , - of which there is a mapping to the original curve, by iterating a , b , c in certain ranges, calculating d and e , checking that d and e are obtained as integers, and discarding those quartics on which there are no real points or p-adic points for at least some p .
After the first stage, some quartics may turn out to be equivalent (go into each other by a linear fractional change of X ). At the second stage, the method leaves one quartic from each equivalence class. After that, 2 ^{m + k} -1 quartic remains, where the factor 2 ^k is 1.2 or 4, if there are 0.1.3 points of order 2 on the original curve (which are characterized by y = 0), and m is the upper bound on rank. On the curve y ² = x ³ +877 x there is one point of order 2, and the rank is equal to one, so three quartics are obtained.

First:





(The Habr format does not have to write formulas for all coefficients, but nevertheless they are: all coefficients in the coordinate transformation are calculated using a , b , c , d , e , for example, 12321 is obtained as d ²-8 e c / 3. )



The second:






Third:





(On a quartic, there can be 0 or 2 infinitely distant points depending on whether a is an exact square; in the first two quarts there were none, here are two of them, both go to (0,0). Replacing reduces them to normal points.)

At the third stage, the method searches for a rational point on the quarticles left after the second stage. The numerator and the denominator x -coordinates on the original curve are in order approximately equal to the fourth degree of the numerator and denominator X -coordinates on the quartic (multiplication by n on the elliptic curve is an operation of degree n ², for 2-descent we get degree 4). If the point could be found, then it is possible to calculate a point on the original curve. If the point could not be found, then a problem arises: this may mean either that there really isn’t (and there are non-trivial elements of the Tate-Shafarevich group) or that they were looking badly. The theoretical solution is to simultaneously launch a higher-order descent and search for points with large numerators and denominators. A good practical solution is unknown.

For the case of a curve y ² = x ³ +877 x, the coordinates on the quarticles look less impressive than the coordinates of the points of the original curve, but are still too large for brute force. However, with a quart You can go further. (It is difficult, but not necessary, to continue working with the first quartic: one generating point is enough.) The right-hand side is biquadratic, that is, expressed in X ²; it always happens when there is a second-order point on the original curve (with y = 0). If we take the pair ( X ², Y ) as variables, we get a quadratic equation, which can already be solved:

Further, any common divisor of the numerator and denominator of the expression for X² should also divide ; their quotient can be a square only if they both have the form " d multiplied by a square", where d is a divisor of 3508, free from squares, that is, one of the numbers 1,2,877,1754 (negative d are excluded, because the denominator is always positive). The numerator cannot be an exact square in 2-adic (and even more so in rational) numbers. Try d = 2: . Substituting the obtained expression for u into the numerator and demanding that the numerator be doubled square, we get a new quartic .
It already has a point available for iteration: .

Brute force quarting can still be reduced by using the same idea of ​​considering an equation modulo prime numbers and powers of primes. When searching for k , m such that - an exact square, for each prime number p approximately half of the p ² possible pairs of residues ( k , m ) do not square modulo p . (Instead of 2 and 3, it is better to look at their degrees 16 and 9; for large p, increasing the degree does not give anything.) This means that “good” pairs of residues modulo, for example, 16 * 9 * 5 * 7, approximately 1/16 of the total the number of possible pairs; it suffices to search only pairs ( k , m ) with “good” residues. The aforementioned mwrank, using similar considerations, finds the point P with CDRP in less than a second.

Birch's hypothesis - Swinnerton-Dyer

In the first article, “Notes on elliptic curves. I »Birch and Swinnerton-Dyer collected statistics on the ranks of a pair of thousands of elliptic curves. In the second article, “Notes on elliptic curves. II ”the time has come to analyze the accumulated statistics, and the title hypothesis was also proposed there.

Working with an elliptic curve over rational numbers is problematic: the coordinates of points grow rapidly, the procedure for calculating the rank does not always give an answer. On the other hand, having an equation with integer coefficients, we can consider it modulo different prime numbers. The number of points modulo small primes is easy to calculate directly, but it’s not at all clear what relation the numbers obtained have to rational points.

If for each prime number we count the number of points on a curve modulo this prime number, we get some infinite set of integers. The Hasse – Weil zeta function of an elliptic curve, traditionally denoted by the letter L , is obtained if we “glue” this set in a certain way into a single function of a complex variable. Since it contains information on the modulo behavior of all primes at once, it is difficult to prove anything about it. (In this, it is similar to the Riemann zeta function - and the Riemann hypothesis regarding the properties of the Riemann zeta function is another one of the “millennium problems”.) The Birch – Swinnerton-Dyer hypothesis goes further and connects the behavior of a curve over rational numbers (specifically, rank) with the properties of the zeta function, which is calculated exclusively from the behavior of the curve by finite modules.