The proof of the correctness of the Birch and Swinnerton-Diyer conjecture
UUDC 511.3 The proof of the correctness of the Birch and Swinnerton–Diyer conjecture. S.V.Matnyak Khmelnitsky, Ukraine
Abstract . The proof of the conjecture of the Birch and Swinnerton-Dyer is presented in the paper . The Riemann's hypothesis on the distribution of non-trivial zeroes function of Riemann, previously proven, is word to prove this hypothesis. The theorem proved about the behavior of the L -function curve E for s . It is shown that the L -function of the curve E tends to zero for any prime unpaired integers. It is shown that the function can be expanded in a power series of the holomorphic field. The theorem is proved on conformity of the basis of the Galois group and the number of zero coefficients of the power series. The result proved the conjecture of Birch and Swinnerton-Dyer. Introduction . In the beginning of 1960s, Birch and Swinnerton-Dyer set that dimension r of the group of an elliptic curve E over Q is equal to the order of zero of Hasse – Weil zeta function sEL , in the point s . A more detailed conjecture states that there is no zero border set rsE s sELB , where value E B depends on refined arithmetic invariants of curves. The most important partial result for 2011 is be statement in 1977 by John Coates and Andrew Wiles state that is true for a large class of elliptic curves, which cays that if the curve F has infinite amount of rational points, than sEL . The conjecture is the only relatively simple overall method of calculating the rank of elliptic curves. 1. The problem (conjecture).
Let us consider that E is a certain elliptic curve, stated off Q . Then the group rank E , E r is equal to zero order of L - function sEL , , in point s . Solution.
Let us assume that E is the elliptic curve, stated off Q with equation xBxxAxxx , QBA , . (1) We obtain affine equation, when we set xxx and xxy : BxAxy . (2) The transformation ycxcyx ,, turns this equation into BcAxcxy . (3) o, from the beginning we can assume that
ZBA , . The Number BA is denominated as discriminant of curve E . As we can see . Let the Zp be some prime number, then we consider the equation pBAxxy , That is the equivalent to the equation BxAxy , p FpZZBA /, . (4) This equation determines the elliptical curve p E off p F , only when p . Later will be considered only such primes, when the approved is not determined. The curve p E is called the reduction of curve E on the module p . Let the m p N denotes the amount of points in m pp FE . Than we can analyze zeta function muNuEZ mm pmp exp, . (5) Using the Riemann-Roch theorem, we can write puu puuauEZ mpp , Za p . (6) We write down for pa p uupuua p , (7) Where is complexly conjugated with . It is clear that p , p a . Besides, p . This is Riemann’s conjecture for elliptical curve off p F . Logarithmically differentiating (5) and (6), and taking into account (7) and comparing the coefficients, we will have mmmpm pN . (8) Especially, pp apN In such way calculating p N , we determine p a . Considering and are roots of the equation pTaT p , than the equation (8) determines pm N for all m We change variable u for s p and will get ss sspp pp ppasE , (9) e have determined sE p , for prime numbers p . When / p , we consider ssp ppsE
11 1, . Now, having introduced in the local zeta function for all primes р , we determine the global zeta function simply as the product of local zeta functions sEПsE pз ,, . (10) From the definition we can see that, sE sssEL ,1, . (11) We write down the function (11) as sspр ssр ppaП ppПksE sssEL
21 11 , (12) where k is the value of function The function coincides with the theorem 2 [3, p.2] The theorem 1.The function sEL , at s will be equal to zero sEL with all values of p . Proof.
We determine n n nn nnnc cnnnMk c cnnc cnn nc c kk , where .3,1;2,1;1,1 c The value of the function k is situated in the range cc . Then we write down that if s sspр ssр ppaП ppПsE sssEL
21 1 , (13) And if s and when cck we will have papp pppПsEL pз ; Because from lemma 1 [9, p. 33]: ‘ When s , N Nn N sss duu usNsNns " we will deduce ; when we will deduce pap pПppapp ppПsEL pрpр · Because when p , and
11 1 p ap p and if p
11 1lim pp ap p The theorem is proved. 2.
Order of a zero. When the function sEL . , which is identically not equal to zero, is holomorphic in domain D , and equal to zero in point a of this domain, then its decomposition for somewhat domain of the point is the following ...1...11, nn scscscsEL , (14) by virtue of the fact that ELc . Obviously, all coefficients n c of decomposition (14) cannot be equal to zero, by virtue of the fact that the function sEL , is equal to zero everywhere in some neighborhood of the point a , by the theorem of unique solution will be identical zero in domain D . So, among coefficients n c ,...3,2,1 n we have different from zero; let us denote by n , if n -the lowest number of this coefficients. Then we will get: n ccc , n c . Now the decomposition (14) will look as follows: ...11, nnnn scscsEL , (15) where n c . In this case point a will be zero in order m for sEL , function in point s .
3. Constructing the Galois group of an elliptic curve.
Let the elliptic curve E be defined over the field K and let the L be the expansion of field K . It denotes following, is the somorphism of the field L , not certainly identical to K [4, p. 27]. It defines the curve E , obtained using to equation coefficients, which set the curve E . For example, when the curve E is set by equation BAxxy , Than E is defined by equation BxAxy . When P , Q are points on the curve E in domain L , we can have the formula QPQP . The Total in by left side is relative to the addition to E , and the total in right side is related to addition to E . Obviously the equality follows from the fact that the algebraic addition formula is given by rational functions of the coordinates with coefficients from the field K . Also, when yxP , , than yxP , is obtained using to coordinates. Especially, let us assume that P is a point of finite order, because NP . Since the point O is rational on K , then any isomorphism of the field L over K we get NP and, so, P also is a point of order N . Then as the number of points of order N is finite, it follows that they are algebraic over K (that is their coordinates are algebraic over K ). When yxP , , we set yxKPK , field expansion K , obtained by addition of coordinates of point P , Similarly, N Ek the denote the composite the fields PK for all n EP . We emphasize that we consider all points of finite order, as points with coordinates with fixed algebraic closure of the field K , which we denote as K a or a K . The above remark shows that the Galois group Gal KK / acts as a group of elements of the set N E . So, N EK is a normal expansion of the field and K is the Galois expansion, if N is not divisible by the characteristics of the field K . We set N EK as the field of points in the order N of the curve A over the field K . Besides when is automorphism of the field N EK over K and if t , , tt ,..., r ttt ,...,, ‒bases N E over nZZ / , than can be set as matrixes a , aa aa , ..., rrr r aa aa ,1, ,11,1 ...... such as t , tttata tatatt and etc. So we have obtained injective homomorphism: QnGLKEKGal N ,/ . Theorem 2. (Mordell) [1,p. 367]. Let E be some elliptical curve, defined over Q . Then QE is be finitely generated abelian group. Theorem 3. [ ] . Let E be some elliptical curve, defined over Q . Then QE is isomorphic to one of following groups mZZ / if m or m , mZZZZ if m .
4. Theorem 4 (consistency between the group and the rank and the order of zero).
Let KL / is the finite Galois expansion of degree n and n ,...,, ‒ are elements of the set G , where s , s ,..., nn s . Then there is element L , such as, n ,...,, create basis L over K , then the elements of the Galois group use up( n ) first coefficients of the series (14) into zeroes of series (15). So the rank of the Galois group will equal to the number of zeroes of order. sEL ,. . Proof.
For any G let the X be the variable and , Xt . We set i XX i . Where i sX i , and st . Let the jin txxxf ,det,...,, . Then f is not identical to zero, that is evident, in the theorem 19 [ ] the determinant can not be equal to zero with all Lx , when we in f substitute x i instead of i X . That’s why there exists the element L , which is ji . We set coefficients of a power series (14) by ,...,, n ccc . And let us assume that the elements (coefficients of power series) Kccc n ,...,, such as nn ccc . We use i according to the expression for each ni . Since Kc ji , , we get a system of simple linear equations for unknown quantity j c and receive that, j c for ni . And so, will be the sought for element, in this case for s . According to the Corollary of Lemma 2.3 [10 st.144]:"Let the L be the finite expansion of the field K with Abelian Galois group G of power which divides n . Then the group G is a direct product of cyclic subgroups r GGG ...,, ,21 . Suppose that for each i over i L ill be set the subfield, fixed for the subgroups r GGG ... ; then ii GKLG / , ii KL , where Ka ini and n KL ,..., ". And Lemmas 2.4[10, p.144]. When L is the normal algebraic expansion of K with the Galois group G , then LGH
Then using Theorem 1 and Lemma 2.4, we can write that in the normal expansion of the Galois group, change series (14) into series ...111, nnnn scscEL . (16) Using the formula ( ) we will have, if E rn nr nnnnsrsEn cs scscs ELC EE . According to Lemma 1 and the result [9, p.33] the function sEL , is analytical in whole domain ,0 s , it can be expanded into a Taylor series in powers of s and with coefficients !! 1, ,, kBkELC EnnEn , where sEL n , –is derivative by the order n from power of Hasse-Weil function sEL , . Thus, the rank order of the Galois group is equal to zero Hasse-Weil function. Theorem is proved.
So the Birch and Swinnerton-Dyer conjecture is fair.
Key words : the hypothesis of Birch and Swinnerton-Dyer, function of Hasse-Weil , Riemann's hypothesis, the Galois group, the complex power series. 05 September 2013 Rewriten 14 May 2014 References [1] K.Ayerland , M. Rouzen . Introduction to the modern classic theory of numbers. Moscow: Mir , 1987 - 415s . [2] S.Lenh . Algebra. Moscow: Mir, 1968 - 564s . [3] S.V.Matnyak . Proof of the Riemann's hypothesis .// arXiv:1404.5872 [math.GM],22 Apr 2014. [4] S.Lenh . Elliptic functions. Moscow: Nauka, 1987 - 311s . [5] N. Koblyts . Introduction to elliptic curves and modular forms. Moscow: Mir, 1988 , 318p. 6] I.I. Privalov . Introduction to the theory of functions of complex variable . Moscow: Nauka , 1984 - 432p. [7] N. Koblyts . p- adic numbers , p- adic analysis and zeta functions. Moscow: Mir , 1982, 192p. [8] B.L. Van der Waerden . Algebra. Moscow: Moscow 1976 - 648s . [9] A.A Karatsuba . Bases of the analytical theory of numbers. Moscow: URSS , 2004 - 182s . [10]: the hypothesis of Birch and Swinnerton-Dyer, function of Hasse-Weil , Riemann's hypothesis, the Galois group, the complex power series. 05 September 2013 Rewriten 14 May 2014 References [1] K.Ayerland , M. Rouzen . Introduction to the modern classic theory of numbers. Moscow: Mir , 1987 - 415s . [2] S.Lenh . Algebra. Moscow: Mir, 1968 - 564s . [3] S.V.Matnyak . Proof of the Riemann's hypothesis .// arXiv:1404.5872 [math.GM],22 Apr 2014. [4] S.Lenh . Elliptic functions. Moscow: Nauka, 1987 - 311s . [5] N. Koblyts . Introduction to elliptic curves and modular forms. Moscow: Mir, 1988 , 318p. 6] I.I. Privalov . Introduction to the theory of functions of complex variable . Moscow: Nauka , 1984 - 432p. [7] N. Koblyts . p- adic numbers , p- adic analysis and zeta functions. Moscow: Mir , 1982, 192p. [8] B.L. Van der Waerden . Algebra. Moscow: Moscow 1976 - 648s . [9] A.A Karatsuba . Bases of the analytical theory of numbers. Moscow: URSS , 2004 - 182s . [10]