(Arithmetic Invariant Theory) 1 (Geometric Invariant Theory, GIT) GIT

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Zbigniew Skiba
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1 ( 1 2) Gauss ( 3) 1 (Arithmetic Invariant Theory) 1 (Geometric Invariant Theory, GIT) GIT 2 2 V = Sym 2 C 2 := { f(x, y) = ax 2 + bxy + cy 2 a, b, c C } G = GL 2 (C) ( ) g = ( p r ) q G, s f(x, y) = ax 2 + bxy + cy 2 (g f)(x, y) = 1 f(px + ry, qx + sy) det(g) Disc(f) := b 2 4ac G a, b, c ( ) Disc(f) 2 G V G- Disc(f) 0 ( ) : f(x, y) = x 2 + y 2. Disc(f) 0 0 : f(x, y) = x Disc(f) ( ) C R ( ) 2 G GL 2 (R) x 2 + y 2 x 2 y 2 yasu-ishi@math.kyoto-u.ac.jp 1 2 G- C- Disc(f) 1

2 2 Q, Z Gauss, Dedekind Gauss 200 Z ( ) Dedekind ? 5? 2 Gauss Dedekind Gauss, Dedekind 2 2 Q 2 2 Q( 5) = Q + Q 5 2 Q( 5) Q Z Z[ 5] = Z + Z 5 ( ) 6 = 2 3 = ( ) ( 1 5 ) ( ), ( 1 5 ) 2 Z[ 5] (6) = (2) (3) = ( ) (1 5 ) = ( 2, ) ( 2, 1 5 ) ( 3, ) ( 3, 1 5 ) ( ( 2, ) Z[ 5] ) Q( 5) Z[ 5]- M Z[ 5] 0( ) I x K M = xi ( : ) ( : 1, J Z[ ) 5] ) J P ( Z[ 5] Cl(Z[ 5]) J/P Cl(Z[ 5]) ( 2, ) Cl(Z[ 5]) Z/2Z 2 Z[ 5] M Z- Z 2 M {α, β} : Q M (x, y) := N Q( 5)/Q (αx βy) N(M). N Q( 5)/Q : Q( 5) Q N: J Z αx βy Q( 5) Q- M Z[ 5] Q( 5) L N(M) := [L : M] [L : Z[ 5]]

3 3 Z[ 5] {1, 5} Z- Q Z[ 5] (x, y) = N Q( 5)/Q (x 5y) N ( Z[ 5] ) = x 2 + 5y 2 ( 2, ) {2, 1 + 5} Z- Q (2,1+ 5) (x, y) = N ( ) Q( 5)/Q 2x (1 + 5)y N (( 2, )) = 2x 2 2xy + 3y Z[ 5] 2 GL 2 (Z) Z- Z- 20 Z- GL 2 (Z)- Z[ 5] Gauss, Dedekind Q( 5) 2 K Z[ 5] K O K 3 2 K d K = Q( d) D d (d 1 (mod 4)) D := 4d (d 2, 3 (mod 4)) O K = Z + Z D + D 2 Cl(O K ) 2.2 ( [15], [13, ]). K, O K, d, D, Cl(O K ) { } Cl(O K ) 1:1 D Sym 2 Z 2. GL 2 (Z)- 1 2 Z Cl(O K ) (= )2 Gauss Cohen Lenstra [20, 6 ] Cohen Lenstra [26] 3 Q n n n Q O K K

4 4? Z[2 5], Z Z, Z[x]/(x 2 ) Z 2 1 ( [23], [22, Chapter2] ) Z n n Gauss 2 Sym 2 Z 2 GL 2 - n 2.4 ( [1]; [12], [14], [2]; [3]; [6]). { }/ = 1:1 Z 2 Z 2 Z 2 GL 2 (Z) 3 -. { }/ = 1:1 Z 2 Z 3 Z 3 GL 2 (Z) GL 3 (Z) 2 -. { 4 }/ = 1:1 Z 2 Sym 2 Z 3 GL 2 (Z) GL 3 (Z)-. { 5 }/ = 1:1 Z 4 2 Z 5 GL 4 (Z) GL 5 (Z)-. n ([4, 11]) 2.5. n n 2 Sym 2 Z 3 Z 3 Z 3 2 Z 5 Z 5 Z 5 [27] n R Disc(R) ) Disc(R) := det ((Tr(τ i τ j )) 0 i,j n 1 {τ i } 0 i n 1 R Z- Tr: R Z R End(R) = M n (Z) 2 Q( d) D 0 n n 2 3 (balanced) 3 (resp. 2 ) Galois [24] 4 [25] n Selmer 3 Selmer Selmer [21] [28]

5 5 3.1 Jacobi g 1 n = 2g + 1 ( ) Q y 2 = f(x), f(x) = x n + a 4 x n 2 + a 6 x n a 2n (a i Z) (3.1) Q g 4 g = 1 C P 2 y 2 z = f(x, z), f(x, z) = x 3 + a 4 xz 2 + a 6 z 3 (a i Z) (3.2) O = (0 : 1 : 0) 1 O Q C(Q) := {(a : b : c) C a, b, c Q} Mordell Weil 3.1. Mordell Weil C(Q) = Z r ( ) r C Mordell Weil rank Mordell Weil 3.2. Mordell Weil C (3.1) (3.1) C g 2 C g J C Abel Jacobi Jacobi J C J C (Q) 3.1 Mordell Weil Mordell Weil Selmer 3.2 Selmer Q Archimedes R p p Q p Q 3.3. Q X Q p R (Q p - R- ) Q- x 2 + y 2 + z 2 = 1 R Q Q p R Q- Selmer 4 Q- Weierstrass g Q [17, 7.4.3]

6 n 2 Q E n-selmer S n (E) ( S n (E) := ker H 1 (Gal(Q/Q), E[n]) ) H 1 (Gal(Q v /Q v ), E)[n] v Q v 0 E Jacobi J C Mordell Weil E(Q) E(Q)/nE(Q) E(Q)/nE(Q) n r S n (E) Mordell Weil r ([9, Theorem5.6], [10, Proposition28]). { 2-Selmer }/ = 1:1 Sym 4 Q 2 PGL 2 (Q)-. { 3-Selmer }/ = 1:1 Sym 3 Q 3 PGL 3 (Q)-. 4-Selmer 5-Selmer Mordell Weil 1 BSD ([10]) [28] Sym 4 Q 2 GL 2 (Q) 2 PGL 2 (Q) Sym 3 Q 3 2 R, Q p (3.2) Bhargava Ho 2-Selmer ([7], [8]) : 3.7 ([8, Corollary 2, Corollary 4], [19]). Weierstrass Q g g Jacobi Mordell Weil 3 2 g 3 20 g 3 C C [16] n-selmer H 1 (Gal(Q/Q), J C [2])

7 7 3.8 (I.). ( ) { } { Weierstrass g C Q 2 Sym 2 Q 2g+1 +H 1 (Gal(Q/Q), J C [2]) { 1:1 1:1 GL / 2 (Q) GL 2g+1 (Q) - } P 2g J C (Q)/2J C (Q) Q 2 Sym 2 Q 2g+1 GL 2 (Q) GL 2g+1 (Q) H 1 (Gal(Q/Q), J C [2]) H 1 (Gal(Q/Q), J C [2]) (3.1) f(x) J C [2] Gauss [20, 5, 6 ] [27] Gauss ( ) Bhargava [5] [18] } / =. [1] Manjul Bhargava. Higher composition laws i: A new view on gauss composition, and quadratic generalizations. Annals of mathematics, pp , [2] Manjul Bhargava. Higher composition laws ii: On cubic analogues of gauss composition. Annals of mathematics, pp , [3] Manjul Bhargava. Higher composition laws iii: The parametrization of quartic rings. Annals of mathematics, pp , [4] Manjul Bhargava. The density of discriminants of quartic rings and fields. Annals of Mathematics, pp , [5] Manjul Bhargava. Higher composition laws and applications. In Proceedings oh the International Congress of Mathematicians: Madrid, August 22-30, 2006: invited lectures, pp , [6] Manjul Bhargava. Higher composition laws iv: The parametrization of quintic rings. Annals of Mathematics-Second Series, Vol. 167, No. 1, pp , [7] Manjul Bhargava and Benedict H Gross. Arithmetic invariant theory. arxiv preprint arxiv: , [8] Manjul Bhargava and Benedict H Gross. The average size of the 2-selmer group of jacobians of hyperelliptic curves having a rational weierstrass point. arxiv preprint arxiv: , [9] Manjul Bhargava and Arul Shankar. Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves. arxiv preprint arxiv: , [10] Manjul Bhargava and Arul Shankar. Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0. arxiv preprint arxiv: , 2010.

8 8 [11] Manjul Bhargava and Melanie Matchett Wood. The density of discriminants of s 3-sextic number fields. PROCEEDINGS-AMERICAN MATHEMATICAL SOCIETY, Vol. 136, No. 5, p. 1581, [12] Boris Nikolaevich Delone, Dmitriĭ Konstantinovich Faddeev, Emma Lehmer, and Sue Ann Walker. The theory of irrationalities of the third degree, Vol. 10. American Mathematical Society Rhode, Island, [13] Peter Gustav Lejeune Dirichlet. Vorlesungen über zahlentheorie. F. Vieweg und sohn, [14] Wee Teck Gan, Benedict Gross, and Gordan Savin. Fourier coefficients of modular forms on g2. Duke Mathematical Journal, Vol. 115, No. 1, p. 105, [15] Carl Friedrich Gauss. Disquisitiones arithmeticae, Vol Yale University Press, [16] Yasuhiro Ishitsuka. Complete intersections of two quadrics and galois cohomology. arxiv preprint arxiv: , [17] Qing Liu and Reinie Erné. Algebraic geometry and arithmetic curves, Vol. 9. Oxford university press, [18] Bjorn Poonen. Average rank of elliptic curves (after manjul bhargava and arul shankar). Séminaire Bourbaki, Vol. 1049,, [19] Bjorn Poonen and Michael Stoll. Chabauty s method proves that most odd degree hyperelliptic curves have only one rational point. arxiv preprint arxiv: , [20] Paulo Ribenboim.., 2003., ( : My numbers, my friends: popular lectures on number theory). [21] Joseph H Silverman. The arithmetic of elliptic curves, Vol Springer, [22] Melanie Eggers Matchett Wood. Moduli spaces for rings and ideals. Princeton University, [23] Melanie Matchett Wood. Gauss composition over an arbitrary base. Advances in Mathematics, Vol. 226, No. 2, pp , [24] Melanie Matchett Wood. Rings and ideals parameterized by binary n-ic forms. Journal of the London Mathematical Society, Vol. 83, No. 1, pp , [25] Melanie Matchett Wood. Quartic rings associated to binary quartic forms. International Mathematics Research Notices, Vol. 2012, No. 6, pp , [26]. Cohen lenstra heuristics. 10, pp. 1 16, [27]. (algebraic number theory and related topics and related topics 2009)., Vol. 25, pp , [28]. Q rank. 8, 2012.

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