MODULAR FORMS-page66

# MODULAR FORMS-page66 - y 2 = 4 x 3-c 4 g 2 x-c 6 g 3 for...

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62 LECTURE 6. MODULAR FORMS 6.5 Show that det 0 @ ( z 1 ) 0 ( z 1 ) 1 ( z 2 ) 0 ( z 2 ) 1 r℘ ( z 3 ) 0 ( z 3 ) 1 1 A = 0 whenever z 1 + z 2 + z 3 = 0. Deduce from this an explicit formula for the group law on the projective cubic curve y 2 t = 4 x 3 - g 2 xt 2 - g 3 t 3 . 6.6 ( Weierstrass ζ -function ) It is deﬁned by Z ( z ;Λ) = 1 z + X ω Λ \{ 0 } 1 - ω + 1 ω + z ω 2 « . Let Λ = Z ω 1 + Z ω 2 . Show that (i) Z 0 ( z ) = - ( z ); (ii) Z ( z + ω i ) = Z ( z ) + η i ,i = 1 , 2 where η i = Z ( ω i / 2); (iii) η 1 ω 2 - η 2 ω 1 = 2 πi ; (iv) Z ( λz ; λ · Λ) = λ - 1 Z ( z ; · Λ) , where λ is any nonzero complex number. 6.7 Let φ ( z ) be a holomorphic function satisfying φ ( z ) 0 ( z ) = Z ( z ) , (i) Show that φ ( - z ) = - φ ( z ); (ii) φ ( z + ω i ) = - e η i ( z + ω i 2 φ ( z ); (iii) φ ( z ) = σ ( z ) , where σ ( z ) is the Weierstrass σ -function. 6.8 Using the previous exercise show that the Weierstrass σ -function σ ( z ) admits an in ﬁnite product expansion of the form σ ( z ) = z Y ω Λ \{ 0 } (1 - z ω ) e z ω + 1 2 ( z ω ) 2 which converges absolutely, and uniformly in each disc | z | ≤ R . 6.9 Let E τ be an elliptic curve and y 2 = 4 x 3 - g 2 x - g 3 be its Weierstrass equation. Show that any automorphism of E τ is obtained by a linear transformation of the variables ( x,y ) which transforms the Weierstrass equation to the form
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Unformatted text preview: y 2 = 4 x 3-c 4 g 2 x-c 6 g 3 for some c 6 = 0. Show that E is harmonic (resp. anharmonic) if and only if g 3 = 0 (resp. g 2 = 0). 6.10 Let k be an even integer and let L R k be a lattice with a basis ( e 1 ,...,e k ). Assume that || v || 2 is even for any v L . Let D be the determinant of the matrix ( e i e j ) and N be the smallest positive integer such that N || v * || 2 2 Z for all v * R k satisfying v * w Z for all w L . Dene the theta series of the lattice L by L ( ) = X n =0 # { v L : || v || 2 = 2 n } e 2 in . (i) Show that L ( ) = P v L e i || v || 2 ; (ii) Show that the functions (0 , ) k discussed in the beginning of Lecture 6 are special cases of the function L ....
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## This note was uploaded on 01/08/2012 for the course MATH 300 taught by Professor Ontonkong during the Fall '09 term at SUNY Stony Brook.

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