A COMPUTATION OF MINIMAL POLYNOMIALS OF SPECIAL VALUES OF SIEGEL MODULAR FUNCTIONS

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1 MATHEMATICS OF COMPUTATION Volume 7 Number 4 Pages S Article electronically published on March A COMPUTATION OF MINIMAL POLYNOMIALS OF SPECIAL VALUES OF SIEGEL MODULAR FUNCTIONS TSUYOSHI ITOH Abstract. Recently Fukuda and Komatsu constructed units of a certain abelian extension of Qexpπ 1/5 using special values of Siegel modular functions. In this paper we determine the minimal polynomials of these units. 1. Introduction We put ζ n =expπ 1/n for a positive integer n andk = Qζ 5. We explain about Siegel modular forms. Let H be the set of all complex symmetric matrices of degree with positive definite imaginary parts. For u C z H and r s R we define the theta series by Θu z r s = x Z e 1 t x + rzx + r+ t x + ru + s where eξ denotes expπ 1ξ. Moreover we consider the following function on H : Φz r s r s Θz r s = Θz r s. Let Sp Z ={α GL 4 Z t αjα = J} be the symplectic group of degree and put Γ N = {α Sp Z α I 4 mod N} for a positive integer N wherei n is the unit matrix of degree n and I J =. I For any element α = CD ABofSp Z we define an action of α on z H by αz =Az+B Cz+D 1 as usual. If N is a positive integer and r s r s 1 N Z we know that Φz r s r s is a Siegel modular function with respect to Γ N [S] Prop Let σ be the element of the Galois group Galk/Q defined by ζ5 σ = ζ5.welet O k be the ring of integers in k and put L = { ξ ξ ξ σ Ok }. We define a Riemann form E on C /L for u i v i C as follows: E u1 u v1 v = ρu 1 v 1 ū 1 v 1 +ρ σ u v ū v Received by the editor March 15 and in revised form April 1 1. Mathematics Subject Classification. Primary 11G15 11R7 11Y c American Mathematical Society

2 97 TSUYOSHI ITOH where ρ =ζ 5 ζ5 4 /5. Moreover for ζ5 ζ 4 5 ω 1 = ζ 5 ω = ζ 5 ω = ζ 5 + ζ 4 5 ζ ζ 5 ω 4 = we can see that {ω 1 ω ω ω 4 } is a free basis of L over Z and Eω i ω j ij={14} = J. Hence we see that ζ z = 5 + ζ5 4 ζ5 1 ζ5 ζ ζ ζ 5 ζ 5 ζ5 ζ5 is a CM-point of H corresponding to the polarized abelian variety C /L E. By [S] Prop.. if N is a positive integer and r s r s 1 N Z Φz r s r s is contained in some abelian extension k of k. Let ω be an integer of k which is prime to N. WedenotebyRω M 4 Z the regular representation of ω with respect to an O k -basis { ζ 5 ζ5 4ζ 5 + ζ 5 ζ 5 }. We put v = N k/q ω. Then there exists a matrix β in Sp Z satisfying Rϕω I β mod N vi where ϕ is an endomorphism of k defined by ϕa =a 1+σ.Ifr s are in 1 N Z we define Φ Rϕωz r s r s =Φβz r vs r vs. Then by Shimura s reciprocity law [S] we have Φz r s r s k /k ω =Φ Rϕωz r s r s. ζ 5 We put ζ = ζ 5. Fukuda and Komatsu showed the following theorem. Theorem [FK]. We put ε 1 = Φ z 1 z 1 Φ Rϕ+ζ ζ 5 and ε = Φ z 1 1 z 1 1. Φ Rϕ+ζ Then ε 1 and ε are units in k 6. We will construct the minimal polynomials of ε 1 and ε by numerically computing the conjugates of ε i over Q with high precision. These polynomials allow us to compute the approximate values of these units with arbitarily high precision. Our method will be useful to compute the rank of the unit group generated by special values of Siegel modular functions and so on.

3 A COMPUTATION OF MINIMAL POLYNOMIALS 971. Computation of conjugates For a positive integer N k N denotes the maximal ray class field of k modulo N. We consider the structure of the Galois group Galk 6 /k. Let M be the subgroup of k generated by integers of k which are prime to 6. We also let M be the set of all principal ideals of k which have a generator in M. Similarly we put S 6 = {a k a 1mod6} and S 6 = {a a S 6 }. Then by class field theory Galk 6 /k = M/ S 6 = M/Uk /S 6 U k /U k = M/S 6 U k = Z/1Z where U k is the full unit group of k. Soby[FK]weknowthat k6 /k τ = ζ 5 + generates the Galois group of k 6 over k. Weextendσ to k 6. By Shimura s theory we know that Φz r s r s is contained in k 18 if r s r s 1 Z [S] see also [K]. Since we will treat the case r s r s 1 6 Z later we choose β Sp Z which satisfies I Rϕω β mod 7. vi In the case ω = ζ 5 +wechoose β = Sp Z. Then we can compute Φz r s τ n by using β. But the theta series Θβ n z r s converges slowly if n is large because the imaginary part of β n z tends to quickly. So we determine Φz r s τ n using the fact that ζ6 τ = ζ6 11 and the following lemma recursively. Lemma cf. [S] Prop. 1.. If r s are in 1 6 Z and r s 1 Z then there exist r 1 s Z r s 1 Z and an integer t such that Φβz r s r s =ζ t 6 Φz r 1 s 1 r s Next we shall determine Φz r s σ. We know that Φz r s is an algebraic integer in k 6 cf. [FK] and hence N k6/q 5 Φz r s = 9 Φ Rϕ+ζnz r s. n= Furthermore we know that Φz r s σ is given by ξφz r 1 s 1 r s for some r 1 s Z r s 1 Z and some 6-th root of unity ξ. So we computed 1 N k6/q 5 Φz r s N k6/q 5 Φz r 1 s 1 r s for all r 1 s 1 r s. Fortunately there is only one integral value for 1. Hence we can determine Φz r s σ up to conjugation by τ n and multiplication by an 18-th root of unity.

4 97 TSUYOSHI ITOH Next we shall determine Φz r s σ. We know that σ acts as complex conjugation on k. So we computed Tr k6/kφz r s + Tr k6/kξφz r 1 s 1 r s for all r 1 s 1 r s and a 6-th root of unity ξ and we found only one value which is in R. We can determine Φz r s σ in a similar way. We note that Φz r s σ and Φz r s σ are also determined up to conjugation by τ n and multiplication by an 18-th root of unity. By the above argument we can determine all Φz r s σi τ j and hence ε σi τ j 1 i j 9 up to multiplication by an 18-th root of unity. We note that ε 1 k 6 andgalk 6 /Q ={σ i τ j i j 9} as a set. We were able to determine the 18-th root of unity ξ i for each σ i so that the all fundamental symmetric forms of {ξ i ε σi 1 τ j } are close to rational integers. Consequently the set {ε ρ 1 ρ Galk 6/Q} was determined. The same method is applicable to ε.. Minimal polynomials Now it is easy to compute the minimal polynomials of ε i using approximate values of the conjugates of ε i. We know that the coefficients of these polynomials are in Z. So we computed these fundamental symmetric forms with a precision of digits by using TC on Sparc Station 5. These values are close to integers which is enough to recognize the coefficients of minimal polynomials. Φz 1 The case of ε 1 = 1/ Φ Rϕζ+ z 1/. Φz 1/6 ε σ up to conjugation by τ i ζ 1/ 1/ 1/ Φ Rϕζ+4 z 1/6 1/ 1/ 1/ ε σ up to conjugation by τ i Φ Rϕζ+9z 1/ Φ Rϕζ+7 z 1/ Φ Rϕζ+5z 1/6 ε σ up to conjugation by τ i ζ 1/ 1/ 1/ Φ Rϕζ+z 1/6 1/ 1/ 1/ 1/ 1/ 1/ minimal polynomial 1 1x +66x + 17x + 748x x x x 7 +5x x x x x x x x x x x x x +4689x x x + 11x x x x x x x x 1 + 5x + 141x +71x x x x 7 +66x 8 1x 9 + x 4

5 A COMPUTATION OF MINIMAL POLYNOMIALS 97 The case of ε = Φz 1/ / 1/ Φ Rϕζ+ z 1/ / 1/. Φz 5/6 1/6 ε σ up to conjugation by τ i 1/ 1/ 1/ ζ Φ Rϕζ+4 z 5/6 1/6 1/ 1/ 1/ 1/ Φ Rϕζ+ 9z 1/ ε σ up to conjugation by τ i / 1/ Φ Rϕζ+7 z 1/ / 1/ Φ Rϕζ+ 5z 5/6 1/6 ε σ up to conjugation by τ i 1/ 1/ 1/ ζ Φ Rϕζ+z 5/6 1/6 1/ 1/ 1/ minimal polynomial 1+5x +1x +9x + 1x x x x x x x x x x x x x x x x x +1859x x x x x x x x x x x x + 588x +94x x 5 + 1x 6 +9x 7 +1x 8 +5x 9 + x 4 Acknowledgments The author would like to thank Professor K. Komatsu for giving useful advice and Professor T. Fukuda for instruction about computational techniques for theta series. The computations are mainly due to TC which is an interpreter of multiprecision C language developed by T. Fukuda which is available from ftp://tnt.math.metro-u.ac.jp/pub/math-packs/tc/. References [FK] T. Fukuda and K. Komatsu: On a unit group generated by special values of Siegel modular functions Math. Comp MR j:1189 [K] K. Komatsu: Construction of normal basis by special values of Siegel modular functions Proc. Amer. Math. Soc MR d:1119 [S] G. Shimura: Theta functions with complex multiplicationduke Math. J MR 54:1664 Department of Mathematical Sciences School of Science and Engineering Waseda University -4-1 Okubo Shinjuku-ku Tokyo Japan address: tsitoh@mn.waseda.ac.jp