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Unformatted text preview: ON SIEGEL’S MODULAR CURVE OF LEVEL 5 AND THE CLASS NUMBER ONE PROBLEM IMIN CHEN Abstract. Another derivation of an explicit parametrisation of Siegel’s mod- ular curve of level 5 is obtained with applications to the class number one problem. 1. Introduction In a not so well-known paper , Siegel obtained an explicit parametrisation of the modular curve X + ns (5) / Q by constructing a uniformiser out of η-functions (see the end of Section 3 for a brief description of the modular curves X + ns ( p ) / Q ). In modern terms, he then applied this to solve the class number one problem as follows. Let Q be the field of algebraic numbers of C . The modular curves X + ns ( p ) / Q classify isomorphism classes of elliptic curves with a certain type of ”non-split” level p structure. Now, to every order O of class number one in an imaginary quadratic field K , there is an associated elliptic curve E O / Q , unique up to Q-isomorphism, with the property that E O / Q has complex multiplication by O . When p is inert in O , the properties of E O imply that E O has one or more of the above ”non-split” level p structures which are defined over Q . Thus, by the modular interpretation of X + ns ( p ) / Q above, E O gives rise to one or more Q-rational points of X + ns ( p ) / Q if p is inert in O . The case p = 3 is special because the weaker condition that p is not ramified in O is sufficient to imply that E O has a ”non-split” level p structure. It is well-known and not too difficult to check that the imaginary quadratic orders with discriminant d O =- 3,- 4,- 7,- 8,- 11,- 12,- 16,- 19,- 27,- 28,- 43,- 67,- 163 have class number one and that these are the only ones for d O ≥ - 163 (see for instance Appendix A.2. of ). From this list of imaginary quadratic orders of class number one, those with discriminant d O =- 7 ,- 8 ,- 28 ,- 43 ,- 67 ,- 163 have the property that 3 is unramified and 5 is inert in O . These imaginary quadratic orders therefore give rise to at least 6 distinct Q-rational points on X + ns (3) / Q and X + ns (5) / Q . An explicit parametrisation of X + ns (3) / Q shows that an elliptic curve E/K de- fined over a field K ⊂ Q gives rise to a Q-rational point on X + ns (3) (i.e. has a “non-split” level p structure defined over Q ) if and only if j ( E ) is a cube in Q . This is a folklore fact, but we shall prove this later for completeness (see Appendix A.6. of , Section 5.3(b) of , and ). In the case that E/K has complex multipli- cation, then E/K gives rise to a Q-rational point on X + ns (3) / Q if and only if j ( E ) is an integer cube since one knows in such a case that j ( E ) is an algebraic integer. Date : 1 August 1998....
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This note was uploaded on 12/03/2011 for the course MATH 8430 taught by Professor Clark during the Fall '11 term at UGA.
- Fall '11