Fields generated by sums and products of singular moduli

Abstract

We show that the field Q(x, y), generated by two singular moduli x and y, is generated by their sum x + y, unless x and y are conjugate over Q, in which case x + y generates a subfield of degree at most 2. We obtain a similar result for the product of two singular moduli. CDC, the IRN GandA (CNRS) and the ALGANT Program. Our calculations were performed using the PARI/GP package [9]. The sources are available from the second author.

Equations in three singular moduli: the equal exponent case

“…Section 3 contains some primitive element theorems for singular moduli, which will be central to our proofs. These generalise results of Faye and Riffaut [11]. Theorem 1.1 is proved in Sections 4 and 5.…”

“…In this section, we prove the following two results, which can be seen as combining the results of [11] with Proposition 3.3. The proofs of Theorems 3.4 and 3.5 both follow the approach of Faye and Riffaut.…”

“…The proof of Theorem 3.5 is then almost exactly the same as the proof of Theorem 3.2 in [11], thanks to the elementary fact that…”

“…(3)-( 5) are proved in [12,Lemma 2.1]. ( 6)- (11) then follow by the same argument as in [12], which uses the fact that b 2 1 − b 2 2 ≡ 0 mod 4a for any two tuples (a, b 1 , c 1 ), (a, b 2 , c 2 ) ∈ T ∆ . The singular modulus corresponding to the unique tuple (a, b, c) ∈ T ∆ with a = 1 is called the "dominant" singular modulus of discriminant ∆.…”

“…In [11], Faye and Riffaut proved the following primitive element theorems for sums and products of singular moduli. See also the generalisation of Theorem 3.1 obtained in [3].…”