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Threefield identities and simultaneous representations of primes by binary quadratic forms
Abstract
Kaplansky [2003] proved a theorem on the simultaneous representation of a
prime p by two different principal binary quadratic forms. Later, Brink found
five more like theorems and claimed that there were no others. By putting
Kaplansky-like theorems into the context of threefield identities after
Andrews, Dyson, and Hickerson, we find that there are at least two similar
results not on Brink's list. We also show how such theorems are related to
results of Muskat on binary quadratic forms
We study intrinsic Galois structure behind theorems by Kaplansky, Brink, and Mortenson on simultaneous representation of primes by binary quadratic forms with different discriminants. This study brings new theorems like theirs also in indefinite quadratic forms.
We give a new proof of a recent theorem of Kaplansky and use it to revive an old, seemingly forgotten result of Glaisher.
Let p ≡ ± 1 (mod 8) be a prime which is a quadratic residue modulo 7. Then p = M2 + 7N2, and knowing M and N makes it possible to “predict” whether p = A2 + 14B2 is solvable or p = 7C2 + 2D2 is solvable. More generally, let q and r be distinct primes, and let an integral solution of H2p = M2 + qN2 be known. Under appropriate assumptions, this information can be used to restrict the possible values of K for which K2q = A2 + qrB2 is solvable and the possible values of K′ for which K′2p = qC2 + rD2 is solvable. These restrictions exclude some of the binary quadratic forms in the principal genus of discriminant −4qr from representing p.
By developing a connection between partial theta functions and Appell-Lerch
sums, we find and prove a formula which expresses Hecke-type double sums in
terms of Appell-Lerch sums and theta functions. Not only does our formula prove
classical Hecke-type double sum identities such as those found in work Kac and
Peterson on affine Lie Algebras and Hecke modular forms, but once we have the
Hecke-type forms for Ramanujan's mock theta functions our formula gives
straightforward proofs of many of the classical mock theta function identities.
In particular, we obtain a new proof of the mock theta conjectures. Our formula
also applies to positive-level string functions associated with admissable
representations of the affine Lie Algebra as introduced by Kac and
Wakimoto.
It is a theorem of Kaplansky that a prime p ≡ 1 (mod 16) is representable by both or none of x² + 32y² and x² + 64y², whereas a prime p ≡ 9 (mod 16) is representable by exactly one of these binary quadratic forms. In this paper five similar theorems are proved, and a heuristic argument is given why there are no other results of the same kind. The latter argument relies on the (plausible) conjecture that the list of 485 known negative discriminants Δ such that the class group C(Δ) has exponent 4 is complete. The methods are purely classical.
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The forms x + 32y 2 and x + 64y 2
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4072 E-mail address: etmortenson@gmail
- K S Williams
K. S. Williams, Private communication.
Department of Mathematics, The University of Queensland, Brisbane, Australia 4072
E-mail address: etmortenson@gmail.com