Initial definitions

Our first goal in this miniproject is the definition of these spaces of quaternionic modular forms. We start with some preliminary work towards this.

Let \(K\) be a field. Recall that a quaternion algebra over \(K\) is a central simple \(K\)-algebra of \(K\)-dimension 4.

A fundamental fact about central simple algebras is that if \(D/K\) is a central simple \(K\)-algebra and \(L/K\) is an extension of fields, then \(D\otimes_KL\) is a central simple \(L\)-algebra. In particular if \(D\) is a quaternion algebra over \(K\) then \(D\otimes_KL\) is a quaternion algebra over \(L\). Some Imperial undergraduate students have established this fact in ongoing project work.

A totally real field is a number field \(F\) such that the image of every ring homomorphism \(F\to\mathbb{C}\) is a subset of \(\mathbb{R}\). We fix once and for all a totally real field \(F\) and a quaternion algebra \(D\) over \(F\). We furthermore assume that \(D\) is totally definite, that is, that for all field embeddings \(\tau:F\to\mathbb{R}\) we have \(D\otimes_{F,\tau}\mathbb{R}\cong\mathbb{H}\). Because \(F\) has at least one real place, the totally definite hypothesis is enough to show that \(D\) is not a matrix algebra and thus must be a division algebra. Thus Fujisaki's theorem (theorem NumberField.FiniteAdeleRing.DivisionAlgebra.units_cocompact from the Fujisaki miniproject) applies, and we know that \(D^\times\backslash (D\otimes_F\mathbb{A}_F)^{(1)}\) is compact.

The high-falutin' explanation of what is about to happen is that the units \(D^\times\) of \(D\) can be regarded as a connected reductive algebraic group over \(F\), and we are going to define certain spaces of automorphic forms for this algebraic group. For a general connected reductive algebraic group, part of the definition of an automorphic form is that it is satisfies some differential equations (for example modular forms are automorphic forms for the algebraic group \(\operatorname{GL}_2\) over \(\mathbb{Q}\), and the definition of a modular form involves holomorphic functions, which are solutions to the Cauchy–Riemann equations). However under the assumption that \(F\) is totally real and \(D/F\) is totally definite, the “associated symmetric space is a 0-dimensional manifold”, meaning in practice that the part of the definition of an automorphic form involving differential equations is vacuously satisfied in this setting. As a consequence, the definitions we're about to give have a far more algebraic flavour. Crucially, in stark contrast to the general theory, the fact that we do not need to talk about differential equations at all means that one does not need to assume that our automorphic forms are \(\mathbb{C}\)-valued; our definition makes sense for forms valued in an arbitrary additive commutative group. In particular, it is possible to talk about mod \(p^n\) and \(p\)-adic automorphic forms in this setting without doing any complicated algebraic geometry.

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