By Alexandru Buium

ISBN-10: 270566226X

ISBN-13: 9782705662264

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**Extra info for Differential algebra and diophantine geometry**

**Sample text**

The ﬁrst statement of Proposition 12 is purely algebraic, and purely algebraic proofs are known, but they are not as simple or as elegant as the modular proof just given. No non-modular proof of the asymptotic formula (38) is known. Before continuing with the theory, we look at some examples, starting in rank 8. Deﬁne the lattice Λ8 ⊂ R8 to be the set of vectors belonging to either Z8 or (Z+ 12 )8 for which the sum of the coordinates is even. This is unimodular because the lattice Z8 ∪ (Z + 12 )8 contains both it and Z8 with the same index 2, and is even because x2i ≡ xi (mod 2) for xi ∈ Z and x2i ≡ 14 (mod 2) for xi ∈ Z + 12 .

1, the deﬁnition of Tm is very simple: it sends a homogeneous function F of degree −k on lattices Λ ⊂ C to the function Tm F deﬁned (up to a suitable normalizing constant) by Tm F (Λ) = F (Λ ), where the sum runs over all sublattices Λ ⊂ Λ of index m. The sum is ﬁnite and obviously still homogeneous in Λ of the same degree −k. Translating from the language of lattices to that of functions in the upper half-plane by the usual formula f (z) = F (Λz ), we ﬁnd that the action of Tm is given by (cz + d)−k f Tm f (z) = mk−1 a b ∈Γ1 \Mm c d az + b cz + d (z ∈ H) , (40) where Mm denotes the set of 2 × 2 integral matrices of determinant m and where the normalizing constant mk−1 has been introduced for later convenience (Tm normalized in this way will send forms with integral Fourier coeﬃcients to forms with integral Fourier coeﬃcients).

In general one must require the vanishing of the constant term of the Fourier expansion of f , suitably deﬁned, at every cusp of the group Γ , in which case it again follows that f can be estimated as in (25). Actually, it is easier to simply deﬁne cusp forms of weight k as modular forms for which y k/2 f (x + iy) is bounded, a deﬁnition which is equivalent but does not require the explicit knowledge of the Fourier expansion of the form at every cusp. ♠ Congruences for τ (n) As a mini-application of the calculations of this and the preceding sections we prove two simple congruences for the Ramanujan tau-function deﬁned by 24 D.

### Differential algebra and diophantine geometry by Alexandru Buium

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