By Goro Shimura

This is a sophisticated publication on modular types. whereas there are lots of books released approximately modular varieties, they're written at an hassle-free point, and never so fascinating from the point of view of a reader who already is familiar with the fundamentals. This publication deals whatever new, that can fulfill the will of any such reader. notwithstanding, we nation each definition and each crucial truth referring to classical modular varieties of one variable.

One of the significant new positive factors of this e-book is the idea of modular different types of half-integral weight, one other being the dialogue of theta capabilities and Eisenstein sequence of holomorphic and nonholomorphic kinds. hence the e-book is gifted in order that the reader can research such theories systematically. eventually, we be aware of the subsequent topics:

(I) The correspondence among the different types of half-integral weight and people of imperative weight.

(II) The arithmeticity of varied Dirichlet sequence linked to modular different types of essential or half-integral weight.

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**Example text**

5) ca (ξ)σ (πy)−a e(ξz). 5) is convergent and deﬁnes an element of Xk . (ii) For two positive integers M and N put Γ (M, N ) = γ ∈ Γ (1) bγ ∈ M Z, cγ ∈ N Z . 6) Xk (M, N ; ψ) = f ∈ Xk f kγ = ψ(dγ )f for every γ ∈ Γ (M, N ) . Then (iia) Xk (M, N ; ψ)σ = Xk (M, N ; ψ σ ); (iib) Xk (M, N ; ψ) is spanned by its Qab -rational elements. 42 III. THE RATIONALITY AND EISENSTEIN SERIES (iii) For every subﬁeld Φ of C containing Qab , the set Xk (Φ) is stable under f → (cα z + dα )−k f (αz) for every α ∈ GL+ 2 (Q).

11) ∞ η(z) = θϕ (z/12) = m=1 3 m and 3 e(m2 z/24). 9) we obtain η 1/2 ι = η. Also, m2 − 1 ∈ 24Z for every integer m prime to 6, and so η(z + 1) = e(1/24)η(z). 12) 1 − e(nz) , η(z) = Δ(z)1/24 = e(z/24) n=1 ∞ 24 where Δ(z) = e(z) n=1 1 − e(nz) . We leave the proof to the reader. In fact, it was explained in [S07, p. 19]. We also note an easy fact: 5. 12a) η 1/2 29 γ = η for every γ ∈ Γ (1) such that bγ , cγ ∈ 24Z. 7. For every congruence subgroup Γ of Γ θ and k ∈ 2−1 Z we have M k (Γ ) = {0} if k < 0 and M 0 (Γ ) = C.

8. For every subﬁeld Φ of C and every congruence subgroup / 2Z, the following Γ of Γ (1), which is assumed to be contained in Γ θ if k ∈ assertions hold: m+p (i) The operator Dkp sends N km (Γ, Φ) into N k+2p (Γ, Φ); in particular, it p (Γ, Φ). Moreover, (Dkp f )σ = Dkp f σ for every σ ∈ sends M k (Γ, Φ) into N k+2p Aut(C). (ii) Every element f (z) of N k (Γ, Φ) can be written in the form (k/2)−1 p Dk−2p gp + f (z) = 0≤p≤k/2 cD2 0 E2 if k ∈ 2Z, if k ∈ / 2Z 46 III. THE RATIONALITY AND EISENSTEIN SERIES with gp ∈ M k−2p (Γ, Φ) and c ∈ Φ.