G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions (Princeton University Press, Princeton, 1971)  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the linear form 0 ) U 0 ) 1 U i by the general formula of Proposition 6.4 e) U ; k (7:11) and U (compare with Proposition 4.5 of [Hi85] the equality (7. 11) is deduced from the action of double cosets (see [Shi1], 3.4.5) Since we have that Let us use the action of the involution W Np D 1 ; g and the de nition (6.1) of f 0 which implies 0 W Np 1 = p WNpWNp = 1) 1 = ....

Shimura G.: Introduction to the Arithmetic Theory of Automorphic Functions, Princeton Univ. Press, 1971

....z i# where # is some positive constant. The solutions of this eigenvalue problem can be identified with Maass waveforms [14] The identification is worthwhile, since much is known about Maass waveforms from number theory and harmonic analysis which will simplify their computation, see e.g. [15, 16, 17, 18, 19, 20, 13, 21, 22, 23]. 3 The Picard group In the three dimensional case one considers the upper half space, x 0 , x 1 , y) y 0 equipped with the hyperbolic metric 0 dx 1 dy . The geodesics of a particle moving freely in the upper half space are straight lines and semicircles ....

G. Shimura. Introduction to the Arithmetic Theory of Automorphic Functions. Princeton Univ. Press, 1971.

..... 13 3 L functions and modular forms In this section we want to study L functions and modular forms, and their relation to the arithmetic questions addressed in the last section. The basic references are Tate s papers [83, 84] Silverman s books [77, 78] Koblitz s book [52] and Shimura s book [75], and my paper [95] We will start with a definition of L series using integral models of elliptic curves or the Galois representations on Tate modules. The Taniyama Shimura conjecture, or the modularity conjecture implies the analytic continuations already conjectured by Hasse. Then we state the ....

....quadratic field K, 3. ord s=k L(s, f) 1, and that # does not divide 2N . Then the following equality holds: Q l Im(# f,Q ) ord s=k L(s, f) 4 Complex multiplications In this section, we study elliptic curves with complex multiplications. The basic references are Shimura s books [75, 76] and Silverman s book [78] The main results include the algebraicity of j invariants which will be used to construct algebraic points on Shimura varieties, and the computation of Lfunctions which provides examples of modular elliptic curves in terms of theta series. By an elliptic curve with ....

G. Shimura, Introduction to the arithmetic theory of automorphic functions, Princeton University Press (1971)

....series E . If is odd, there are no SL(2; Z=N ) invariant elements of E (N ) reflecting the fact that there are no level 1 Eisenstein series of odd weight. Eichler and Shimura have calculated the cohomology of the sheaves H n . This calculation is explained in Verdier [26] and Shimura [23]. The mixed Hodge structure on this cohomology may be calculated by the same technique that Deligne uses in [5] to calculate the action of the Frobenius operator on the etale cohomology groups. Define the Hodge structure S n 2 (N) to be gr c (Y (N) H n ) These polynomials have ....

G. Shimura, "Introduction to the arithmetic theory of automorphic functions," (Publications of the Mathematical Society of Japan, vol. 11) Princeton University Press, Princeton, N.J., 1971.

....then it is not possible to go up and so we say that E is on the surface at l . Similarly, if If l 6 j[End(E) Z[ then it is not possible to go down and we say that E is on the oor at l . For each possible endomorphism ring O, the theory of complex multiplication (see Lang [15] or Shimura [33]) combined with Deuring s theory (see Lang [15] about lifting endomorphisms from characteristic p to characteristic zero, shows that the number of isomorphism classes of elliptic curves over F p which have endomorphism ring isomorphic to O is equal to hO (the number of elements of the group P ....

....assuming the Riemann hypothesis for the zeta function of K, the class group is generated by the prime ideals of norm less than 6(ln jDj) Class numbers of orders are closely related to the class number hK of the maximal order by the following formula (see Lang [15] Theorem 8. 7 or Shimura [33] Exercise 4.12) Suppose O is an order of conductor c in OK . Let O K and O be the units (i.e. multiplicative subgroup of invertible elements) De ne ( to be 1; 0; 1 depending on whether the prime p is inert, rami ed or split in K respectively. Then the relation between the class ....

G. Shimura, Introduction to the arithmetic theory of automorphic functions, Iwanami/Princeton (1971)

....so called Riemann surface with g holes or handles. In Figure 1 we saw a genus 1 curve (an elliptic curve) The complex points of a genus 5 curve are shown in Figure 4. Figure 4: A Riemann surface with 5 handles. The exact formulas for the genus of modular curves are a little complicated (see [19]) Asymptotically the genus of X 0 ( is 1 12 , and the genus of X 1 ( is 1 24 2 . Thus, if is a 160 bit prime, the curve X 1 ( considered over the complex numbers, looks like a Riemann surface with about 2 315 handles It is this daunting object that would have to have a ....

G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton Univ. Press, 1971.

....about H 2 (1 ; Z) Every bilinear form B : D D Z de nes a 2 cocycle 2 of 1 acting trivially on Z. We denote this cocycle by B, too. It is given by B( a; t; b] a 0 ; t 0 ; b 0 ] B(a; b 0 ) 2 Throughout we use the inhomogeneous standard complex of group cohomology as in [Sh] chapter 8. 10 In fact, it is easily checked that this is a 2 cocycle. Hence we get a map Bil(D) H 2 (1 ; Z) from the group Bil(D) of bilinear forms on D to H 2 (1 ; Z) Only the image is important for our applications. It is a basic fact that this map is not injective: Proposition 3.5. ....

G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton University Press (1971).

....Y 1 (N ) or Y (N) One compacti es Y by adjoining a nite set of cusps which correspond to orbits of 1 ( f1g under . Call X the corresponding compact Riemann surface. For more details, notably on the de nition of the analytic structure on X at the cusps, see for example [Kn] p. 311, or [Shi2], ch. 1. It follows from the de nition of this analytic structure that the eld K of meromorphic functions on X is equal to the set of meromorphic functions on H satisfying (Transformation property) f( f( for all 2 ; Behaviour at the cusps) For all 2 SL 2 ( the function ....

.... H, and that e = 1 for all but nitely many in H= Using the Riemann Hurwitz formula (cf. Ki] sec. 4.3) show that the genus of X is given by g( 1 [P SL 2 ( 1 2 X 2 H= e 1) Use this to compute the genus of X 0 (p) X 1 (p) and X(p) for p prime. For details, see [Shi2], sec. 1.6 or [Ogg] 3. For = 2) show that X is isomorphic to 1 , and that Y is isomorphic to 1 f0; 1; 1g. Show that =h 1i is the free group on the two generators g 1 = 1 2 0 1 and g 2 = 1 0 2 1 . 4. De ne a homomorphism (2) n =n , by sending g 1 to (1; 0) and g 2 to ....

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G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton Univ. Press, Princeton, 1971.

....congruence modular groups and in section 8 we present our results for congruence subgroups of Hecke type. 2 Modular groups and modular surfaces We will give here a brief introduction to modular groups and modular surfaces. Our references for this material are the books of Miyake [11] and Shimura [12]. Consider the upper half plane IH = fz 2 CI j= z) 0g with the Poincare metric given by ds 2 (z) dz d z = z) 2 : 2 The special linear group SL(2; IR) f a b c d ja; b; c; d 2 IR and ad bc = 1g acts on the upper half plane IH by the Mbius transformation a b c d z = az ....

G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton University Press (1971).

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G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions (Princeton University Press, Princeton, 1971)

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G. Shimura, Introduction to the arithmetic theory of automorphic functions, Iwanami Shoten and Princeton University Press, 1971.

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G. Shimura, Introduction to the arithmetic theory of automorphic forms, Publications of the Mathematical Society of Japan 11, Iwanami Shoten, Tokyo; University Press, Princeton, 1971.

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G. Shimura, Introduction to the arithmetic theory of automorphic functions, Princeton University Press, Princeton, NJ, 1994

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G. Shimura, Introduction to the arithmetic theory of automorphic functions, Princeton University Press, Princeton, NJ, 1994

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G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Publications of the Math. Soc. of Japan, vol. 11, Princeton University Press, Princeton, NJ, 1971.

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G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton University Press, Princeton N.J., 1971.

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Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo, 1971, Kano Memorial Lectures, No. 1.

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G. Shimura. Introduction to the Arithmetic Theory of Automorphic Functions. Princeton University Press, Princeton, New Jersey, 1971.

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G. Shimura, Introduction to the arithmetic theory of automorphic functions, Publ. Math. Soc. Japan 11, Iwanami Shoten, Tokyo, and Princeton University Press, 1971.

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G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton University Press & Iwanami Shoten, Princeton, 1971.

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G. Shimura. Introduction to the Arithmetic Theory of Automorphic Functions. Princeton Univ. Press, 1971.

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G. Shimura, Introduction to the arithmetic theory of auto- morphic functions, Iwanami Shoten and Princeton University

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G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Publications of the Math. Soc. of Japan, vol. 11, Princeton University Press, Princeton, NJ, 1971. HYPERGEOMETRIC EQUATION AND RAMANUJAN FUNCTIONS 15

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G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions (Publ. Mat. Soc. Japan, num. 11, 1971).

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G. Shimura, Introduction to the arithmetic theory of automorphic functions, Princeton University Press, 1994. MR 95e:11048

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