Arithmeticity in the Theory of Automorphic Forms

Shimura, Goro

doi:10.1090/surv/082

Cited by 99 publications

(433 citation statements)

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“…Thus, as we said, the number of (21) These are due to Hecke when k ∈ Z. The proof for an arbitrary k ∈ 2 −1 Z, even in the Hilbert modular case, can be found in [20,Proposition A6.4] and [16].…”

Section: Theorem 11 (I) F Is An Eisenstein Seriesmentioning

confidence: 83%

Quadratic Diophantine Equations, the Class Number, and the Mass Formula

Shimura¹

2016

Collected Papers V

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The basic setting and two ternary casesWe take a finite-dimensional vector space V over a field F and take also an F -bilinear symmetric form ϕ : V × V → F . We then put ϕ[x] = ϕ(x, x) for x ∈ V, thus using the same letter ϕ for the quadratic form and the corresponding symmetric form. By a quadratic Diophantine equation we mean an equation of the typewith a given q ∈ F × . In particular, in the classical case with F = Q and V = Q n , we usually assume that ϕ is Z-valued on Z n and q ∈ Z. The purpose of the present article is to present some new ideas on various arithmetical questions on such an equation. We start with some of our basic symbols and terminology. For a set X we denote by #X or #{X} the number (≤ ∞) of elements of X. For an associative ring R with identity element, we denote by R × the group of invertible elements of R and by M n (R) the ring of all square matrices of size n with entries in R. We then put GL n (R) = M n (R) × and denote by 1 n the identity element of M n (R). For two square matrices A and B of size m and n we denote by diag[A, B] the square matrix of size m + n with A and B in diagonal blocks and zeros in the remaining blocks. Now, given (V, ϕ) as above, we always assume that ϕ is nondegenerate. We also put n = dim(V ) and define, as usual, the orthogonal group O ϕ (V ) and the special orthogonal group SO ϕ (V ) by written also SO(ϕ) and SO(V, ϕ).We let GL(V ) act on V on the right, so that xα is the image of x under α.As the base field F we take, for the moment, an algebraic number field or its completion at a nonarchimedean prime. We denote by g the ring of algebraic integers in the former case and the ring of local integers in the latter case. Those who are not much interested in the general case may assume that F is Q or the p-adic number field Q p for any prime number p, and g is Z or the ring Z p of p-adic integers. By a g-lattice (simply a lattice) in V we mean a finitely generated

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Section: Theorem 11 (I) F Is An Eisenstein Seriesmentioning

confidence: 83%

Quadratic Diophantine Equations, the Class Number, and the Mass Formula

Shimura¹

2016

Collected Papers V

Self Cite

View full text Add to dashboard Cite

show abstract

“…For an element σ ∈ θ and z ∈ H we write h σ (z) for the 1/2-weight factor of automorphy as defined for example in [10,Theorem 6.8]. For a k ∈ 1 2 Z we define the factor of automorphy…”

Section: Siegel Modular Formsmentioning

confidence: 99%

“…As we mentioned in our previous paper, the origin of these two works is the work of Shimura on the algebraicity of these special L-values (both in integral and half-integral situation). Indeed, Shimura, in his admirable book "Arithmeticity in the Theory of Automorphic Forms" [10], establishes various algebraicity results of these special L-values. These results are established over an algebraic closure of Q (see for example Theorem 6.1 below), and Shimura leaves it as an "exercise" to the reader to establish more accurate results (i.e.…”

Section: Introductionmentioning

confidence: 99%

“…Notation Our main reference is the book of Shimura [10], and hence we have adopted most of the notation used there. There is only a few notational differences, perhaps the one worth emphasizing, is that we use the L notation for our functions (L-functions) where in [10] the zeta notation, Z , is used.…”

Section: Introductionmentioning

confidence: 99%

“…There is only a few notational differences, perhaps the one worth emphasizing, is that we use the L notation for our functions (L-functions) where in [10] the zeta notation, Z , is used.…”

Section: Introductionmentioning

confidence: 99%

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