By Henryk Iwaniec

Automorphic kinds are one of many vital issues of analytic quantity concept. actually, they take a seat on the confluence of research, algebra, geometry, and quantity conception. during this booklet, Henryk Iwaniec once more screens his penetrating perception, robust analytic suggestions, and lucid writing variety. the 1st variation of this quantity was once an underground vintage, either as a textbook and as a revered resource for effects, rules, and references. The book's attractiveness sparked a starting to be curiosity within the mathematical neighborhood to convey it again into print. The AMS has responded that decision with the ebook of this moment variation. within the ebook, Iwaniec treats the spectral conception of automorphic varieties because the research of the distance $L^2 (H\Gamma)$, the place $H$ is the higher half-plane and $\Gamma$ is a discrete subgroup of volume-preserving alterations of $H$. He combines quite a few options from analytic quantity conception. one of the issues mentioned are Eisenstein sequence, estimates for Fourier coefficients of automorphic varieties, the idea of Kloosterman sums, the Selberg hint formulation, and the speculation of small eigenvalues. Henryk Iwaniec was once presented the 2002 AMS Cole Prize for his basic contributions to analytic quantity concept. additionally on hand from the AMS by means of H. Iwaniec is issues in Classical Automorphic types, quantity 17 within the Graduate reviews in arithmetic sequence. The booklet is designed for graduate scholars and researchers operating in analytic quantity concept

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**Spectral methods of automorphic forms**

Automorphic kinds are one of many primary themes of analytic quantity idea. actually, they sit down on the confluence of research, algebra, geometry, and quantity thought. during this publication, Henryk Iwaniec once more monitors his penetrating perception, robust analytic strategies, and lucid writing sort. the 1st version of this quantity used to be an underground vintage, either as a textbook and as a revered resource for effects, principles, and references.

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**Example text**

Bg , E1 , . . , E , P1 , . . , Ph satisfying the relations [A1 , B1 ] · · · [Ag , Bg ]E1 · · · E P1 · · · Ph = 1 , mj Ej = 1, where Aj , Bj are hyperbolic motions, [Aj , Bj ] stands for the commutator, g is the genus of Γ\H, Ej are elliptic motions of order mj ≥ 2, Pj are parabolic motions and h is the number of inequivalent cusps. 3. Basic examples 43 The symbol (g; m1 , . . 7) 1− j=1 1 mj +h= |F | . 7) guarantees the existence of Γ with the given signature. Of all the ﬁnite volume groups, the most attractive ones for number theory are the arithmetic groups.

Two points ˆ are said to be equivalent if w ∈ Γz; we then write z ≡ w (mod Γ). z, w ∈ C A set F ⊂ H is called a fundamental domain for Γ if i) F is a domain in H, ii) distinct points in F are not equivalent, ˆ iii) any orbit of Γ contains a point in F (the closure of F in the Ctopology). Any Fuchsian group has a fundamental domain, not, of course, unique. However, all fundamental domains have the same positive volume (possibly inﬁnite) |F | = dμz. F A fundamental domain of a Fuchsian group Γ of the ﬁrst kind can be chosen as a convex polygon.

A. 42) g(u) = 0 +∞ P−1/2+it (u) 1 P−1/2+it (v) g(v) dv t tanh (πt) dt . Here Ps (v) = Ps0 (v) denotes the Legendre function of order m = 0. The spherical functions of order zero are special; they depend only on the hyperbolic distance, namely Us0 (z) = P−s (2u + 1) = Fs (u), say. 44) u(u + 1) F (u) + (2u + 1) F (u) + s(1 − s) F (u) = 0 . 7. 45) for functions depending only on the distance variable u. 46) 1 4π Gs (u) = 1 (ξ(1 − ξ))s−1 (ξ + u)−s dξ . 16)): Gs (u) = Γ(s)2 1 . 7. 46) converges absolutely for Re s = σ > 0.