# Automorphic Forms and Geometry of Arithmetic Varieties by K. Hashimoto, Y. Namikawa

By K. Hashimoto, Y. Namikawa

Automorphic varieties and Geometry of mathematics forms offers with the measurement formulation of varied automorphic kinds and the geometry of mathematics types. The relation among basic equipment of acquiring size formulation (for cusp forms), the Selberg hint formulation and the index theorem (Riemann-Rochs theorem and the Lefschetz fastened element formula), is examined.

Comprised of 18 sections, this quantity starts off via discussing zeta services linked to cones and their particular values, by means of an research of cusps on Hilbert modular forms and values of L-functions. The reader is then brought to the size formulation of Siegel modular kinds; the graded earrings of modular types in different variables; and Selberg-Iharas zeta functionality for p-adic discrete teams. next chapters concentrate on zeta services of finite graphs and representations of p-adic teams; invariants and Hodge cycles; T-complexes and Ogatas zeta 0 values; and the constitution of the icosahedral modular workforce.

This publication can be an invaluable source for mathematicians and scholars of arithmetic.

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Ramanujan Birth Centenary Int. , TIFR, Bombay, 1988. H. Shimizu, On discontinuous groups operating on the product of upper half planes, Ann. , 77 (1963), 33-71. T. Shintani, On zeta-functions associated with the vector space of quadratic forms, J. Fac. , Univ. Tokyo, 22 (1975), 25-65. , On evaluation of zeta functions of totally real algebraic number fields at non-positive integers, J. Fac. , Univ. Tokyo, 23 (1976), 393-417. C. L. Siegel, Über die Zetafunktionen indefiniter quadratischer Formen, Math.

52 R. Tsushima /j is an intersection of Ex and another irreducible component in E which we denote by E2. So we have Since X2(N) has (1/2)JV4 ΠρϋνΟ — P~*) cusps of degree one and X^N) has (l/2)iV 2 n p N v(l-^- 2 ) cusps, there are (1/12)ΛΓ 7 Πρ,*(1-/τ 2 )(1-- / τ 4 ) points in X2(N) where three irreducible components of E intersect and there are (l/8)7V7I~f piiv(l —JP~2)(1 — P~*) rational curves where two irreducible components of E intersect. Therefore the first term in the fourth line in (*) is calculated as \2-iiei)\2-ii

Sczech This number was studied, among others by Ehlers [3] and Satake [5]. In [6], we established the following explicit expression for ψ(Μ, V, x). Theorem 4. Ψ(Μ, v, x)=(-LY 2 (-1)* Σ s; Π cjxT). \ Z / o ρτ>0 τζσ Aside from the missing signature term and the additional factor 2~n, the only difference between this expression and the corresponding expression in Theorem 3 is the slightly different definition of the trigonometric functions Ck(u): CM = φ) +1, Ck(u) = ck(u) foik>l. Despite these differences, we conjecture that Conjecture.