1975 U.S. - JAPAN SEMINAR APPLICATIONS OF AUTOMORPHIC FORMS TO NUMBER THEORY Ann Arbor, Michigan Abstracts and Summaries of Reports by Participants and Observers TABLE OF CONTENTS Page Tetsuya Asai On certain Dirichlet series associated with Hilbert ' modular forms and Rankin s me t h ~ d John Coates On the values of the p-adic zeta functions at the odd positive integers Stephen Gelbart On the construction of modular forms of half integral weight Larry Joel Goldstein The arithmetic of Hecke differentials Hir~akiH ijikata On the theta series obtained from certain orders of quaternion algebras Ya sutaka Ihara Some fundamental groups in arithmetic of algebraic curves over finite fields Masao Koike On the congruences between Eisenstein series and cusp forms Tomio Kubota On a generalized Fourier transformation R. P. Langlands Shimura varieties and the Selberg trace formula Erik A. Lippa Hecke eigenforms of degree 2 : Fourier coefficients, Dirichlet series, and Euler products Yoshi nobu Nakai On a Q-Weyl sum Douglas Niebur The Bessel function expansion of Fourier coefficients of modular forms Takayuki Oda Construction of some transcendental 2-cycles on certain Hilbert wdular surfaces M ~ S ~i I IOIh trl On reductions and zeta functions of varieties obtaj ned from TABLE OF CONTENTS (continued) Page 112 Arnold Pizer Representing modular forms by generalized theta series 122 K. A. Ribet Division points of J*(N) 131 Hiroshi Saito Automorphic forms and algebraic extensions ~f number fields Kuang-yen Shih On certain twisted modular equations Hideo Shimizu Some examples of new forms Goro Shimura On the special values of certain non-holomsrphic derivatives of Hilbert modular forms Takuro Shintani On "liftings" of holomorphic automorphic forms (a representation theoretic interpretation of the recent work of H. ~ a i t o ) - Non-Alphabetically-Arranged Supplement Herve Jacquet From G L ( ~ )t o G L ( ~ ) S. Lang and H. Trotter Distribution of Frobenius automorphisms in GL2-extensions of the rationals A. P. Ogg On the reduction modulo p of x ~ ( ~ M ) A. 0. L. Atkin Modular forms of weight one, and supersingular equations ON CERTAIN DIRICHLET SERIES ASSOCIATED WITH HILBERT MODULAR FORMS AND RANKIN' S METHOD Tetsyya Asai 1, Let F be a real quadratic number field with the class number one and the discriminant D, and @ be the ring of integers, Let be a Dirichlet series ordinarily associated with a Hilbert modular cusp form of weight k with respect to GL~(~?'),a nd so ~(02)'s are the Fourier coefficients of the cusp form, where Ut runs over all non-zero integral ideals of 6, It is well known that D(s) is holomorphically continued to the whole complex s-plane and satisfies a functional equation, Here we are concerned about a certain subseries of D(s), in which the summation is restricted on the rational integers only, This series has also the analytic continuation and a functional equation, Namely, if we put and - C , where is the Riemann zeta function and s1 s-k+l then we get Theorem 1. G*(s) can be holomorp-h ically continued t o the whole s- plane except possible simple poles at s k, k-1, and'satisf ies the functional - equation G*(s) C*(2k-1-s) , This can be proved by so-called Rankin's method, -1- 2. Let us assume that D(s) has the Euler product expansion g w , where runs over all prime ideals of then we can obtain the Euler product expansion of G(s), too. For this purpose, let us put , $ , where the roots d depend on and so we can rewrite (3) as follows r - - k-l Here pes are rational primes and we put V p*S ( and we also put v p for later use ). Then we can see Theorem 2. If D(s) has the Euler product expansion (4), then G(s) has also its ~ulerp-r oduct expansion 3. Rankines method ( or convolution ) gives also the analytic continuation of the Dirichlet series , (Re s ) k ) where SF is the Dedekind zeta function of F, s1 = s-k+l , and denotes m. the conjugate of In fact, the function can be holomorphically continued t o the whole s-plane except possible simple - poles at s k, kbl, and satisfies H*(s) = H*(2k-1-S), Further, if ~ ( s ) has the Euler product expansion (4), H(s) has also its product expansion There is a relation between G(s) and ~(s). Namely, under the assumption that the discriminant D is odd, we can prove Theorem 3. If ~ ( sh)a s the Euler product expansion, H(s) splits as - . follows r ~ ( s ) G(s) G1(s) Here G (s), which is naturally defined by 1 the Euler product expansion, is also analytically continued and satisfies a functional equation. In fact, both functions ~ ( s a)n d Gl(s) are holomorphic - but allowing the possibility that at most one of them has a simple pole at s k. 7 The holomorphy of G (s), which can be also expressed by the series 1 4. Lastly we give a remark on the case of the cusp form derived by Doi- Naganma's map. There are at least two cases. Not only an elliptic modular cusp form induces a Hilbert cusp form relative t o CL ( 8 )( Case 1, 2 (L) say ), but also a cusp form of Neben type ( ( D ) ) induces a Hilbert cusp form relative to C L ~w( ) ( Case /2 , say ). And we assume D(s) has the Euler product expansion. Then we can see the followings. In Case 1, - G(s) is entire ( and Gl(s) has a simple pole at s k ), and in Case 2, G(s) - has a simple pole at s k ( and Gl(s) is entire ). Consequently, these two cases are not overlapped each other. The phenmena of theorems 1 and 2 were observed by Professor G. Shimura almost a decade ago, and the author has completed some details only. The author wishes to express his hearty thanks to Professor G. Shimura for his suggestion and encouragement. The proofs of these and other details w i l l appear elsewhere. Department of Mathematics Faculty of Science Nagoya University ON THE VALUES OF THE p-ADIC ZETA FUNCTIONS AT THE ODD FOSITIVE INTEGERS John Coates Introduction. Let F be a totally real number field, and let G(F,S) be the classical complex zeta function of F. The arithmetic meaning of the values of G(F,S) at the odd positive integers remains one of the most puzzling mysteries of number theory. Of course, it is classical that G(F,S) has a simple pole at s = 1 with residue , where d is the degree of F over the rational field $ h is the class number, Rw the regulator, and A the absolute value of the discriminant of . F over $ Recently, Kchtenbaum [8] asked whether K-theory might provide .. . . an analogous formula for <(F, s) at s = 3,5, More precisely, let 8 , .. be the ring of integers of F and let Km B (m = 91,. ) be the K-groups of 8 in the sense of Quillen [g] ; in particular, Ko8 is the ideal class group and K @ the group of units of 8. For each integer m >- 1, let 1 w,(F) be the largest integer r such that the Galois group of F($) wer F has exponent dividing m ; here is a primitive r-th root of unity. cr If A is a finite group, we write # (A) for the order of A . Then, for n even and positive, Lichtenbaum asked (essentially) whether or not a formula oi' the type is valid, where ~,(n) is a generalized regulator formed from the torsion ! . free subgroup of K2 n+lQ The precise definition of ~ ~ ( nis) du e to Borel, who earlier had sho-i that K2,8 is finite, and that the torsion I % free subgroup of n+l (9 has rank d (for n even and positive). More- over, Borel has recently proven that the right and left hand sides of (1) do in fact only differ by a multiple of a ratfonal number. But it seems un- likely that his methods w i l l be able to determine this rational nwnber pre- cisely. LRt p be a prime number. In the present lecture, we propose a p-adic analogue of Iiichtenbaumts conjecture, in which G(F, 1 +n) is replaced by the n l u e of an appropriate p-adic L-function at l + n and ~,(n) is re- placed by a higher p-adic regulator RP(n). That such an analogue might be plausible is suggested by kopoldt 's theorem that, for abelian extensions F , of $ the p-adic zeta function of F has a simple pole at s = 1 with residue vhere R is Leopoldt 's p-adic regulator (see [6], $4 ), S is the set P . of primes of F above p , and Np is the absolute norm of P One suspects that the same result is true for a l l F, but this is unknown at present. While we cannot prove this p-adic analogue of (1) in any single case yet, we do give some evidence in favour of it by using Iwasawa's theory of 2 -extensions of number fields. We should also point out that our P higher p-adic regulator is only defined up t o a p-adic unit, so that our conjecture really only concerns the power of p occuring in the value of the p-adic L-function at l + n . , l- 2-- adic regulators. Let 2 Qp be the p-adic integers and the p-adic nuxibers, respectively. If a, #3 are non-zero p-adic numbers, a- f3 will . mean that a/#3 is a unit in ;I P As in § 1, F w i l l denote an arbitrary totally real number field. We - /fi , begin by giving an algebraic interpretation of R up to assuming that R f 0. For each P E S , let U P be the units of the completion of P , F at p and let U be the elements of U which are 1 mod p . P,1 P Then U is a profinite p-group, and hence a 22 -module. kt E be the P, 1 P , units of the ring of integers (9 of F and let E c E be those units . which are 1 mod P for a l l p e S Let