Abstract not found

We prove that, assuming GRH, there are only finitely many newforms with rational Fourier coefficients and complex multiplication for fixed weight up to twisting. In the sequel, we produce tables of such forms for weights 3 and 4, where this holds unconditionally. We also comment on geometric realizations.

Abstract not found

The classical class number problem of Gauss asks for a classification of all imaginary quadratic fields with a given class number N. The first complete results were for N = 1 by Heegner, Baker, and Stark. After the work of Goldfeld and Gross-Zagier, the task was a finite decision problem for any N. Indeed, after Oesterlé handled N = 3, in 1985 Serre wrote, “No doubt the same method will work for other small class numbers, up to 100, say.” However, more than ten years later, after doing N =5, 6, 7, Wagner remarked that the N = 8 case seemed impregnable. We complete the classification for all N ≤ 100, an improvement of four powers of 2 (arguably the most difficult case) over the previous best results. The main theoretical technique is a modification of the Goldfeld-Oesterlé work, which used an elliptic curve L-function with an order 3 zero at the central critical point, to instead consider Dirichlet L-functions with low-height zeros near the real line (though the former is still required in our proof). This is numerically much superior to the previous method, which relied on work of Montgomery-Weinberger. Our method is still quite computer-intensive, but we are able to keep the time needed for the computation down to about seven months. In all cases, we find that there is no abnormally large “exceptional modulus” of small class number, which agrees with the prediction of the Generalised Riemann Hypothesis.

. In a quadratic number field Q( Ö" D ), D a squarefree integer, with class number 1 any algebraic integer can be decomposed uniquely into primes but for only 21 domains Euclidean algorithms are known. It was shown by Cohn [5] that for D --19 even remainder sequences with possibly non-decreasing norms cannot determine the GCD of arbitrary inputs. We extend this result by showing that there does not even exist an input in these domains for which the GCD computation becomes possible by allowing non-decreasing norms or remainders whose norms are not as small as possible. We then provide two algorithms for computing the GCD of algebraic integers in quadratic number fields Q( Ö" D ). The first applies only to complex quadratic number fields with class number 1, and it is based on a short vector construction in a lattice. Its complexity is O (S 3 ), where S is the number of bits needed to encode the input. The second allows to compute GCDs of algebraic integers in arbitrary number fields ...

Abstract not found

this paper we show that there are analogues to all three statements (a), (b), and (c) if we introduce the appropriate "real" structure and level structure on the abelian varieties. Consider a2 -dimensional abelian variet A with a principal polarization H and with a homomorphism # : O d # End R (A) such that# = #( # d) acts as an anti-holomorphic endomorphism of A. Such a triple (A, H,# ) will be referred to (in 4) as a principally polarized abelian surface with anti-holomorphic multiplication by O d There is an associated notion of a level N structure on such a triple. (See 7.) If N is a positive integer, let #N = SL(2 , O d )[N]and#(N)=Sp(4,Z)[N ] be the principal congruence subgroups of SL(2 , O d )andSp(4,Z) respectively, with level N. Fix d<0 and assume that Q( # d) has class number one. Suppose N # 3 and, if d # 1(mod 4), then assume also that N is even. For simplicity(and for the purposes of this introduction only), assume that d is invertible (mod N). In Theorems 6.3 and 7.5 we prove the following analogues of statements (a) and (b) above. 1. School of Mathematics, Institute for Advanced Study, PrincetonN] . Research partially supported by N SF grants # DMS 9626616 and DMS 9900324. 2. Dept. of Mathematics, Haverford College, Haverford PA.. 1 1.1.Theorem.Given d, N as above, the moduli space V (d, N) of principally polarized abelian surfaces with anti-holomorphic multiplication by O d and level N structure consists of finitely many copies of the arithmetic quotient #N \H 3 . These copies are indexed by a certain (nonabelian cohomology) set H 1 (C/R, #(N)). oreover this moduli space V (d, N) coincides with the set of real points X R of a quasi-projective complex algebraic variety which is defined over Q. The keyobservation is t...

A nearly zero-free region for L(s, χ), and Siegel’s theorem We used positivity of the logarithmic derivative of ζq to get a crude zero-free region for L(s, χ). Better zero-free regions can be obtained with some more effort by working with the L(s, χ) individually. The situation is most satisfactory for

A nearly zero-free region for L(s, χ), and Siegel’s theorem We used positivity of the logarithmic derivative of ζq to get a crude zero-free region for L(s, χ). Better zero-free regions can be obtained with some more effort by working with the L(s, χ) individually. The situation is most satisfactory for

The dissertation investigates the arithmetic of singular K3 surfaces. It also gives an extensive analysis of Hecke eigenforms with rational coefficients and complex multiplication. The thesis starts with a brief introduction on K3 surfaces. I emphasize the notion of singular K3 surfaces over C, i.e. K3 surfaces with maximal Picard number ϱ = 20. In particular, I recall the theory of Shioda-Inose [S-I] and Livné’s theorem that a singular K3 surface over Q is modular [L]. I then formulate the question which newforms might actually be associated to singular K3 surfaces over Q. This motivates the study of Hecke eigenforms with rational Fourier coefficients and complex multiplication (CM) of arbitrary weight. This investigation is the content of the second chapter of the thesis. The main result is the following Theorem For fixed weight k, the newforms with rational Fourier coefficients and CM are, up to twisting, in 1:1-correspondence with the imaginary quadratic fields K with exponent