We define generalized Mellin transforms of mixed cusp forms, show their convergence, and prove that the function obtained by such a Mellin transform of a mixed cusp form satisfies a certain functional equation. We also prove that a mixed cusp form can be identified with a holomorphic form of the highest degree on an elliptic variety.

Hecke operators are used to investigate part of the $\E_2$-term of the Adams spectral sequence based on elliptic homology. The main result is a derivation of $\Ext^1$ which combines use of classical Hecke operators and $p$-adic Hecke operators due to Serre.

Analogues of van der Waerden's theorem on arithmetic progressions are considered where the family of all arithmetic progressions, $\AP$, is replaced by some subfamily of $\AP$. Specifically, we want to know for which sets $A$, of positive integers, the following statement holds: for all positive integers $r$ and $k$, there exists a positive integer $n= w'(k,r)$ such that for every $r$-coloring of $[1,n]$ there exists a monochromatic $k$-term arithmetic progression whose common difference belongs to $A$. We will call any subset of the positive integers that has the above property {\em large}. A set having this property for a specific fixed $r$ will be called {\em $r$-large}. We give some necessary conditions for a set to be large, including the fact that every large set must contain an infinite number of multiples of each positive integer. Also, no large set $\{a_{n}: n=1,2,\dots\}$ can have $\liminf\limits_{n \rightarrow \infty} \frac{a_{n+1}}{a_{n}} > 1$. Sufficient conditions for a set to be large are also given. We show that any set containing $n$-cubes for arbitrarily large $n$, is a large set. Results involving the connection between the notions of ``large'' and ``2-large'' are given. Several open questions and a conjecture are presented.

We study the splitting of Fermat Jacobians of prime degree $\ell$ over an algebraic closure of a finite field of characteristic $p$ not equal to $\ell$. We prove that their decomposition is determined by the residue degree of $p$ in the cyclotomic field of the $\ell$-th roots of unity. We provide a numerical criterion that allows to compute the absolutely simple subvarieties and their multiplicity in the Fermat Jacobian.

We consider the asymptotic behavior of the moments of the sum-of-digits function of canonical number systems in number fields. Using Delange's method we obtain the main term and smaller order terms which contain periodic fluctuations.

We study metaplectic coverings of the adelized group of a split connected reductive group $G$ over a number field $F$. Assume its derived group $G'$ is a simply connected simple Chevalley group. The purpose is to provide some naturally defined sections for the coverings with good properties which might be helpful when we carry some explicit calculations in the theory of automorphic forms on metaplectic groups. Specifically, we \begin{enumerate} \item construct metaplectic coverings of $G({\Bbb A})$ from those of $G'({\Bbb A})$; \item for any non-archimedean place $v$, show the section for a covering of $G(F_{v})$ constructed from a Steinberg section is an isomorphism, both algebraically and topologically in an open subgroup of $G(F_{v})$; \item define a global section which is a product of local sections on a maximal torus, a unipotent subgroup and a set of representatives for the Weyl group.

Over a decade ago, this author produced class number one criteria for real quadratic fields in terms of prime-producing quadratic polynomials. The purpose of this article is to revisit the problem from a new perspective with new criteria. We look at the more general situation involving arbitrary real quadratic orders rather than the more restrictive field case, and use the interplay between the various orders to provide not only more general results, but also simpler proofs.

In the present paper we consider the problem of finding power integral bases in number fields which are composits of two subfields with coprime discriminants. Especially, we consider imaginary quadratic extensions of totally real cyclic number fields of prime degree. As an example we solve the index form equation completely in a two parametric family of fields of degree $10$ of this type.

The aim of this article is to obtain an upper bound for the exponential sums $\sum e(f(x)/q)$, where the summation runs from $x=1$ to $x=q$ with $(x,q)=1$ and $e(\alpha)$ denotes $\exp(2\pi i\alpha)$. We shall show that the upper bound depends only on the values of $q$ and $s$, %% where $s$ is the number of terms in the polynomial $f(x)$.

The splitting pattern of a quadratic form $q$ over a field $k$ consists of all distinct Witt indices that occur for $q$ over extension fields of $k$. In small dimensions, the complete list of splitting patterns of quadratic forms is known. We show that {\it all\/} splitting patterns of quadratic forms of dimension at most nine can be realized by trace forms.

If $f(x_1,\dots,x_k)$ is a polynomial with complex coefficients, the Mahler measure of $f$, $M(f)$ is defined to be the geometric mean of $|f|$ over the $k$-torus $\Bbb T^k$. We construct a sequence of approximations $M_n(f)$ which satisfy $-d2^{-n}\log 2 + \log M_n(f) \le \log M(f) \le \log M_n(f)$. We use these to prove that $M(f)$ is a continuous function of the coefficients of $f$ for polynomials of fixed total degree $d$. Since $M_n(f)$ can be computed in a finite number of arithmetic operations from the coefficients of $f$ this also demonstrates an effective (but impractical) method for computing $M(f)$ to arbitrary accuracy.

We discuss the $q$-exponential functions $e_q$, $E_q$ for $q$ on the unit circle, especially their continuity in $q$, and analogues of the limit relation $ \lim_{q\rightarrow 1}e_q((1-q)z)=e^z$.

Garrison [3], Forman [2], and Abel and Siebert [1] showed that for all positive integers $k$ and $N$, there exists a positive integer $\lambda$ such that $n^k+\lambda$ is prime for at least $N$ positive integers $n$. In other words, there exists $\lambda$ such that $n^k+\lambda$ represents at least $N$ primes. We give a quantitative version of this result. We show that there exists $\lambda \leq x^k$ such that $n^k+\lambda$, $1\leq n\leq x$, represents at least $(\frac 1k+o(1)) \pi(x)$ primes, as $x\rightarrow \infty$. We also give some related results.

This paper is a study of summability methods that are based on Dirichlet convolution. If $f(n)$ is a function on positive integers and $x$ is a sequence such that $\lim_{n\to \infty} \sum_{k\le n} {1\over k}(f\ast x)(k) =L$, then $x$ is said to be {\it $A_f$-summable\/} to $L$. The necessary and sufficient condition for the matrix $A_f$ to preserve bounded variation of sequences is established. Also, the matrix $A_f$ is investigated as $\ell - \ell$ and $G-G$ mappings. The strength of the $A_f$-matrix is also discussed.

Elliptic units of global function fields were first studied by D.~Hayes in the case that $\deg\infty$ is assumed to be $1$, and he obtained some class number formulas using elliptic units. We generalize Hayes' results to the case that $\deg\infty$ is arbitrary.

This paper studies how the local root numbers and the Weil additive characters of the Witt ring of a number field behave under reciprocity equivalence. Given a reciprocity equivalence between two fields, at each place we define a local square class which vanishes if and only if the local root numbers are preserved. Thus this local square class serves as a local obstruction to the preservation of local root numbers. We establish a set of necessary and sufficient conditions for a selection of local square classes (one at each place) to represent a global square class. Then, given a reciprocity equivalence that has a finite wild set, we use these conditions to show that the local square classes combine to give a global square class which serves as a global obstruction to the preservation of all root numbers. Lastly, we use these results to study the behavior of Weil characters under reciprocity equivalence.

In this paper, we consider Dirichlet series with Euler products of the form $F(s) = \prod_{p}{\bigl(1 + {a_p\over{p^s}}\bigr)}$ in $\Re(s) > 1$, and which are regular in $\Re(s) \geq 1$ except for a pole of order $m$ at $s = 1$. We establish criteria for such a Dirichlet series to be non-vanishing on the line of convergence. We also show that our results can be applied to yield non-vanishing results for a subclass of the Selberg class and the Sato-Tate conjecture.

Let $H$ be the split, adjoint group of type $E_6$ over a $p$-adic field. In this paper we study the restriction of the minimal representation of $H$ to the closed subgroup $PGL_3 \times G_2$.

The primary purpose of this paper is to provide necessary and sufficient conditions for certain quadratic polynomials of negative discriminant (which we call Euler-Rabinowitsch type), to produce consecutive prime values for an initial range of input values less than a Minkowski bound. This not only generalizes the classical work of Frobenius, the later developments by Hendy, and the generalizations by others, but also concludes the line of reasoning by providing a complete list of all such prime-producing polynomials, under the assumption of the generalized Riemann hypothesis ($\GRH$). We demonstrate how this prime-production phenomenon is related to the exponent of the class group of the underlying complex quadratic field. Numerous examples, and a remaining conjecture, are also given.

We generalize Siegel modular forms and construct an exact sequence for the cohomology of locally symmetric varieties which plays the role of the Eichler-Shimura isomorphism for such generalized Siegel modular forms.

Resum\'e. On caract\'erise les corps $K$ satisfaisant le th\'eor\`eme de l'axe principal \`a l'aide de propri\'et\'es des formes carac\-t\'erisation de ces m\^emes corps due \`a Waterhouse, on retrouve \`a partir de l\`a, de fa\c{c}on \'el\'ementaire, un r\'esultat de Becker selon lequel un pro-$2$-groupe qui se r\'ealise comme groupe de Galois absolu d'un tel corps $K$ est engendr\'e par des involutions. ABSTRACT. We characterize general fields $K$, satisfying the Principal Axis Theorem, by means of properties of trace forms of the finite extensions of $K$. From this and Waterhouse's characterization of the same fields, we rediscover, in quite an elementary way, a result of Becker according to which a pro-$2$-group which occurs as the absolute Galois group of such a field $K$, is generated by