Search

The systole of a closed Riemannian manifold is the minimal length of a non-contractible closed loop. We give a uniform lower bound for the systole for large classes of simple arithmetic locally symmetric orbifolds. We establish new bounds for the translation length of a semisimple element x in SL_n(R) in terms of its associated Mahler measure. We use these geometric methods to prove the existence of extensions of number fields in which fixed sets of primes have certain prescribed splitting behavior.

The Arakelov-Zhang pairing ⟨ ψ , ϕ ⟩ \langle \psi ,\phi \rangle is a measure of the “dynamical distance” between two rational maps ψ \psi and ϕ \phi defined over a number field K K . It is defined in terms of local integrals on Berkovich space at each completion of K K . We obtain a simple expression for the important case of the pairing with a power map, written in terms of integrals over Julia sets. Under certain disjointness conditions on Julia sets, our expression simplifies to a single canonical height term; in general, this term is a lower bound. As applications of our method, we give bounds on the difference between the canonical height h ϕ h_\phi and the standard Weil height h h , and we prove a rigidity statement about polynomials that satisfy a strong form of good reduction.

When ordered by discriminant, it is known that about 83% of quartic fields over Q have associated Galois group S_4, while the remaining 17% have Galois group D_4. We study these proportions over a general number field F. We find that asymptotically 100% of quadratic number fields have more D_4 extensions than S_4 and that the ratio between the number of D_4 and S_4 quartic extensions is biased arbitrarily in favor of D_4 extensions. Under GRH, we give a lower bound that holds for general number fields., Fixed a typo with Theorem 1.3 that is present in the published version of the paper. The main results remain unchanged

In this article, we study the Piltz divisor problem, which is sometimes called the generalized Dirichlet divisor problem, over number fields. We establish an identity akin to Voronoï’s formula concerning the error term in the Dirichlet divisor problem.

WOS: 000454742000015, We show that the height density of a finite sum of fractions is zero. In fact, we give quantitative estimates in terms of the height function. Then, we focus on the unit fraction solutions in the ring of integers of a given number field. In particular, we prove that finitely many representations of 1 as a sum of unit fractions determines the field of rational numbers among all real number fields. Finally, using non-standard methods, we prove some density and finiteness results on a finite sum of fractions., Nesin Mathematics Village, The authors would like to thank Nesin Mathematics Village for their support and warm hospitality.

Let Λ ⊂ R n \Lambda \subset \mathbb R^n be an algebraic lattice coming from a projective module over the ring of integers of a number field K K . Let Z ⊂ R n \mathcal Z \subset \mathbb R^n be the zero locus of a finite collection of polynomials such that Λ ⊈ Z \Lambda \nsubseteq \mathcal Z or a finite union of proper full-rank sublattices of Λ \Lambda . Let K 1 K_1 be the number field generated over K K by coordinates of vectors in Λ \Lambda , and let L 1 , … , L t L_1,\dots ,L_t be linear forms in n n variables with algebraic coefficients satisfying an appropriate linear independence condition over K 1 K_1 . For each ε > 0 \varepsilon > 0 and a ∈ R n \boldsymbol a \in \mathbb R^n , we prove the existence of a vector x ∈ Λ ∖ Z \boldsymbol x \in \Lambda \setminus \mathcal Z of explicitly bounded sup-norm such that ‖ L i ( x ) − a i ‖ > ε \begin{equation*} \| L_i(\boldsymbol x) - a_i \| > \varepsilon \end{equation*} for each 1 ≤ i ≤ t 1 \leq i \leq t , where ‖ ‖ \|\ \| stands for the distance to the nearest integer. The bound on sup-norm of x \boldsymbol x depends on ε \varepsilon , as well as on Λ \Lambda , K K , Z \mathcal Z , and heights of linear forms. This presents a generalization of Kronecker’s approximation theorem, establishing an effective result on density of the image of Λ ∖ Z \Lambda \setminus \mathcal Z under the linear forms L 1 , … , L t L_1,\dots ,L_t in the t t -torus R t / Z t \mathbb R^t/\mathbb Z^t .

In this paper we investigate the distribution of the number of primes which ramify in number fields of degree d ≤ 5 d \leq 5 . In analogy with the classical Erdős-Kac theorem, we prove for S d S_d -extensions that the number of such primes is normally distributed with mean and variance log ⁡ log ⁡ X \log \log X .

Let $\Phi^\infty(d)$ denote the set of finite abelian groups that occur infinitely often as the torsion subgroup of an elliptic curve over a number field of degree $d$. The sets $\Phi^\infty(d)$ are known for $d\le 4$. In this article we determine $\Phi^\infty(5)$ and $\Phi^\infty(6)$., Comment: Minor edits. 10 pages

Fix a number field K K . For each nonzero α ∈ Z K \alpha \in \mathbf {Z}_K , let ν ( α ) \nu (\alpha ) denote the number of distinct, nonassociate irreducible divisors of α \alpha . We show that ν ( α ) \nu (\alpha ) is normally distributed with mean proportional to ( log ⁡ log ⁡ | N ( α ) | ) D (\log \log |N(\alpha )|)^{D} and standard deviation proportional to ( log ⁡ log ⁡ | N ( α ) | ) D − 1 / 2 (\log \log {|N(\alpha )|})^{D-1/2} . Here D D , as well as the constants of proportionality, depend only on the class group of K K . For example, for each fixed real λ \lambda , the proportion of α ∈ Z [ − 5 ] \alpha \in \mathbf {Z}[\sqrt {-5}] with \[ ν ( α ) ≤ 1 8 ( log ⁡ log ⁡ N ( α ) ) 2 + λ 2 2 ( log ⁡ log ⁡ N ( α ) ) 3 / 2 \nu (\alpha ) \le \frac {1}{8}(\log \log {N(\alpha )})^2 + \frac {\lambda }{2\sqrt {2}} (\log \log {N(\alpha )})^{3/2} \] is given by 1 2 π ∫ − ∞ λ e − t 2 / 2 d t \frac {1}{\sqrt {2\pi }} \int _{-\infty }^{\lambda } e^{-t^2/2}\, \mathrm {d}t . As further evidence that “irreducibles play a game of chance”, we show that the values ν ( α ) \nu (\alpha ) are equidistributed modulo m m for every fixed m m .

The endomorphism algebra X f X_f attached to a non-CM primitive cusp form f f of weight at least two is a 2-torsion element in the Brauer group of a number field F F . We give formulas for the ramification of X f X_f locally at primes lying above the odd supercuspidal primes of f f . We show that the local Brauer class is determined by the underlying local Galois representation together with an auxiliary Fourier coefficient.

— Let E be an elliptic curve defined over a number field F and K/F a quadratic extension. For a point P ∈ E(F) that is a local trace for every completion of K/F, we find necessary and sufficient conditions for P to lie in the image of the global trace map. These conditions can then be used to determine whether a quadratic twist of E, as a genus one curve, has rational points. In the case of quadratic twists of genus one modular curves X0(N) with squarefree N , the existence of rational points corresponds to the existence of Q-curves of degree N defined over K.

In this article we construct for each integer $n\ge 2$ an abelian variety $A$ of dimension $n$ defined over a number field for which there exists a symmetric slope sequence of length $2n$ that does not appear as the slope sequence of $\widetilde A$ for all good reductions $\widetilde A$ of $A$.

Given a number field $K$ and a finite set $S$ of places of $K$, the first main result of this paper shows that the quadratic rational maps $\phi:{\mathbb P}^1\to{\mathbb P}^1$ defined over $K$ which have good reduction at all places outside $S$ comprise a Zariski-dense subset of the moduli space ${\mathcal M}_2$ parametrizing all isomorphism classes of quadratic rational maps. We then consider quadratic rational maps with double unramified fixed-point structure, and our second main result establishes a geometric Shafarevich-type non-Zariski-density result for the set of such maps with good reduction outside $S$. We also prove a variation of this result for quadratic rational maps with unramified 2-cycle structure., Comment: This version fixes a few LaTeX errors

Let ϕ \phi be a morphism of P N \mathbb {P}^N defined over a number field K . K. We prove that there is a bound B B depending only on ϕ \phi such that every twist of ϕ \phi has no more than B B K K -rational preperiodic points. (This result is analogous to a result of Silverman for abelian varieties.) For two specific families of quadratic rational maps over Q \mathbb {Q} , we find the bound B B explicitly.

Let K / Q K/\mathbf {Q} be an abelian extension and let D D be the absolute value of the discriminant of K K . We show that for each ε > 0 \varepsilon > 0 , the smallest rational prime that splits completely in K K is O ( D 1 4 + ε ) O(D^{\frac 14+\varepsilon }) . Here the implied constant depends only on ε \varepsilon and the degree of K K . This generalizes a theorem of Elliott, who treated the case when K / Q K/\mathbf {Q} has prime conductor.

We will prove that if the unit rank of a number field with cyclic class group is large enough and if the Galois group of its Hilbert class field over Q \mathbb {Q} is abelian, then every generator of its class group is a Euclidean ideal class. We use this to prove the existence of a non-principal Euclidean ideal class that is not norm-Euclidean by showing that Q ( 5 , 21 , 22 ) \mathbb {Q}(\sqrt {5}, \sqrt {21}, \sqrt {22}) has such an ideal class.

We construct many families of nonquadratic algebraic Laurent series with continued fractions having a bounded partial quotients sequence (the diophantine approximation exponent for approximation by rationals is thus 2, agreeing with the Roth value) and with the diophantine approximation exponent for approximation by quadratics being arbitrarily large. In contrast, the Schmidt value (analog of the Roth value for approximations by quadratics, in the number field case) is 3. We calculate diophantine approximation exponents for approximations by rationals for function field analogs of π, e and Hurwitz numbers (which are transcendental) and also give an interesting lower bound (which may be the actual value) for the exponent for approximation by quadratics for the latter two. We do this exploiting the situation when 'folding' or 'negative reversal' patterns of the relevant continued fractions become 'repeating' or 'half-repeating' in even or odd characteristic respectively.

We show that the vector of period ratios of a cubic surface is rational over $Q(\omega)$, where $\omega = \exp(2\pi i/3)$ if and only if the associate abelian variety is isogeneous to a product of Fermat elliptic curves. We also show how to construct cubic surfaces from a suitable totally real quintic number field $K_0$. The ring of rational endomorphisms of the associated abelian variety is $K = K_0(\omega)$., Comment: Implemented the referees thoughtful comments and suggestions. Thankyou!

For any integer n >= 2 and any nonnegative integers r,s with r+2s = n, we give an unconditional construction of infinitely many monic irreducible polynomials of degree n with integer coefficients having squarefree discriminant and exactly r real roots. These give rise to number fields of degree n, signature (r,s), Galois group S_n, and squarefree discriminant; we may also force the discriminant to be coprime to any given integer. The number of fields produced with discriminant in the range [-N, N] is at least c N^(1/(n-1)). A corollary is that for each n \geq 3, infinitely many quadratic number fields admit everywhere unramified degree n extensions whose normal closures have Galois group A_n. This generalizes results of Yamamura, who treats the case n = 5, and Uchida and Yamamoto, who allow general n but do not control the real place., 10 pages; v2: refereed version, minor edits only

Let N ( n , A n , X ) N(n, A_n, X) be the number of number fields of degree n n whose Galois closure has Galois group A n A_n and whose discriminant is bounded by X X . By a conjecture of Malle, we expect that N ( n , A n , X ) ∼ C n ⋅ X 1 2 ⋅ ( log ⁡ X ) b n N(n, A_n, X)\sim C_n\cdot X^{\frac {1}{2}} \cdot (\log X)^{b_n} for constants b n b_n and C n C_n . For 6 ≤ n ≤ 84393 6 \leq n \leq 84393 , the best known upper bound is N ( n , A n , X ) ≪ X n + 2 4 N(n, A_n, X) \ll X^{\frac {n + 2}{4}} , by Schmidt’s theorem, which implies there are ≪ X n + 2 4 \ll X^{\frac {n + 2}{4}} number fields of degree n n . (For n > 84393 n > 84393 , there are better bounds due to Ellenberg and Venkatesh.) We show, using the important work of Pila on counting integral points on curves, that N ( n , A n , X ) ≪ X n 2 − 2 4 ( n − 1 ) + ϵ N(n, A_n, X) \ll X^{\frac {n^2 - 2}{4(n - 1)}+\epsilon } , thereby improving the best previous exponent by approximately 1 4 \frac {1}{4} for 6 ≤ n ≤ 84393 6 \leq n \leq 84393 .

Let K be a number field with unit rank at least four, containing a subfield M such that K/M is Galois of degree at least four. We show that the ring of integers of K is a Euclidean domain if and only if it is a principal ideal domain. This was previously known under the assumption of the generalized Riemann Hypothesis for Dedekind zeta functions. We prove this unconditionally.

We prove a conjecture of Yamauchi which states that the level N N for which the new part of J 0 ( N ) J_0(N) is Q \mathbb {Q} -isogenous to a product of elliptic curves is bounded. We also state and partially prove a higher-dimensional analogue of Yamauchi’s conjecture. In order to prove the above results, we derive a formula for the trace of Hecke operators acting on spaces S n e w ( N , k ) S^{new}(N,k) of newforms of weight k k and level N . N. We use this trace formula to study the equidistribution of eigenvalues of Hecke operators on these spaces. For any d ≥ 1 , d\geq 1, we estimate the number of normalized newforms of fixed weight and level, whose Fourier coefficients generate a number field of degree less than or equal to d . d.

Let $K$ be a fixed number field, and assume that $K$ is Galois over $\qq$. Previously, the author showed that when estimating the number of prime ideals with norm congruent to $a$ modulo $q$ via the Chebotar\"ev Density Theorem, the mean square error in the approximation is small when averaging over all $q\le Q$ and all appropriate $a$. In this article, we replace the upper bound by an asymptotic formula. The result is related to the classical Barban-Davenport-Halberstam Theorem in the case $K=\qq$., Comment: Preprint of an old paper

We prove that if at least one of the prime divisors of an odd integer U ≥ 3 U\geq 3 is equal to 3 3 mod 4 4 , then the ideal class group of the imaginary quadratic field Q ( 1 − 4 U n ) \mathbf {Q}(\sqrt {1-4U^n}) contains an element of order n n .

Suppose that f (resp. g) is a modular form of integral (resp. half-integral) weight with coefficients in the ring of integers O K of a number field K. For any ideal m C O K , we bound the first prime p for which f | Tp (resp. g | T p 2) is zero (mod m). Applications include the solution to a question of Ono (2001) concerning partitions.

We give a criterion for the vanishing of the Iwasawa A-invariants of totally real number fields K based on the class number of K(ζp) by evaluating the p-adic L-functions at s = -1.

Let K be a number field with r real places and s complex places, and let O K be the ring of integers of K. The quotient [H 2 ] r X [H 3 ] s /PSL 2 (O K ) has h K cusps, where h K is the class number of K. We show that under the assumption of the generalized Riemann hypothesis that if K is not Q or an imaginary quadratic field and if i ∉ K, then PSL 2 (O K ) has infinitely many maximal subgroups with h K cusps. A key element in the proof is a connection to Artin's Primitive Root Conjecture.

Let A={a_s+n_sZ}_{s=1}^k be a finite system of arithmetic sequences which forms an m-cover of Z (i.e., every integer belongs at least to m members of A). In this paper we show the following sharp result: For any positive integers m_1,...,m_k and theta in [0,1), if there is a subset I of {1,...,k} such that the fractional part of sum_{s in I}m_s/n_s is theta, then there are at least 2^m such subsets of {1,...,k}. This extends an earlier result of M. Z. Zhang and an extension by Z. W. Sun. Also, we generalize the above result to m-covers of the integral ring of any algebraic number field with a power integral basis., Comment: 7 pages, to appear in Proc. Amer. Math. Soc

We prove an asymptotic formula for the number of sextic number fields with Galois group S 3 and absolute discriminant < X. In addition, we give an interpretation of the constant in the formula in terms of the asymptotic densities of given local completions among these sextic fields. Our proof gives analogous results when we count S 3 -sextic extensions of any number field, and also when finitely many local completions have been specified for the sextic extensions.

Let K K be a number field, K ¯ \overline {K} an algebraic closure of K K , G K G_K the absolute Galois group Gal ⁡ ( K ¯ / K ) \operatorname {Gal}(\overline {K}/K) , K a b K_{ab} the maximal abelian extension of K K and E / K E/K an elliptic curve defined over K K . In this paper, we prove that if all 2-torsion points of E / K E/K are K K -rational, then for each σ ∈ G K \sigma \in G_K , E ( ( K a b ) σ ) E((K_{ab})^{\sigma }) has infinite rank, and hence E ( K ¯ σ ) E(\overline {K}^{\sigma }) has infinite rank.

Some necessary conditions are given on infinitely oscillating real functions and infinite discrete sets of real numbers so that first-order expansions of the field of real numbers by such functions or sets do not define N \mathbb N . In particular, let f : R → R f\colon \mathbb R\to \mathbb R be such that lim x → + ∞ f ( x ) = + ∞ \lim _{x\to +\infty }f(x)=+\infty , f ( x ) = O ( e x N ) f(x)=O(e^{x^N}) as x → + ∞ x\to +\infty for some N ∈ N N\in \mathbb N , ( R , + , ⋅ , f ) (\mathbb R, +,\cdot ,f) is o-minimal, and the expansion of ( R , + , ⋅ ) (\mathbb R,+,\cdot ) by the set { f ( k ) : k ∈ N } \{\,f(k):k\in \mathbb {N}\,\} does not define N \mathbb N . Then there exist r > 0 r>0 and P ∈ R [ x ] P\in \mathbb R[x] such that f ( x ) = e P ( x ) ( 1 + O ( e − r x ) ) f(x)=e^{P(x)}(1+O(e^{-rx})) as x → + ∞ x\to +\infty .

If D D is a division algebra with its center a number field K K and with an involution of the second kind, it is unknown if the group S U ( 1 , D ) / [ U ( 1 , d ) SU(1,D)/[U(1,d) , U ( 1 , D ) ] U(1,D)] is trivial. We show that, by contrast, if K K is a function field in one variable over a number field, and if D D is an algebra with center K K and with an involution of the second kind, the group S U ( 1 , D ) / [ U ( 1 , d ) , U ( 1 , D ) ] SU(1,D)/[U(1,d),U(1,D)] can be infinite in general. We give an infinite class of examples.

Let N ( d , G , X ) N(d, G, X) be the number of degree d d number fields K K with Galois group G G and whose discriminant D K D_K satisfies | D K | ≤ X |D_K| \le X . Under standard conjectures in diophantine geometry, we show that N ( 4 , A 4 , X ) ≪ ϵ X 2 / 3 + ϵ N(4, A_4, X) \ll _\epsilon X^{2/3+\epsilon } , and that there are ≪ ϵ N 3 + ϵ \ll _\epsilon N^{3+\epsilon } monic, quartic polynomials with integral coefficients of height ≤ N \le N whose Galois groups are smaller than S 4 S_4 , confirming a question of Gallagher. Unconditionally we have N ( 4 , A 4 , X ) ≪ ϵ X 5 / 6 + ϵ N(4, A_4, X) \ll _\epsilon X^{5/6 + \epsilon } , and that the 2 2 -class groups of almost all Abelian cubic fields k k have size ≪ ϵ D k 1 / 3 + ϵ \ll _\epsilon D_k^{1/3+\epsilon } . The proofs depend on counting integral points on elliptic fibrations.

We find full invariants for detecting non--zero elements in ${L}_0(F(t))\otimes \Q$, this group plays an important role in topology in the work done by Casson and Gordon., Comment: 8 pages

In this paper, we investigate the Hansen-Mullen conjecture with the help of some formal series similar to the Artin-Hasse exponential series over p-adic number fields and the estimates of character sums over Galois rings. Given n we prove, for large enough q, the Hansen-Mullen conjecture that there exists a primitive polynomial f(x) = x n - a 1 x n-1 + ... + (-1) n a n over Fq of degree n with the m-th (0 < m < n) coefficient a m fixed in advance except when m = n+1/2 if n is odd and when m = n/2, n/2 + 1 if n is even.

This paper describes the Hochschild cohomology ring of a selfinjective algebra Λ \Lambda of finite representation type over an algebraically closed field K K , showing that the quotient HH ∗ ⁡ ( Λ ) / N \operatorname {HH}^*(\Lambda )/\mathcal {N} of the Hochschild cohomology ring by the ideal N {\mathcal N} generated by all homogeneous nilpotent elements is isomorphic to either K K or K [ x ] K[x] , and is thus finitely generated as an algebra. We also consider more generally the property of a finite dimensional algebra being selfinjective, and as a consequence show that if all simple Λ \Lambda -modules are Ω \Omega -periodic, then Λ \Lambda is selfinjective.

The subject of this paper is a quaternion algebra over a quadratic extension of a totally real algebraic number field. We generalize an important technical lemma of Shimura, on which a certain period of automorphic forms depends.

We modify the abc-conjecture for number fields K in order to make the support (like the height) well-behaved under field extensions. We show further that the exponent μ > 1 of the absolute value D K of the discriminant cannot be replaced by μ = 1, and even that an arbitrarily large power of log D K must be present.

Let N be an imaginary abelian number field. We know that h-, the relative class number of N, goes to infinity as fN, the conductor of N, approaches infinity, so that there are only finitely many imaginary abelian number fields with given relative class number. First of all, we have found all imaginary abelian number fields with relative class number one: there are exactly 302 such fields. It is known that there are only finitely many CMfields N with cyclic ideal class groups of 2-power orders such that the complex conjugation is the square of some automorphism of N. Second, we have proved in this paper that there are exactly 48 such fields.

We construct a quadratic relation between cusp forms of weight two on four genus 1 subgroups of SL2 (Z). Two of the subgroups are congruence and two are noncongruence. We then generalize this to subgroups of F(N) of index 2. Let A be a finite-index subgroup of SL2(Z), and let XA be the associated Riemann surface (*/A. If K is a number field and C/K is a projective nonsingular curve, we say that C/K is a model for XA if there exists a morphism : C * JPl defined over K and a complex analytic isomorphism p: XA 0(QC) such that p(oo) is rational over K and XA P 0 (C) (1) 1 jjc * /SL2 (Z) i PI1 (C) commutes. (Here, the left hand arrow is the natural projection map.) It is known that for any A one can find a C/K as above, and, in fact, Belyi's theorem [B] tells us that the converse is true as well. In light of this, one may wish to study the usual arithmetic objects associated with congruence groups (i.e., Hecke operators, Fourier coefficients, and so forth) for arbitrary subgroups of finite index in SL2(Z). While it is true that on a general noncongruence subgroup, Hecke operators and Fourier coefficients are not related, there are nevertheless results concerning both. (See [Be] for Hecke operators on noncongruence subgroups and [S] for Fourier coefficients. From [Be], we see that Hecke operators on noncongruence subgroups give essentially no new information, but the Atkin-Swinnerton-Dyer congruences of [S] are highly nontrivial.) What has not been previously studied, however, is the relationship between cusp forms on related congruence and noncongruence groups. Since cusp forms on congruence groups are to some extent understood, any explicit information we obtain may be viewed as shedding light on the noncongruence forms. In particular, an actual polynomial relationship between forms would lead to an infinite number of relations between Fourier coefficients. One may view this as a Received by the editors November 16, 1998. 1991 Mathematics Subject Classification. Primary lIF1I; Secondary 1F30.

This paper presents an indecomposable finite-dimensional division algebra of p-power index that decomposes over a prime-to-p degree field extension, obtained by adjoining p-th roots of unity to the base. This shows that the theory of decomposability has an arithmetic aspect. Suppose F is a field and D is an indecomposable F-division algebra, that is, a division algebra that cannot be expressed as the tensor product of two nontrivial F-division algebras. It is easy to see that the (Schur) index of D must be a power of some prime p. In "Problem 6" of [Sa], Saltman asks if in general D remains indecomposable upon arbitrary prime-to-p extension. At issue is the nature of indecomposability, in particular whether or not it is "geometric". For example in [K], Karpenko showed a certain generic class of division algebras are indecomposable by computing the degrees of cycles on their Brauer-Severi varieties. As noted in [K], it is immediate from the geometric nature of the proof that these algebras remain indecomposable over all prime-to-p extensions. This paper presents an indecomposable division algebra that decomposes over a prime-to-p extension, namely the cyclotomic extension defined by p-th roots of unity. Thus it is proved that (in)decomposability can have an arithmetic aspect. Let p be an odd prime of Q, let k be a number field that does not contain a pth root of unity, and let k[s, t] be the polynomial ring in two variables over k. Define v :k[s, t] -+Z fDlZ, f --* (a, b) where b is smallest such that f E (tb) and a is smallest such that f E (Sa,tb+l). The map v is a valuation, with value group Z 20 Z ordered reverse lexicographically, so (a, b) < (a', b') if b < b', or if b = b' and a < a'. The field of iterated power series F =k((s))((t)) is Henselian with respect to v, with valuation ring R = k[[s]] +tk((s))[[t]] c k((s))[[t]] . R is a non-Noetherian 2-dimensional Henselian local ring, with maximal ideal (s) = (s, t) and residue field k. The ideal (s) properly contains the (infinitely generated) prime ideal tk((s))[[t]] = (t, t, I,. t Received by the editors June 10, 1998 and, in revised form, October 6, 1998. 1991 Mathematics Subject Classification. Primary 16K20; Secondary 1R37. ?)2000 American Mathematical Society

We prove a complex function field analogue of Szpiro’s conjecture for hyperelliptic curves and some applications. The cases of function fields of positive characteristic and number fields are discussed briefly.

We show that a conjecture by Lang and Vojta regarding integral points on varieties of logarithmic general type implies the existence of a uniform bound on the number of stably S S -integral points on an elliptic curve over a degree- d d number field.

For given elements a1,... , ak and 3 belonging to the ring of integers A of a number field we consider the set of all k-tuples (al,... , ak) in Nk for which Ek l caiai divides Ekcaiai for any z E A, and prove under some reasonable assumptions that the set of solutions is finite. The original motivation for this work comes from a problem raised by J. L. Selfridge (see Guy [1], problem B47) who asks for what pairs (a, b) does 2a-2b divide nanb for all integers n. A related (but more difficult) problem proposed by H. Ruderman asks to show that if 2a 2b divides 3a -3b, then 2a 2b divides n nb for all integers n. This was investigated by B. Velez in [6]. While Ruderman's problem is still open, Selfridge's problem was solved by Pomerance [2], who combined results of Schinzel [4] with Velez's work. It turns out that there are exactly 14 solutions. The problem was also solved by Sun Qi and Zhang Ming Zhi [5]. In this paper we show that the above finiteness result is a particular case of a more general phenomenon. Let IC be a number field, A = AKC its ring of integers and U = UKc its group of units. Let c1,... , ak and p be nonzero elements of A. We consider the set of all k-tuples (al,... , ak) in Nk for which

We use A. Weil’s criterion to prove that all finite dimensional unitary representations of a discrete Kazhdan group are locally rigid. It follows that any such representation is unitarily equivalent to a unitary representation over some algebraic number field. Let Γ be a discrete group satisfying property (T) of Kazhdan, i.e. the trivial representation of Γ is isolated in its unitary dual Γ (cf. [3] and [10], Ch. III); such groups will be referred to as Kazhdan groups in the sequel. It was shown in [15] that for a Kazhdan group Γ every finite dimensional unitary representation is isolated in Γ as well, implying that in effect Γ has only a finite number of inequivalent unitary representations in each (finite) dimension; in other words, Γ has finite representation type with respect to unitary representations. This result was reproved in [16] by a different method which was subsequently used in [4] to obtain an explicit bound on the number of inequivalent n-dimensional unitary representations. Since most examples of Kazhdan groups are lattices in higher rank Lie groups (recently, however, examples of (apparently) different nature have been found, cf. [2], [18]) which enjoy superrigidity (cf. [10]) and hence have finite representation type with respect to all (finite dimensional) representations, it is a natural question to ask if any kind of rigidity property for finite dimensional representations can be inferred directly from property (T). Since an arbitrary Kazhdan group may have an infinite family of pairwise inequivalent noncompletely reducible representations (cf. Example below), one should probably ask about SS-rigidity, i.e. finiteness of representation type with respect to completely reducible representations. We point out that the SS-rigidity of Kazhdan groups would have a number of interesting applications, e.g., combined with Platonov’s conjecture∗ on arithmeticity of SSrigid linear groups (cf. [13, 7.5]) , it would imply that every linear Kazhdan group is a group of arithmetic type, i.e. it is commensurable with the direct product of a finite family of S-arithmetic groups (possibly, for different S). Since a discrete Kazhdan group Γ is necessarily finitely generated, one can consider the corresponding varieties Rn(Γ) and Xn(Γ) of n-dimensional (complex) representations of Γ and of their characters (cf. [7]) ; let θΓ : Rn(Γ) → Xn(Γ) be Received by the editors July 18, 1997 and, in revised form, September 3, 1997. 1991 Mathematics Subject Classification. Primary 22D10; Secondary 22E40, 20G15.

Let F q \mathbb {F}_q be the finite field with q q elements, (2 ⧸ | q {\not |}q ), k = F q ( x ) k= \mathbb {F}_q(x) , K = k ( D ) K=k(\sqrt {D}) where D = D ( x ) = A ( x ) 2 + a D=D(x) =A(x)^2+a is a square-free polynomial in F q [ x ] \mathbb {F}_q[x] with deg ⁡ A ( x ) ≥ 1 \deg A(x)\geq 1 and a ∈ F q ∗ a\in \mathbb { F}_q^* . In this paper several equivalent conditions for the ideal class number h ( O K ) h(O_K) to be one are presented and all such quadratic function fields with h ( O K ) = 1 h(O_K)=1 are determined.

It is known that there are only finitely many imaginary abelian number fields with class numbers equal to their genus class numbers. Here, we determine all the imaginary cyclic fields of 2-power degrees with class numbers equal to their genus class numbers.

In this note, we apply the method of Rodriguez Villegas and Yang (1996) to construct a family of infinite many theta series over the Hilbert-Blumenthal modular surfaces with a common zero. We also relate the non- vanishing of the central L-values of certain Hecke characters of non-biquadratic CM number fields of degree 4 to the nonvanishing of theta functions at CM points in the Hilbert-Blumenthal modular surfaces.

Two famous counterexamples in algebra and number theory are Wang’s counterexample to Grunwald’s Theorem and Amitsur’s noncrossed product division algebra. In this paper we use Wang’s counterexample to construct a noncrossed product division algebra. In the 30’s, Grunwald’s Theorem was used in the proof of a major result of class field theory, that all division algebras over number fields are (cyclic) crossed products. It is ironic that now Grunwald-Wang’s Theorem is the decisive factor in a noncrossed product construction.