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We prove the rank of the group of signatures of the circular units (hence also the full group of units) of ${\mathbb Q}( \zeta_m)^+$ tends to infinity with $m$. We also show the signature rank of the units differs from its maximum possible value by a bounded amount for all the real subfields of the composite of an abelian field with finitely many odd prime-power cyclotomic towers. In particular, for any prime $p$ the signature rank of the units of ${\mathbb Q}( \zeta_{p^n})^+$ differs from $\varphi(p^n)/2$ by an amount that is bounded independent of $n$. Finally, we show conditionally that for general cyclotomic fields the unit signature rank can differ from its maximum possible value by an arbitrarily large amount., Comment: to appear in Pacific Journal of Mathematics

We examine whether it is possible to realize finite groups G as Galois groups of minimally tamely ramified extensions of $$\mathbb {Q}$$ and also specify both the inertia groups and the further decomposition of the ramified primes.

We examine whether it is possible to realize finite groups G as Galois groups of minimally tamely ramified extensions of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Q}$$\end{document}Q and also specify both the inertia groups and the further decomposition of the ramified primes.

Let be the L-function of an elliptic curve E defined over the rational field . Assuming the Birch–Swinnerton-Dyer conjectures, we examine special values of the rth derivatives, L (r)(E, 1, χ), of twists by Dirichlet characters of when L(E, 1, χ)=⋅⋅⋅=L (r−1)(E, 1, χ)=0.

We construct a collection of matrices defined by quadratic residue symbols, termed "quadratic residue matrices", associated to the splitting behavior of prime ideals in a composite of quadratic extensions of $\mathbb{Q}$, and prove a simple criterion characterizing such matrices. We also study the analogous classes of matrices constructed from the cubic and quartic residue symbols for a set of prime ideals of $\mathbb{Q}(\sqrt{-3})$ and $\mathbb{Q}(i)$, respectively., Comment: 10 pages. Comments welcome!

Let K be a field, Br(K) its Brauer group. If L/K is a field extension, then the relative Brauer group Br(L/K) is the kernel of the restriction map resL/K : Br(K)→ Br(L). Relative Brauer groups have been studied by Fein and Schacher. Every subgroup of Br(K) is a relative Brauer group Br(L/K) for some extension L/K, and the question arises as to which subgroups of Br(K) are algebraic relative Brauer groups, i.e. of the form Br(L/K) with L/K an algebraic extension. For example if L/K is a finite extension of number fields, then Br(L/K) is infinite, so no finite subgroup of Br(K) is an algebraic relative Brauer group. In [1] the question was raised as to whether or not the n-torsion subgroup Brn(K) of the Brauer group Br(K) of a field K is an algebraic relative Brauer group. For example, if K is a (p-adic) local field, then Br(K) ∼= Q/Z, so Brn(K) is an algebraic relative Brauer group for all n. A counterexample was given in [1] for n = 2 and K a formal power series field over a local field. For global fields K, the problem is a purely arithmetic one, because of the fundamental local-global description of the Brauer group of a global field. In particular, for a Galois extension L/K of global fields, if the local degree of L/K at every finite prime is equal to n, and is equal to 2 at the real primes for n even, then Br(L/K) = Brn(K). In [1], it was proved that Brn(Q) is an algebraic relative Brauer group for all squarefree n. In [2], the arithmetic criterion above was verified for any number field K Galois over Q and any n prime to the class number of K, so in particular, Brn(Q) is an algebraic relative Brauer group for all n. In [3], Popescu proved that for a global function field K of characteristic p, the arithmetic criterion holds for n prime to the order of the non-p part of the Picard group of K. In this paper we settle the question completely, by verifying the arithmetic criterion for all n and all global fields K. In particular, the n-torsion subgroup of the Brauer group of K is an algebraic relative Brauer group for all n and all global fields K. The proof, an extension of the ideas in [2], reduces to the case n a prime power `. We first carry out the proof for number fields K. The proof for the function field case when ` 6= char(K) is essentially the same as the proof in the number field case. The proof for ` = char(K) appears in [3].

Suppose that E1 and E2 are elliptic curves over the rational field, Q , such that ord s=1 L(E 1 /K,s)≡ ord s=1 L(E 2 /K,s) ( mod 2) for all quadratic fields K/ Q . We prove that their conductors N(E1), and N(E2) are equal up to squares. If rank Z (E 1 (K))≡ rank Z (E 2 (K)) ( mod 2) for all quadratic fields K/ Q , then the same conclusion holds, provided the 2-parts of their Tate–Shafarevich groups are finite.

Suppose that L1(s) and L2(s) are two L-functions whose twists by a set of Dirichlet characters simultaneously vanish (vanish mod p) at a critical point. We examine the extent to which this property determines the L-functions in the cases of L-functions of elliptic curves, of number fields, and of curves over finite fields.

Let E be an elliptic curve over q with L-function {\small $L_E(s)$}. We use the random matrix model of Katz and Sarnak to develop a heuristic for the frequency of vanishing of the twisted L-functions {\small $L_E(1, \chi)$}, as {\small $\chi$} runs over the Dirichlet characters of order 3 (cubic twists). The heuristic suggests that the number of cubic twists of conductor less than X for which {\small $L_E(1, \chi)$} vanishes is asymptotic to {\small $b_E X^{1/2} \log^{e_E}{X}$} for some constants {\small $b_E, e_E$} depending only on E. We also compute explicitly the conjecture of Keating and Snaith about the moments of the special values {\small $L_E(1, \chi)$} in the family of cubic twists. Finally, we present experimental data which is consistent with the conjectures for the moments and for the vanishing in the family of cubic twists of {\small $L_E(s)$}.

We show that substantially more than a quarter of the odd integers of the form $pq$ up to $x$, with $p,q$ both prime, satisfy $p\equiv q\equiv 3 \pmod 4$.

Proof. An+1 is a finite Gn+1-module so that An+1 = B ∪ C where B is the set of those elements in An+1 not fixed by any non-trivial element of Gn+1, and C = An+1 \ B. Since Gn+1 is a cyclic group it follows that every element of C is fixed by the the subgroup of order p in Gn+1, and so C ⊆ An = An. The opposite inclusion is clear so C = An. Counting we have, |An+1| = |B|+ |An|. Since B is a union of orbits each of which contains p elements it follows that |An+1| ≡ |An| (mod p).

Olga Taussky’s interest in class field theory began when she was a student with Ph. Furtwangler at the University of Vienna (she received her doctorate under his supervision in 1930). At this time, E. Artin, using his general reciprocity law, reformulated the “Principal Ideal Theorem” as a purely grouptheoretic question regarding the triviality of the “Verlagerung” homomorphism from a finite group to its commutator subgroup. Since Furtwangler had been developing group theoretic methods in his own number theoretic investigations, Artin communicated this new idea to him. This was considered to be one of the central problems in the subject at the time, and so there was a good deal of excitement when Furtwangler eventually succeeded in proving the result using this new approach. Olga Taussky had been looking for a thesis topic and Furtwangler suggested several questions related to the finer structure of the Principal Ideal Theorem. The optimistic atmosphere, and the prospect of contributing at the frontiers of class field theory, sparked in Olga a deep interest in this problem which lasted her entire career. She returned to the questions of “capitulation”, a term coined by one of her co-authors Arnold Scholz, several times in her life, always with the sense that these were questions of deep arithmetic significance. I believe that it was the experience of her early contact with class field theory which dominated her research interests in number theory and which provided her a fertile source of inspiration for most of her career. I will describe some of her work in this area below. I would like to take this opportunity to thank the referee for many valuable suggestions. For a number field F , (i.e., an extension of finite degree over the rational field Q) its Hilbert class field H = H(F ) is the maximal Galois extension of F which is everywhere unramified and whose Galois group Gal(H/F ) is abelian. It is a consequence of Artin’s reciprocity law that Gal(H/F ) is isomorphic to the ideal class group C(F ) of F. The Principal Ideal Theorem is the statement that every ideal of F becomes a principal ideal when considered

Let E ∕ ℚbe an elliptic curve defined over the rational field ℚ. We examine the rank of the Mordell–Weil group E(K) as Kranges over cubic extensions ofℚ.

We consider the problem of determining whether a set of primes, or, more generally, prime ideals in a number field, can be realized as a finite union of residue classes, or of Frobenius conjugacy classes. We give criteria for a set to be realized in this manner, and show that the subset of primes consisting of every other prime cannot be expressed in this way, even if we allow a finite number of exceptions., Comment: minor changes

Let $L(E/\Q,s)$ be the $L$-function of an elliptic curve $E$ defined over the rational field $\Q$. We examine special values of the derivatives $L^{\prime}(E,1,\chi)$ of twists by Dirichlet characters of $L(E/\Q,s)$ when $L(E,1,\chi)=0$.

It is now known that for any prime p and any finite semiabelian p-group G, there exists a (tame) realization of G as a Galois group over the rationals Q with exactly d = d(G) ramified primes, where d(G) is the minimal number of generators of G, which solves the minimal ramification problem for finite semiabelian p-groups. We generalize this result to obtain a theorem on finite semiabelian groups and derive the solution to the minimal ramification problem for a certain family of semiabelian groups that includes all finite nilpotent semiabelian groups G. Finally, we give some indication of the depth of the minimal ramification problem for semiabelian groups not covered by our theorem., Comment: 13 pages

Let p be a prime number. It is not known if every finite p-group of rank n>1 can be realized as a Galois group over Q with no more than n ramified primes. We prove that this can be done for the family of finite p-groups which contains all the cyclic groups of p-power order, and is closed under direct products, wreath products, and rank preserving homomorphic images. This family contains the Sylow p-subgroups of the symmetric groups and of the classical groups over finite fields of characteristic not p. On the other hand, it does not contain all finite p-groups., 8 pages. Note added at the end

Let L(E/Q,s) be the L-function of an elliptic curve E defined over the rational field Q. We examine the vanishing and non-vanishing of the central values L(E,1,\chi) of the twisted L-function as \chi ranges over Dirichlet characters of given order., Comment: To appear in Journal of the London Mathematical Society

Let E be an elliptic curve over , and let r be an integer. According to the Lang-Trotter conjecture, the number of primes p such that ap(E) = r is either finite, or is asymptotic to CE,r√x/log x where CE,r is a non-zero constant. A typical example of the former is the case of rational ℓ-torsion, where ap(E) = r is impossible if r ≡ 1 (mod ℓ). We prove in this paper that, when E has a rational ℓ-isogeny and ℓ ≠ 11, the number of primes p such that ap(E) ≡ r (mod ℓ) is finite (for some r modulo ℓ) if and only if E has rational ℓ-torsion over the cyclotomic field (ζℓ). The case ℓ = 11 is special, and is also treated in the paper. We also classify all those occurences.

Following Ax 's method a lower bound for the p -adic rank of the group of units in the general case of a Galois number field over the rationals is given. In some nontrivial special cases the result gives Leopoldt's conjecture. The same method is also applied to the case of p -units.

Let F be a perfect field of characteristic p. Let k be a function field in one variable over F. Let v be a discrete rank one valuation for k which is trivial on F, and let k be the associated completion. Using only the separability of the extension k/k, we prove a strong form of Leopoldt′s Conjecture for k.

Let m ( d ) m(d) denote the exponent of the ideal class group of the complex quadratic field Q ( √ d ) Q(\surd d) , where d > 0 d > 0 is a fundamental discriminant. It is shown that there are only finitely many d d for which m ( d ) = 3 m(d) = 3 . Assuming the extended Riemann Hypothesis, it is shown that m ( d ) → ∞ as | d | → ∞ m(d) \to \infty {\text { as }}|d| \to \infty .