arXiv subjects
Publications and source records attributed to Hershy Kisilevsky.
We consider heuristic predictions for small non-zero algebraic central values of twists of the $L$-function of an elliptic curve $E/\mathbb{Q}$ by Dirichlet characters. We provide computational evidence for these predictions and consequences of them for instances of an analogue of the Brauer-Siegel theorem associated to $E/\mathbb{Q}$ extended to chosen families of cyclic extensions of fixed degree.
For an elliptic curve $E/\mathbb{Q}$ we show that there are infinitely many cyclic sextic extensions $K/\mathbb{Q}$ such that the Mordell-Weil group $E(K)$ has rank greater than the subgroup of $E(K)$ generated by all the $E(F)$ for the proper subfields $F \subset K$. For certain curves $E/\mathbb{Q}$ we show that the number of such fields $K$ of conductor less than $X$ is $\gg\sqrt X$.
We consider the signature rank of the units in real multiquadratic fields. When the three quadratic subfields of a real biquadratic field $K$ either (a) all have signature rank 2 (that is, fundamental units of norm $-1$), or (b) all have signature rank 1 (that is, have totally positive fundamental units), we provide explicit examples to show there exist infinitely many $K$ having each of the possible unit signature ranks (namely signature rank 3 or 4 in case (a) and signature rank 1,2, or 3 in case (b)). We make some additional remarks for higher rank real multiquadratic fields, in particular proving the rank of the totally positive units modulo squares in such extensions (hence also in the totally real subfield of cyclotomic fields) can be arbitrarily large.
We prove the rank of the group of signatures of the circular units (hence also the full group of units) of ${\mathbb Q}( ζ_m)^+$ 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 ${\mathbb Q}( ζ_{p^n})^+$ differs from $φ(p^n)/2$ 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.
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 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.
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.
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 $p\equiv q\equiv 3 \pmod 4$.
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,χ) of the twisted L-function as χranges over Dirichlet characters of given order.
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.
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.
Given a global field K and a positive integer n, there exists an abelian extension L/K (of exponent n) such that the local degree of L/K is equal to n at every finite prime of K, and is equal to two at the real primes if n=2. As a consequence, the n-torsion subgroup of the Brauer group of K is equal to the relative Brauer group of L/K.
Let $E$ be an elliptic curve over $\mathbb{Q}$, with L-function $L_E(s)$. For any primitive Dirichlet character $χ$, let $L_E(s, χ)$ be the L-function of $E$ twisted by $χ$. In this paper, we use random matrix theory to study vanishing of the twisted L-functions $L_E(s, χ)$ at the central value $s=1$. In particular, random matrix theory predicts that there are infinitely many characters of order 3 and 5 such that $L_E(1, χ)=0$, but that for any fixed prime $k \geq 7$, there are only finitely many character of order $k$ such that $L_E(1, χ)$ vanishes. With the Birch and Swinnerton-Dyer Conjecture, those conjectures can be restated to predict the number of cyclic extensions $K/\mathbb{Q}$ of prime degree such that $E$ acquires new rank over $K$.