arXiv subjects
Publications and source records attributed to Julian Rosen.
In this paper, we develop an explicit method to express finite algebraic numbers (in particular, certain idempotents among them) in terms of linear recurrent sequences, and give applications to the characterization of the splitting primes in a given finite Galois extension over the rational field.
Let $L$ be a Galois algebra with Galois group $G$ and let $x$ be a normal element of $L$. The moduli space $\mathcal X$ of pairs $(L,x)$ is isomorphic to an open subset of the quotient variety $\mathbb P/G$, where $\mathbb P$ is the projective space of the regular representation of $G$. We provide a formula for the height of any pair $(L,x) \in \mathcal X(\mathbb Q)$ in terms of algebraic invariants of $L$ and $x$ with respect to a natural adelic metric on the anticanonical divisor of $\mathbb P/G$.
We construct an analogue of the ring of algebraic numbers, living in a quotient of the product of all finite fields of prime order. We use this ring to deduce some results about linear recurrent sequences.
There is a notion of $p$-adic period coming from the crystalline Frobenius automorphism of the de Rham cohomology of an algebraic variety. In this paper, we consider sequences of $p$-adic periods, one for each prime. We study the sequences using motivic periods, and we formulate an analogue of the Grothendieck period conjecture.
We prove a general family of congruences for Bernoulli numbers whose index is a polynomial function of a prime, modulo a power of that prime. Our family generalizes many known results, including the von Staudt--Clausen theorem and Kummer's congruence.
By linking widely separated radio dishes, the technique of very long baseline interferometry (VLBI) can greatly enhance angular resolution in radio astronomy. However, at any given moment, a VLBI array only sparsely samples the information necessary to form an image. Conventional imaging techniques partially overcome this limitation by making the assumption that the observed cosmic source structure does not evolve over the duration of an observation, which enables VLBI networks to accumulate information as the Earth rotates and changes the projected array geometry. Although this assumption is appropriate for nearly all VLBI, it is almost certainly violated for submillimeter observations of the Galactic Center supermassive black hole, Sagittarius A* (Sgr A*), which has a gravitational timescale of only ~20 seconds and exhibits intra-hour variability. To address this challenge, we develop several techniques to reconstruct dynamical images ("movies") from interferometric data. Our techniques are applicable to both single-epoch and multi-epoch variability studies, and they are suitable for exploring many different physical processes including flaring regions, stable images with small time-dependent perturbations, steady accretion dynamics, or kinematics of relativistic jets. Moreover, dynamical imaging can be used to estimate time-averaged images from time-variable data, eliminating many spurious image artifacts that arise when using standard imaging methods. We demonstrate the effectiveness of our techniques using synthetic observations of simulated black hole systems and 7mm Very Long Baseline Array observations of M87, and we show that dynamical imaging is feasible for Event Horizon Telescope observations of Sgr A*.
A supercongruence is a congruence between rational numbers modulo a power of a prime. In this paper, we give a technique for finding and algorithmically proving supercongruences by expressing terms as infinite series involving certain generalizations of the harmonic numbers. We apply the technique to derive many new supercongruences. We also provide software for finding and proving supercongruences using our technique.
Proofs that an arbitrary field has a separable closure are necessarily non-constructive, and separable closures are unique only up to non-canonical isomorphism. This means that the absolute Galois group of a field is defined only up to inner automorphism. Here we construct a profinite algebraic group which is an inner form of the absolute Galois group. Our construction uses no form of the axiom of choice, and the group is defined up to canonical isomorphism. We also show that the Frobenius associated with a prime of a number field unramified in an extension, which is classically defined only up to conjugation, has a uniquely-defined analogue in terms of our group. We give a construction of the category of Artin motives with coefficients in an arbitrary field, and we give an interpretation of our absolute Galois group in terms of this category.
A period is a complex number arising as the integral of a rational function with algebraic number coefficients over a rationally-defined region. Although periods are typically transcendental numbers, there is a conjectural Galois theory of periods coming from the theory of motives. This paper formalizes an analogy between a class of periods called multiple zeta values, and congruences for rational numbers modulo prime powers (called supercongruences). We construct an analogue of the motivic period map in the setting of supercongruences, and use it to define a Galois theory of supercongruences. We describe an algorithm using our period map to find and prove supercongruences, and we provide software implementing the algorithm.
There are several formulas for the number of orbits of the projective line under the action of subgroups of $GL_2$. We give an interpretation of two such formulas in terms of the geometry of elliptic curves, and prove a more general formula for a large class of congruence subgroups of Bianchi groups. Our formula involves the number of walks on a certain graph called an isogeny volcano. Underlying our results is a complete description of the group of extensions of a pair of CM elliptic curves, and of a pair of lattices in a quadratic field.
For each positive integer n, we determine the set of symmetric functions f for which the congruence f(p/1,p/2,...,p/(p-1)) \equiv 0 mod p^n holds for all sufficiently large primes p. Our determination is conditional on a conjecture regarding the modulo p independence of Bernoulli numbers. In a recent work the author introduced a new truncated analogue of the multiple zeta function and investigated a class of relations among values of this function at positive integers. The question answered in the present work is equivalent to the determination of the relations satisfied by values of the corresponding analogue of the ordinary Riemann zeta function.
We give a natural parameterization of the Néron-Severi group of a product $A = E\times E'$ of two elliptic curves in terms of quadratic forms. As an application, we determine (in the non-CM case) whether $A$ contains a smooth curve of any fixed genus. We also determine whether $A$ admits a very ample line bundle of any fixed degree. In particular, we determine which of these abelian surfaces embed in $\mathbb{P}^4$, i.e. which come from the Horrocks-Mumford bundle.
Multiple zeta values are real numbers defined by an infinite series generalizing values of the Riemann zeta function at positive integers. Finite truncations of this series are called multiple harmonic sums and are known to have interesting arithmetic properties. When the truncation point is one less than a prime $p$, the mod $p$ values of multiple harmonic sums are called finite multiple zeta values. The present work introduces a new class of congruence for multiple harmonic sums, which we call weighted congruences. These congruences can hold modulo arbitrarily large powers of $p$. Unlike results for finite multiple zeta values, weighted congruences typically involve harmonic sums of multiple weights, which are multiplied by explicit powers of $p$ depending on weight. We also introduce certain formal weighted congruences inolving an infinite number of terms, which we call asymptotic relations. We define a weighted analogue of the finite multiple zeta function, and give an algebraic framework for classifying weighted congruences and asymptotic relations.
We give a family of congruences for the binomial coefficients ${kp-1\choose p-1}$ in terms of multiple harmonic sums, a generalization of the harmonic numbers. Each congruence in this family (which depends on an additional parameter $n$) involves a linear combination of $n$ multiple harmonic sums, and holds $\mod{p^{2n+3}}$. The coefficients in these congruences are integers depending on $n$ and $k$, but independent of $p$. More generally, we construct a family of congruences for ${kp-1\choose p-1} \mod{p^{2n+3}}$, whose members contain a variable number of terms, and show that in this family there is a unique "optimized" congruence involving the fewest terms. The special case $k=2$ and $n=0$ recovers Wolstenholme's theorem ${2p-1\choose p-1}\equiv 1\mod{p^3}$, valid for all primes $p\geq 5$. We also characterize those triples $(n, k, p)$ for which the optimized congruence holds modulo an extra power of $p$: they are precisely those with either $p$ dividing the numerator of the Bernoulli number $B_{p-2n-k}$, or $k \equiv 0, 1 \mod p$.