Visualizing elements of Sha[3] in genus 2 jacobians

Mazur proved that any element xi of order three in the Shafarevich-Tate group of an elliptic curve E over a number field k can be made visible in an abelian surface A in the sense that xi lies in the kernel of the natural homomorphism between the cohomology groups H^1(k,E) -> H^1(k,A). However, the abelian surface in Mazur's construction is almost never a jacobian of a genus 2 curve. In this paper we show that any element of order three in the Shafarevich-Tate group of an elliptic curve over a number field can be visualized in the jacobians of a genus 2 curve. Moreover, we describe how to get explicit models of the genus 2 curves involved.Comment: 12 page

The SKA view of the Neutral Interstellar Medium in Galaxies

Two major questions in galaxy evolution are how star-formation on small scales leads to global scaling laws and how galaxies acquire sufficient gas to sustain their star formation rates. HI observations with high angular resolution and with sensitivity to very low column densities are some of the important observational ingredients that are currently still missing. Answers to these questions are necessary for a correct interpretation of observations of galaxy evolution in the high-redshift universe and will provide crucial input for the sub-grid physics in hydrodynamical simulations of galaxy evolutions. In this chapter we discuss the progress that will be made with the SKA using targeted observations of nearby individual disk and dwarf galaxies.Comment: 22 pages, 4 figures, to appear as part of 'Neutral Hydrogen' in Proceedings 'Advancing Astrophysics with the SKA (AASKA14)', PoS(AASKA14)12

On higher congruences between cusp forms and Eisenstein series

In this paper we present several finite families of congruences between cusp forms and Eisenstein series of higher weights at powers of prime ideals. We formulate a conjecture which describes properties of the prime ideals and their relation to the weights. We check the validity of the conjecture on several numerical examples.Comment: 20 page

A Practical Cryptanalysis of the Algebraic Eraser

Anshel, Anshel, Goldfeld and Lemieaux introduced the Colored Burau Key Agreement Protocol (CBKAP) as the concrete instantiation of their Algebraic Eraser scheme. This scheme, based on techniques from permutation groups, matrix groups and braid groups, is designed for lightweight environments such as RFID tags and other IoT applications. It is proposed as an underlying technology for ISO/IEC 29167-20. SecureRF, the company owning the trademark Algebraic Eraser, has presented the scheme to the IRTF with a view towards standardisation. We present a novel cryptanalysis of this scheme. For parameter sizes corresponding to claimed 128-bit security, our implementation recovers the shared key using less than 8 CPU hours, and less than 64MB of memory.Comment: 15 pages. Updated references, with brief comments added. Minor typos corrected. Final version, accepted for CRYPTO 201

Recognizing hyperelliptic graphs in polynomial time

Recently, a new set of multigraph parameters was defined, called "gonalities". Gonality bears some similarity to treewidth, and is a relevant graph parameter for problems in number theory and multigraph algorithms. Multigraphs of gonality 1 are trees. We consider so-called "hyperelliptic graphs" (multigraphs of gonality 2) and provide a safe and complete sets of reduction rules for such multigraphs, showing that for three of the flavors of gonality, we can recognize hyperelliptic graphs in O(n log n+m) time, where n is the number of vertices and m the number of edges of the multigraph.Comment: 33 pages, 8 figure

Applying Grover's algorithm to AES: quantum resource estimates

We present quantum circuits to implement an exhaustive key search for the Advanced Encryption Standard (AES) and analyze the quantum resources required to carry out such an attack. We consider the overall circuit size, the number of qubits, and the circuit depth as measures for the cost of the presented quantum algorithms. Throughout, we focus on Clifford$+T$ gates as the underlying fault-tolerant logical quantum gate set. In particular, for all three variants of AES (key size 128, 192, and 256 bit) that are standardized in FIPS-PUB 197, we establish precise bounds for the number of qubits and the number of elementary logical quantum gates that are needed to implement Grover's quantum algorithm to extract the key from a small number of AES plaintext-ciphertext pairs.Comment: 13 pages, 3 figures, 5 tables; to appear in: Proceedings of the 7th International Conference on Post-Quantum Cryptography (PQCrypto 2016

Derived equivalence classification of the cluster-tilted algebras of Dynkin type E

We obtain a complete derived equivalence classification of the cluster-tilted algebras of Dynkin type E. There are 67, 416, 1574 algebras in types E6, E7 and E8 which turn out to fall into 6, 14, 15 derived equivalence classes, respectively. This classification can be achieved computationally and we outline an algorithm which has been implemented to carry out this task. We also make the classification explicit by giving standard forms for each derived equivalence class as well as complete lists of the algebras contained in each class; as these lists are quite long they are provided as supplementary material to this paper. From a structural point of view the remarkable outcome of our classification is that two cluster-tilted algebras of Dynkin type E are derived equivalent if and only if their Cartan matrices represent equivalent bilinear forms over the integers which in turn happens if and only if the two algebras are connected by a sequence of "good" mutations. This is reminiscent of the derived equivalence classification of cluster-tilted algebras of Dynkin type A, but quite different from the situation in Dynkin type D where a far-reaching classification has been obtained using similar methods as in the present paper but some very subtle questions are still open.Comment: 19 pages. v4: completely rewritten version, to appear in Algebr. Represent. Theory. v3: Main theorem strengthened by including "good" mutations (cf. also arXiv:1001.4765). Minor editorial changes. v2: Third author added. Major revision. All questions left open in the earlier version by the first two authors are now settled in v2 and the derived equivalence classification is completed. arXiv admin note: some text overlap with arXiv:1012.466

The Metallicity-Luminosity Relation, Effective Yields, and Metal Loss in Spiral and Irregular Galaxies

I present results on the correlation between galaxy mass, luminosity, and metallicity for a sample of spiral and irregular galaxies having well-measured abundance profiles, distances, and rotation speeds. Additional data for low surface brightness galaxies from the literature are also included for comparison. These data are combined to study the metallicity-luminosity and metallicity-rotation speed correlations for spiral and irregular galaxies. The metallicity luminosity correlation shows its familiar form for these galaxies, a roughly uniform change in the average present-day O/H abundance of about a factor 100 over 11 magnitudes in B luminosity. However, the O/H - V(rot) relation shows a change in slope at a rotation speed of about 125 km/sec. At faster V(rot), there appears to be no relation between average metallicity and rotation speed. At lower V(rot), the metallicity correlates with rotation speed. This change in behavior could be the result of increasing loss of metals from the smaller galaxies in supernova-driven winds. This idea is tested by looking at the variation in effective yield, derived from observed abundances and gas fractions assuming closed box chemical evolution. The effective yields derived for spiral and irregular galaxies increase by a factor of 10-20 from V(rot) approximately 5 km/sec to V(rot) approximately 300 km/sec, asympotically increasing to approximately constant y(eff) for V(rot) > 150 km/sec. The trend suggests that galaxies with V(rot) < 100-150 km/sec may lose a large fraction of their SN ejecta, while galaxies above this value tend to retain metals.Comment: 40 pages total, including 7 encapsulated postscript figures. Accepted for publication in 20 Dec 2002 Ap

HST FOC spectroscopy of the NLR of NGC 4151. I. Gas kinematics

We present the results from a detailed kinematic analysis of both ground-based, and Hubble Space Telescope/Faint Object Camera long-slit spectroscopy at sub-arcsec spatial resolution, of the narrow-line region of NGC 4151. In agreement with previous work, the extended emission gas (R > 4") is found to be in normal rotation in the galactic plane, a behaviour that we were able to trace even across the nuclear region, where the gas is strongly disturbed by the interaction with the radio jet, and connects smoothly with the large scale rotation defined by the neutral gas emission. The HST data, at 0.029" spatial resolution, allow us for the first time to truly isolate the kinematic behaviour of the individual clouds in the inner narrow-line region. We find that, underlying the perturbations introduced by the radio ejecta, the general velocity field can still be well represented by planar rotation down to a radius of ~ 0.5" (30 pc), distance at which the rotation curve has its turnover. The most striking result that emerges from our analysis is that the galaxy potential derived fitting the rotation curve changes from a "dark halo" at the ENLR distances to dominated by the central mass concentration in the NLR, with an almost Keplerian fall-off in the 1"< R < 4" interval. The observed velocity of the gas at 0.5" implies a mass of M ~ 10E9 M(sol) within the inner 60 pc. The presence of a turnover in the rotation curve indicates that this central mass concentration is extended. The first measured velocity point (outside the region saturated by the nucleus) would imply an enclosed mass of ~ 5E7 M(sol) within R ~ 0.15" (10 pc) which represents an upper limit to any nuclear point mass.Comment: 30 pages (aaspp4.sty), 14 figures. Fig. 1, 2 and 4 available by anonymous FTP at 143.54.2.51 (cd /pub/winge) as GIF files; or upon request to [email protected]. Accepted for publication in the Astrophysical Journal (part 1