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In this note I give simple proofs of classical results of Euler, Legendre and Sylvester showing that for certain integers M there are no (or only a few) solutions of $x^3 + y^3 = M$, with $x$ and $y$ in $\mathbb{Q}$. The proofs all use a single argument -- infinite 3-descent in the ring $\mathcal{O} = \mathbb{Z}[\omega]$ of Eisenstein integers. (Everything needed about $\mathcal{O}$ is developed from scratch.) The reader only needs the briefest acquaintance with complex numbers, fields and congruence modulo an element of a commutative ring. In particular I never say anything about ideals or elliptic curves (though I do mention cubic reciprocity in passing), and a clever high-school student might well enjoy the note. A few new results with $M$ in $\mathcal{O}$ and $x$ and $y$ in $\mathbb{Q}[\omega]$ are also derived., Comment: 18 pages

The ESA PLATO space mission is devoted to unveiling and characterizing new extrasolar planets and their host stars. This mission will encompass a very large field of view, granting it the potential to survey up to one million stars depending on the final observation strategy. The telemetry budget of the spacecraft cannot handle transmitting individual images for such a huge stellar sample at the right cadence, so the development of an appropriate strategy to perform on-board data reduction is mandatory. We employ aperture photometry to produce stellar light curves in flight. Our aim is thus to find the mask model that optimizes the scientific performance of the reduced data. We considered three distinct aperture models: binary mask, weighted Gaussian mask, and weighted gradient mask giving lowest noise-to-signal ratio, computed through a novel direct method. An innovative criterion was adopted for choosing between different mask models. We designated as optimal the model providing the best compromise between sensitivity to detect true and false planet transits. We determined the optimal model based on simulated noise-to-signal ratio and frequency of threshold crossing events. Our results show that, although the binary mask statistically presents a few percent higher noise-to-signal ratio compared to weighted masks, both strategies have very similar efficiency in detecting legitimate planet transits. When it comes to avoiding spurious signals from contaminant stars however the binary mask statistically collects considerably less contaminant flux than weighted masks, thereby allowing the former to deliver up to $\sim$30\% less false transit signatures at $7.1\sigma$. Our proposed approach for choosing apertures has been proven to be decisive for the determination of a mask model capable to provide near maximum planet yield and substantially reduced occurrence of false positives., Comment: 20 pages, 22 figures, Astronomy & Astrophysics (accepted)

Let $M(\mathit{odd})\subset Z/2[[x]]$ be the space of odd mod~2 modular forms of level $\Gamma_{0}(3)$. It is known that the formal Hecke operators $T_{p}:Z/2[[x]]\rightarrow Z/2[[x]]$, $p$ an odd prime other than $3$, stabilize $M(\mathit{odd})$ and act locally nilpotently on it. So $M(\mathit{odd})$ is an $\mathcal{O} = Z/2[[t_{5},t_{7}, t_{11}, t_{13}]]$-module with $t_{p}$ acting by $T_{p}$, $p\in \{5,7,11,13\}$. We show: (1) Each $T_{p}:M(\mathit{odd})\rightarrow M(\mathit{odd})$, $p\ne 3$, is multiplication by some $u$ in the maximal ideal, $m$, of $\mathcal{O}$. (2) The kernel, $I$, of the action of $\mathcal{O}$ on $M(\mathit{odd})$ is $(A^{2},AC,BC)$ where $A,B,C$ have leading forms $t_{5}+t_{7}+t_{13},\, t_{7},\, t_{11}$. We prove analogous results in level $\Gamma_{0}(5)$. Now $\mathcal{O}$ is $Z/2[[t_{3},t_{7},t_{11},t_{13}]]$, and the leading forms of $A,B,C$ are $t_{3}+t_{7}+t_{11},\, t_{7},\, t_{13}$. Let $\mathit{HE}$, "the shallow mod~2 Hecke algebra (of level $\Gamma_{0}(3)$ or $\Gamma_{0}(5)$)" be $\mathcal{O}/I$. (1) and (2) above show that $\mathit{HE}$ is a 1 variable power series ring over the 1-dimensional local ring $Z/2[[A,B,C]]/(A^{2},AC,BC)$. For another approach to all these results, based on deformation theory, see Deo and Medvedovsky, "Explicit old components of mod-2 Hecke algebras with trivial $\bar{\rho}$.", Comment: 10 pages

In arXiv:1603.03910 [math.NT] we introduced some $C_{n}$ in $Z/2[t]$ defined by a linear recurrence and showed that each $C_{n}$, $n\equiv 0 \bmod{4}$, is a sum of $C_{k}$, $k

Let $F$ be the element $\sum_{n\ \mathit{odd},\ n>0}x^{n^{2}}$ of $Z/2[[x]]$. Set $G=F(x^{5})$, $D=F(x)+F(x^{25})$. For $k>0$, $(k,10)=1$, define $D_{k}$ as follows. $D_{1}=D$, $D_{3}=D^{8}/G$, $D_{7}=D^{2}G$, $D_{9}=D^{4}G$; furthermore $D_{k+10}=G^{2}D_{k}$. Using modular forms of level $\Gamma_{0}(5)$ we show that the space $W$ spanned by the $D_{k}$ is stabilized by the formal Hecke operators $T_{p}:Z/2[[x]]\rightarrow Z/2[[x]]$, $p\ne 2$ or $5$. And we determine the structure of the (completed) shallow Hecke algebra attached to $W$. This algebra proves to be a power series ring in $T_{3}$ and $T_{7}$ with an element of square $0$ adjoined. As Hecke module, $W$ identifies with a certain subquotient of the space of mod~2 modular forms of level $\Gamma_{0}(5)$, and our Hecke algebra result parallels findings in level 1 (by J.-L. Nicolas and J.-P. Serre) and in level $\Gamma_{0}(3)$ by us., Comment: 25 pages

The "Nicolas-Serre code", $(a,b) \leftrightarrow t^{n}$, is a bijection between $N\times N$ and those $t^{n}$, $n$ odd, in $Z/2[t]$. Suppose $A_{n}$, $n$ odd, in $Z/2[t]$ are defined by: $A_{1}= A_{5}= 0$, $A_{3}= t$, $A_{7}= t^{5}$, and $A_{n+8}= t^{8} A_{n} + t^{2} A_{n+2}$. A lemma, Proposition 4.3 of [6], used to study the Hecke algebra attached to the space of mod $2$ level $1$ modular forms, gives information about the codes $(a,b)$ attached to the monomials appearing in $A_{n}$. The unpublished highly technical proof has been simplified by Gerbelli-Gauthier. Our Theorem 3.7 generalizes Proposition 4.3. The proof, in sections 1-3, is a further simplification of Gerbelli-Gauthier's argument. We build up to the theorem with variants involving the same recurrence, but having different sorts of initial conditions. Section 4 treats the recurrence $A_{n+16}= t^{16} A_{n} + t^{4} A_{n+4} + t^{2} A_{n+2}$. Theorem 4.1, the analog to Theorem 3.7 for this recurrence, is used in [2] and [3] to analyze level 3 Hecke algebras. Finally we introduce a variant code, $(a,b) \leftrightarrow w^{n}$ which is a bijection between $N\times N$ and those $w^{n}$, $n \equiv 1,3,7,9 \bmod{20}$, in $Z/2[w]$. We then study the recurrence $A_{n+80}= w^{80} A_{n}+ w^{20} A_{n+20}$, $n \equiv 1,3,7,9 \bmod{20}$, with appropriate initial conditions. Lemma 5.5, derived from the results of sections 1-3, is the precise analog of Proposition 4.3 for this code, this recurrence, and these initial conditions. It is used in [4] and [5] to analyze level 5 Hecke algebras., Comment: References added to the applications of Lemma 5.5 to Hecke algebras. 14 pages

Let $D$ in $Z/2[[x]]$ be $\sum x^{n^{2}}$, $n>0$ and prime to $6$. Let $W$ be spanned by the $D^{k}$, $k>0$ and prime to $6$. Then the formal Hecke operators $T_{p}$, $p>3$, stabilize $W$, and it can be shown that they act locally nilpotently. We show that the completion of the Hecke algebra generated by these $T_{p}$ acting on $W$, with respect to the maximal ideal generated by the $T_{p}$, is a power series ring in $T_{7}$ and $T_{13}$ with an element of square $0$ adjoined. This may be viewed as a level 3 analog of the level 1 results of Nicolas and Serre -- the Hecke stable space they study is spanned by the odd powers of the mod $2$ reduction of $\Delta$, and their resulting completed Hecke algebra is a power series ring in $T_{3}$ and $T_{5}$., Comment: Revised to include a sketch of a new simpler proof of the results of Nicolas and Serre in level 1. A reference to a result from arXiv 1604.02622 that completes our proof has been added. Typos have been corrected. 20 pages

The last century has seen enormous progress in our understanding of the Universe. We know the life cycles of stars, the structure of galaxies, the remnants of the big bang, and have a general understanding of how the Universe evolved. We have come remarkably far using electromagnetic radiation as our tool for observing the Universe. However, gravity is the engine behind many of the processes in the Universe, and much of its action is dark. Opening a gravitational window on the Universe will let us go further than any alternative. Gravity has its own messenger: Gravitational waves, ripples in the fabric of spacetime. They travel essentially undisturbed and let us peer deep into the formation of the first seed black holes, exploring redshifts as large as z ~ 20, prior to the epoch of cosmic re-ionisation. Exquisite and unprecedented measurements of black hole masses and spins will make it possible to trace the history of black holes across all stages of galaxy evolution, and at the same time constrain any deviation from the Kerr metric of General Relativity. eLISA will be the first ever mission to study the entire Universe with gravitational waves. eLISA is an all-sky monitor and will offer a wide view of a dynamic cosmos using gravitational waves as new and unique messengers to unveil The Gravitational Universe. It provides the closest ever view of the early processes at TeV energies, has guaranteed sources in the form of verification binaries in the Milky Way, and can probe the entire Universe, from its smallest scales around singularities and black holes, all the way to cosmological dimensions., Comment: 20 pages; submitted to the European Space Agency on May 24th, 2013 for the L2/L3 selection of ESA's Cosmic Vision program

The analysis of the noise sources perturbing a test mass (TM) geodesic motion is the main scientific objective of the LISA Technology Package experiment (LTP) on board of the LISA Pathfinder space mission. Information on force noise acting on TMs are obtained with a data reduction procedure involving system parameters. Such parameters can be estimated from dedicated experimental runs. Therefore the final estimation of force noise is affected by two sources of uncertainty. One is statistical and connected to the random nature of noisy signals. The other is connected to the uncertainties on the system parameters. The analysis of simulated LTP data is indicating that the major contribution to the force noise power spectral density uncertainties is coming from the statistical properties of the spectrum estimator., Comment: To be published in Journal of Physics: Conference Series, Proceedings of the 8th International LISA Symposium

The precursor ESA mission LISA-Pathfinder, to be flown in 2013, aims at demonstrating the feasibility of the free-fall, necessary for LISA, the upcoming space-born gravitational wave observatory. LISA Technology Package (LTP) is planned to carry out a number of experiments, whose main targets are to identify and measure the disturbances on each test-mass, in order to reach an unprecedented low-level residual force noise. To fulfill this plan, it is then necessary to correctly design, set-up and optimize the experiments to be performed on-flight and do a full system parameter estimation. Here we describe the progress on the non-linear analysis using the methods developed in the framework of the \textit{LTPDA Toolbox}, an object-oriented MATLAB Data Analysis environment: the effort is to identify the critical parameters and remove the degeneracy by properly combining the results of different experiments coming from a closed-loop system like LTP., Comment: 6 pages, 2 figures, 1 table, Proceedings of the 8th International LISA Symposium

Suppose l=2m+1, m>0. We introduce m "theta-series", [1],...,[m], in Z/2[[x]]. It has been conjectured that the n for which the coefficient of x^n in 1/[i] is 1 form a set of density 0. This is probably always false, but in certain cases, for n restricted to certain arithmetic progressions, it is true. We prove such zero-density results using the theory of modular forms, and speculate about what may be true in general., Comment: 16 pages

The complete list of pairs of non-isomorphic finite simple groups having the same order is well-known. In particular for p>3, PSL_2(Z/p) is the "only" simple group of order (p^3-p)/2. It's less well-known that Frobenius proved this uniqueness result in 1902. This note presents a version of Frobenius' argument that might be used in an undergraduate honors algebra course. It also includes a short modern proof, aimed at the same audience, of the much earlier result that PSL_2(Z/p) is simple for p>3; a result stated by Galois in 1832., Comment: 10 pages

This paper presents a quantitative assessment of the performance of the upcoming LISA Pathfinder geodesic explorer mission. The findings are based on the results of extensive ground testing and simulation campaigns using flight hardware and flight control and operations algorithms. The results show that, for the central experiment of measuring the stray differential acceleration between the LISA test masses, LISA Pathfinder will be able to verify the overall acceleration noise to within a factor two of the LISA requirement at 1 mHz and within a factor 6 at 0.1 mHz. We also discuss the key elements of the physical model of disturbances, coming from LISA Pathfinder and ground measurement, that will guarantee the LISA performance., Comment: Proceedings of the 8th LISA Symposium. To appear Class. Quantum Grav. 28 (2011)

Suppose B=F[x,y,z]/h is the homogeneous coordinate ring of a characteristic p degree 3 irreducible plane curve C with a node. Let J be a homogeneous (x,y,z)-primary ideal and n -> e_n be the Hilbert-Kunz function of B with respect to J. Let q=p^n. When J=(x,y,z), Pardue (see R. Buchweitz, Q. Chen. Hilbert-Kunz functions of cubic curves and surfaces. J. Algebra 197 (1997). 246-267) showed that e_n=(7q^2)/3-q/3-R where R=5/3 if q is congruent to 2 (3), and is 1 otherwise. We generalize this, showing that e_n= (mu q^2) + (alpha q) - R where R only depends on q mod 3. We describe alpha and R in terms of classification data for a vector bundle on C. Igor Burban (I. Burban. Frobenius morphism and vector bundles on cycles of projective lines. 2010. arXiv 1010.0399) provided a major tool in our proof by showing how pull-back by Frobenius affects the classification data of an indecomposable vector bundle over C. We are also indebted to him for pointing us towards Y. A. Drozd, G.-M. Greuel, I. Kashuba. On Cohen-Macaulay modules on surface singularities. Mosc. Math. J. 3 (2003). 397-418, 742, in which h^0 is described in terms of these classification data., Comment: 13 pages. Misspellings corrected

I present Franz Lemmermeyer's proof that if p is a prime congruent to 9 mod 16 then the class number of Q(p^(1/4)) is congruent to 2 mod 4., Comment: 4 pages

Let g be the element that is the sum of x^(n^2) for n >= 0 of A=Z/2[[x]], and let B consist of all n for which the coefficient of x^n in 1/g is 1. (The elements of B are the entries 0, 1, 2, 3, 5, 7, 8, 9, 13, ... in A108345; see The On-Line Encyclopedia of Integer Sequences (OEIS).) Cooper, Eichhorn, and O'Bryant [1] have shown that the (upper) density of B is at most 1/4, and it is conjectured that B has density 0. This note uses results of Gauss on sums of 3 squares to show that the subset of B consisting of all n not congruent to 15 mod 16 has density 0. The final section gives some computer calculations, made by Kevin O'Bryant, indicating that, pace [1], B has density 1/32., Comment: 7 pages

A main scientific output of the LISA Pathfinder mission is to provide a noise model that can be extended to the future gravitational wave observatory, LISA. The success of the mission depends thus upon a deep understanding of the instrument, especially the ability to correctly determine the parameters of the underlying noise model. In this work we estimate the parameters of a simplified model of the LISA Technology Package (LTP) instrument. We describe the LTP by means of a closed-loop model that is used to generate the data, both injected signals and noise. Then, parameters are estimated using a Bayesian framework and it is shown that this method reaches the optimal attainable error, the Cramer-Rao bound. We also address an important issue for the mission: how to efficiently combine the results of different experiments to obtain a unique set of parameters describing the instrument., Comment: 14 pages, 4 figures, submitted to PRD

Let p be a prime. The Hilbert-Kunz multiplicity, mu, of the element sum(x_i^(d_i)) of (Z/p)[x_1,..., x_s] depends on p in a complicated way. We calculate the limit of mu as p -> infinity. In particular when each d_i is 2 we show that the limit is 1 + the coefficient of z^(s-1) in the power series expansion of sec z + tan z., Comment: 6 pages

The scientific objectives of the Lisa Technology Package (LTP) experiment, on board of the LISA Pathfinder mission, demand for an accurate calibration and validation of the data analysis tools in advance of the mission launch. The levels of confidence required on the mission outcomes can be reached only with an intense activity on synthetically generated data. A flexible procedure allowing the generation of cross-correlated stationary noise time series was set-up. Multi-channel time series with the desired cross correlation behavior can be generated once a model for a multichannel cross-spectral matrix is provided. The core of the procedure is the synthesis of a noise coloring multichannel filter through a frequency-by-frequency eigendecomposition of the model cross-spectral matrix and a Z-domain fit. The common problem of initial transients in noise time series is solved with a proper initialization of the filter recursive equations. The noise generator performances were tested in a two dimensional case study of the LTP dynamics along the two principal channels of the sensing interferometer., Comment: Accepted for publication in Physical Review D (http://prd.aps.org/)

Suppose that h in F[x,y,z], char F=2, defines a nodal cubic. In earlier papers we made a precise conjecture as to the Hilbert-Kunz functions attached to the powers of h. Assuming this conjecture we showed that a class of characteristic 2 hypersurfaces has algebraic but not necessarily rational Hilbert-Kunz multiplicities. We now show that if the conjecture holds, then transcendental multiplicities exist, and in particular that the number \sum\binom{2n}{n}^{2}/(65,536)^{n}, proved transcendental by Schneider, is a Q-linear combination of Hilbert-Kunz multiplicities of characteristic 2 hypersurfaces., Comment: 5 pages

Let F be a finite field of characteristic 2 and h be the element x^3+y^3+xyz of F[[x,y,z]]. In an earlier paper we made a precise conjecture as to the values of the colengths of the ideals (x^q,y^q,z^q,h^j) for q a power of 2. We also showed that if the conjecture holds then the Hilbert-Kunz series of H=uv+h is algebraic (of degree 2) over Q(w), and that mu(h) is algebraic (explicitly, (4/3)+(5/14)sqrt(7)). In this note, assuming the same conjecture, we use a theory of infinite matrices to rederive this result, and we extend it to a wider class of H; for example H=g(u,v)+h. In a follow-up paper, under the same hypothesis, we will show that transcendental Hilbert-Kunz multiplicities exist., Comment: 11 pages

Let V be an infinite matrix with rows and columns indexed by the positive integers, and entries in a field F. Suppose that v_{i,j} only depends on i-j and is 0 for |i-j| large. Then V^n is defined for all n, and one has a "generating function" G=\sum a_{1,1}(V^n)z^n. Ira Gessel has shown that G is algebraic over F(z). We extend his result, allowing v_{i,j} for fixed i-j to be eventually periodic in i rather than constant. This result and some variants of it that we prove will have applications to Hilbert-Kunz theory., Comment: 12 pages

We give a treatment of the Brenner-Monsky example based on polynomial algebra and linear algebra. No prior knowledge of tight closure theory, Hilbert-Kunz theory, algebraic geometry or local cohomology is assumed.

We give an example showing that tight closure does not commute with localization., Comment: final version

We use the results of our paper "p-Fractals and power series--I" (Journal of Algebra 280, 2004, pp. 505--536) to prove the rationality of the Hilbert-Kunz series of a large family of power series, including those of the form \sum_i f_i(x_i,y_i), where the f_i(x_i,y_i) are power series with coefficients in a finite field. The methods are effective, as we illustrate with examples. In the final section, which can be read independently of the others, we obtain more precise results for the Hilbert-Kunz function of the 3 variable power series z^D-h(x,y).

Let (R,m,k) be an excellent, local, normal ring of characteristic p with a perfect residue field and dim R=d. Let M be a finitely generated R-module. We show that there exists a real number beta(M) such that lambda(M/I^[q]M) = e_{HK}(M) q^d + beta(M) q^{d-1} + O(q^{d-2})., Comment: 11 pages, minor changes, additional references