On certain finiteness questions in the arithmetic of modular forms

We investigate certain finiteness questions that arise naturally when studying approximations modulo prime powers of p-adic Galois representations coming from modular forms. We link these finiteness statements with a question by K. Buzzard concerning p-adic coefficient fields of Hecke eigenforms. Specifically, we conjecture that for fixed N, m, and prime p with p not dividing N, there is only a finite number of reductions modulo p^m of normalized eigenforms on \Gamma_1(N). We consider various variants of our basic finiteness conjecture, prove a weak version of it, and give some numerical evidence.Comment: 25 pages; v2: one of the conjectures from v1 now proved; v3: restructered parts of the article; v4: minor corrections and change

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

Restricted sum formula of alternating Euler sums

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Interpolated sequences and critical LL-values of modular forms

Recently, Zagier expressed an interpolated version of the Ap\'ery numbers for $\zeta(3)$ in terms of a critical $L$-value of a modular form of weight 4. We extend this evaluation in two directions. We first prove that interpolations of Zagier's six sporadic sequences are essentially critical $L$-values of modular forms of weight 3. We then establish an infinite family of evaluations between interpolations of leading coefficients of Brown's cellular integrals and critical $L$-values of modular forms of odd weight.Comment: 23 pages, to appear in Proceedings for the KMPB conference: Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theor

Natural Variability and Warming Signals in Global Ocean Wave Climates

地球全体の波浪特性の変化傾向と自然変動の関係を解明 --地球温暖化の沿岸域への影響を定量化--. 京都大学プレスリリース. 2021-06-15.This paper presents a multivariate classification of the global wave climate into types driven by atmospheric circulation patterns. The primary source of the net long-term variability is evaluated based on historical wave simulations. Results show that the monsoon, extratropical, subtropical, and polar wave climate types of the Pacific and North Atlantic Oceans are dominated by natural variability, whereas the extratropical and subtropical wave climate types in the Indian Ocean, and the tropical wave climate types of the Atlantic and Pacific Oceans exhibit a global warming signal. In the Pacific sector of the Southern Ocean, strong natural variability may mask a global warming signal that is yet to emerge as being statistically significant. In addition, wave climate teleconnections were found across the world that can provide a framework for joint strategies to achieve the goals of climate adaption for resilient coastal communities and environments

On the multiplicativity of quantum cat maps

The quantum mechanical propagators of the linear automorphisms of the two-torus (cat maps) determine a projective unitary representation of the theta group, known as Weil's representation. We prove that there exists an appropriate choice of phases in the propagators that defines a proper representation of the theta group. We also give explicit formulae for the propagators in this representation.Comment: Revised version: proof of the main theorem simplified. 21 page

Shimura varieties in the Torelli locus via Galois coverings of elliptic curves

We study Shimura subvarieties of $\mathsf{A}_g$ obtained from families of Galois coverings $f: C \rightarrow C'$ where $C'$ is a smooth complex projective curve of genus $g' \geq 1$ and $g= g(C)$. We give the complete list of all such families that satisfy a simple sufficient condition that ensures that the closure of the image of the family via the Torelli map yields a Shimura subvariety of $\mathsf{A}_g$ for $g' =1,2$ and for all $g \geq 2,4$ and for $g' > 2$ and $g \leq 9$. In a previous work of the first and second author together with A. Ghigi [FGP] similar computations were done in the case $g'=0$. Here we find 6 families of Galois coverings, all with $g' = 1$ and $g=2,3,4$ and we show that these are the only families with $g'=1$ satisfying this sufficient condition. We show that among these examples two families yield new Shimura subvarieties of $\mathsf{A}_g$, while the other examples arise from certain Shimura subvarieties of $\mathsf{A}_g$ already obtained as families of Galois coverings of $\mathbb{P}^1$ in [FGP]. Finally we prove that if a family satisfies this sufficient condition with $g'\geq 1$, then $g \leq 6g'+1$.Comment: 18 pages, to appear in Geometriae Dedicat

Elliptic logarithms, diophantine approximation and the Birch and Swinnerton-Dyer conjecture

Most, if not all, unconditional results towards the abc-conjecture rely ultimately on classical Baker's method. In this article, we turn our attention to its elliptic analogue. Using the elliptic Baker's method, we have recently obtained a new upper bound for the height of the S-integral points on an elliptic curve. This bound depends on some parameters related to the Mordell-Weil group of the curve. We deduce here a bound relying on the conjecture of Birch and Swinnerton-Dyer, involving classical, more manageable quantities. We then study which abc-type inequality over number fields could be derived from this elliptic approach.Comment: 20 pages. Some changes, the most important being on Conjecture 3.2, three references added ([Mas75], [MB90] and [Yu94]) and one reference updated [BS12]. Accepted in Bull. Brazil. Mat. So

The Evolving Accretion Disc in the Black Hole X-ray Transient XTE J1859+226

We present HST, RXTE, and UKIRT observations of the broad band spectra of the black hole X-ray transient XTE J1859+226 during the decline from its 1999-2000 outburst. Our UV spectra define the 2175A interstellar absorption feature very well and based on its strength we estimate E(B-V)=0.58+/-0.12. Hence we deredden our spectra and follow the evolution of the spectral energy distribution on the decline from outburst. We find that the UV and optical data, and the X-ray thermal component when detectable, can be fit with a simple blackbody model of an accretion disc heated by internal viscosity and X-ray irradiation, and extending to close to the last stable orbit around the black hole, although the actual inner radius cannot be well constrained. During the decline we see the disc apparently evolving from a model with the edge dominated by irradiative heating towards one where viscous heating is dominant everywhere. The outer disc radius also appears to decrease during the decline; we interpret this as evidence of a cooling wave moving inwards and discuss its implications for the disc instability model. Based on the normalisation of our spectral fits we estimate a likely distance range of 4.6-8.0kpc, although a value outside of this range cannot securely be ruled out.Comment: 10 pages including figures. Accepted for publication in MNRA

The factors associated with pain severity in patients with knee osteoarthritis vary according to the radiographic disease severity: a cross-sectional study

SummaryObjectivesKnee osteoarthritis (OA) pain is suggested to be associated with inflammation and detrimental mechanical loading across the joint. In this cross-sectional study, we simultaneously examined the inflammation and alignment of the lower limb and examined how the pain components varied depending on the disease progression.DesignOne-hundred sixty female medial type of early- [n = 74 in Kellgren–Lawrence (K/L) 2] to advanced-stage (n = 96 in K/L >2) knee OA subjects (70.5 years on average) were enrolled. Knee pain was evaluated using a pain visual analog scale (VAS) and the pain-related subcategory of the Japanese Knee Osteoarthritis Measure (JKOM-pain). The serum interleukin (sIL)-6 level reflecting synovitis, and the high sensitivity C-reactive protein (hs-CRP) level were measured to evaluate the severity of inflammation. The anatomical axis angle (AAA) was measured as an alignment index. The β-coefficient was estimated after adjusting for age and the body mass index (BMI) using a multiple linear regression analysis.ResultsMultiple linear regression analyses showed that the sIL-6 levels, but not AAA, associated with the pain VAS [β = 10.77 (95% confidence interval (CI): 4.14–17.40), P < 0.01] and JKOM-pain scores [β = 3.19 (95% CI: 1.93–4.44), P < 0.001] in the early stage. Conversely, AAA, but not the sIL-6 levels, was found to be associated with the pain VAS [β = −1.29 (95% CI: −2.51 to −0.08), P < 0.05] and JKOM-pain scores [β = −0.49 (95% CI: −0.82 to −0.16), P < 0.01] in the advanced stage.ConclusionsThe presence of a higher level of sIL-6 and the varus alignment of the joint is associated with pain in early- and advanced-stage knee OA patients, respectively