Additive decompositions for rings of modular forms

We study rings of integral modular forms for congruence subgroups as modules over the ring of integral modular forms for the full modular group. In many cases these modules are free or decompose at least into well-understood pieces. We apply this to characterize which rings of modular forms are Cohen--Macaulay and to prove finite generation results. These theorems are based on decomposition results about vector bundles on the compactified moduli stack of elliptic curves.Comment: Complete revision. Comments welcome. arXiv admin note: text overlap with arXiv:1609.0926

Schematic estimating different carbon storage zones across South Georgia’s continental shelf.

<p>The shades are from lightest grey (low benthos carbon), mid grey (moderate benthos carbon), dark grey (high carbon immobilization but low conversion to sequestration) and black (moderate carbon immobilization but high conversion to sequestration). The black box outlines indicate areas that were not sampled with respect to benthos Carbon characteristics.</p

Non-metric multidimensional scaling (nMDS) ordination of benthos using different habitat and benthos categories.

<p>Each point represents a site from <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0179735#pone.0179735.g001" target="_blank">Fig 1</a>. Benthos ordinated by functional groups and displayed in Barnes <i>et al</i>. (2016b) habitat categories (colours in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0179735#pone.0179735.g001" target="_blank">Fig 1</a>)(A). Benthos ordinated by functional groups and displayed in Hogg <i>et al</i>. (2016) habitat categories (B). Benthos ordinated by morphospecies in habitat categories from Barnes <i>et al</i>. (2016q) (C). Benthos ordinated by functional groups, carbon storage and biodiversity characteristics and displayed in Barnes <i>et al</i>. (2016q) habitat categories (D).</p

Measures of benthic colonization, seabed carbon stocks and functional group partitioning at South Georgia, Southern Ocean.

<p>The plots are benthos density and richness (A) at sample sites, in which the size of point increases with density and the darkness of point increases with richness. Functional group diversity (B) in which size increases with number of functional groups and colour represents which functional group is numerically dominant. Carbon accumulation in benthos (C) in which size increases with C magnitude and the colours represent habitat category of <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0179735#pone.0179735.g001" target="_blank">Fig 1</a>. Carbon immobilization and estimate of sequestration (D) in which symbol size increases with magnitude of C immobilization (circles) and sequestration estimate (stars). All data are given in supplementary materials <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0179735#pone.0179735.s001" target="_blank">S1 Table</a>.</p

Functional group categorization of benthos on South Georgia’s shelf.

<p>Functional group categorization of benthos on South Georgia’s shelf.</p

Carbonaccumulation across macrobenthic functional groups at South Georgia.

<p>The values are GLM ANOVA output, with most significant factor shown in bold. <i>P</i> values are shown (*<i>P</i>< 0.05 and **<i>P</i>< 0.01).</p

Carbonimmobilisation across macrobenthic functional groups at South Georgia.

<p>The values are GLM ANOVA output, with most significant factor shown in bold. <i>P</i> values are shown (*<i>P</i>< 0.05 and **<i>P</i>< 0.01).</p

The Southern Ocean continental shelf around South Georgia, with study sites and major habitat categories of Barnes <i>et al</i>. (2016b).

<p>The habitats are old, outer sediments (blue), young basin sediments (green), fjord and canyons (yellow) and moraines (red).</p

Benthic carbon, functional groups and substratum relationships at South Georgia.

<p>Increase in carbon accumulation and immobilization with number of benthos functional groups (A and B respectively). Carbon immobilization with the proportion of substratum which is hard (boulder and cobble rubble) (C). Number of benthos part burial observations with the proportion of substratum which is mixed (boulder and cobbles with mud) (D). The associated ANOVA statistics are F = 84.6, 85.8, 69.6 and 153.1 for Fig 3A-D respectively, all p<0.001.</p

2b-RAD STRUCTURE alignment of Ophionotus victoriae (STACKS)

1,739 bi-allelic SNP loci using using the STACKS software package