Equations to predict methane emissions from cows fed at maintenance energy level in pasture-based systems

Ruminant production is a vital part of food industry but it raises environmental concerns, partly due to the associated methane outputs. Efficient methane mitigation and estimation of emissions from ruminants requires accurate prediction tools. Equations recommended by international organizations or scientific studies have been developed with animals fed conserved forages and concentrates and may be used with caution for grazing cattle. The aim of the current study was to develop prediction equations with animals fed fresh grass in order to be more suitable to pasture-based systems and for animals at lower feeding levels. A study with 25 nonpregnant nonlactating cows fed solely fresh-cut grass at maintenance energy level was performed over two consecutive grazing seasons. Grass of broad feeding quality, due to contrasting harvest dates, maturity, fertilisation and grass varieties, from eight swards was offered. Cows were offered the experimental diets for at least 2 weeks before housed in calorimetric chambers over 3 consecutive days with feed intake measurements and total urine and faeces collections performed daily. Methane emissions were measured over the last 2 days. Prediction models were developed from 100 3-day averaged records. Internal validation of these equations, and those recommended in literature, was performed. The existing in greenhouse gas inventories models under-estimated methane emissions from animals fed fresh-cut grass at maintenance while the new models, using the same predictors, improved prediction accuracy. Error in methane outputs prediction was decreased when grass nutrient, metabolisable energy and digestible organic matter concentrations were added as predictors to equations already containing dry matter or energy intakes, possibly because they explain feed digestibility and the type of energy-supplying nutrients more efficiently. Predictions based on readily available farm-level data, such as liveweight and grass nutrient concentrations were also generated and performed satisfactorily. New models may be recommended for predictions of methane emissions from grazing cattle at maintenance or low feeding levels.

Transcendental Brauer groups of products of CM elliptic curves

Let L be a number field and let E/L be an elliptic curve with complex multiplication by the ring of integers O_K of an imaginary quadratic field K. We use class field theory and results of Skorobogatov and Zarhin to compute the transcendental part of the Brauer group of the abelian surface ExE. The results for the odd order torsion also apply to the Brauer group of the K3 surface Kum(ExE). We describe explicitly the elliptic curves E/Q with complex multiplication by O_K such that the Brauer group of ExE contains a transcendental element of odd order. We show that such an element gives rise to a Brauer-Manin obstruction to weak approximation on Kum(ExE), while there is no obstruction coming from the algebraic part of the Brauer group.

Computing the Cassels–Tate pairing on the 3-Selmer group of an elliptic curve

We extend the method of Cassels for computing the Cassels-Tate pairing on the 2-Selmer group of an elliptic curve, to the case of 3-Selmer groups. This requires significant modifications to both the local and global parts of the calculation. Our method is practical in sufficiently small examples, and can be used to improve the upper bound for the rank of an elliptic curve obtained by 3-descent.

Shadow lines in the arithmetic of elliptic curves

Let E/Q be an elliptic curve and p a rational prime of good ordinary reduction. For every imaginary quadratic field K/Q satisfying the Heegner hypothesis for E we have a corresponding line in E(K)\otimes Q_p, known as a shadow line. When E/Q has analytic rank 2 and E/K has analytic rank 3, shadow lines are expected to lie in E(Q)\otimes Qp. If, in addition, p splits in K/Q, then shadow lines can be determined using the anticyclotomic p-adic height pairing. We develop an algorithm to compute anticyclotomic p-adic heights which we then use to provide an algorithm to compute shadow lines. We conclude by illustrating these algorithms in a collection of examples.