Congruences Among Power Series Coefficients of Modular Forms

Many authors have investigated the congruence relations amongst the coefficients of power series expansions of modular forms $f$ in modular functions $t$. In a recent paper, R. Osburn and B. Sahu examine several power series expansions and prove that the coefficients exhibit congruence relations similar to the congruences satisfied by the Ap\'ery numbers associated with the irrationality of $\zeta(3)$. We show that many of the examples of Osburn and Sahu are members of infinite families.Comment: 24 page

Discipleship Goals in the Holy Trinity Brompton [HTB] Network of Churches. An exploration of the theological trajectories of the movement and their implications for the future of the network.

The Holy Trinity Brompton Network is comprised of some of the most influential and well- funded churches within the Church of England, closely linked to the Archbishop of Canterbury, Justin Welby, and known internationally for the Alpha Course which originated at Holy Trinity Brompton [HTB] in the 1980s. As a network it provides a window into contemporary charismatic Christianity in the UK and beyond. This thesis details HTB’s roots in an Upper-Middle Class Anglican Evangelicalism prevalent in the 1950s-1980s showing how those roots shaped its future trajectory. It then explores the impact of three waves of the charismatic renewal movement on discipleship goals in the church, paying attention to unconscious by-products of shifts in praxis, including substantive liturgical changes as John Wimber’s ‘Worship - Teaching - Ministry’ service structure refocused worship. It then considers the impact of a pragmatic consumerism and managerial approach on discipleship goals from developing the network into what Pete Ward calls ‘branded church’. Using the Theological Action Research framework and ethnographic study to build a thick description which draws from interview data and other interactions with the network, it argues that these influences have caused an unacknowledged drift in ‘normative’, ‘espoused’ and ‘operant’ theology as the network has reorientated, and that a widening gap has emerged between what classic HTB network leaders would still see as their normative theology and the operant and espoused theologies evident in many network churches and leaders today. A conversation partner of the 18th Century revivalists John Wesley and George Whitefield is introduced to show the full extent of this shift on discipleship goals when contrasted with the start of the evangelical era. This leads to a reflection on how the HTB network, as a case study in contemporary charismatic Christianity, has unintentionally vacated its evangelical Anglican heritage and how out of David Bebbington’s quadrilateral definition of evangelicals as ‘crucicentric, biblicist, conversionist and activist’, the HTB network can only fully claim to be ‘activist’. This is explored in relation to discipleship goals, and concludes with ways in which the network could reconnect with its evangelical heritage to bring normative, espoused and operant theologies into closer harmony