Stable averages of central values of Rankin-Selberg L-functions: some new variants

As shown by Michel-Ramakrishan (2007) and later generalized by Feigon-Whitehouse (2008), there are "stable" formulas for the average central L-value of the Rankin-Selberg convolutions of some holomorphic forms of fixed even weight and large level against a fixed imaginary quadratic theta series. We obtain exact finite formulas for twisted first moments of Rankin-Selberg L-values in much greater generality and prove analogous "stable" formulas when one considers either arbitrary modular twists of large prime power level or real dihedral twists of odd type associated to a Hecke character of mixed signature.Comment: 25 pages; typos corrected in corollaries to Thm 1.2, substantial details added to Sec 2.2--2.3, minor changes throughou

Fonned and Refonned: What God Effects through the Liturgical Assembly of Christians

(Excerpt) It is a privilege to be invited to offer a keynote address to the Institute of Liturgical Studies, where so much serious reflection and energy for the renewal and study of Christian worship has been generated. As will become clear to you almost immediately, keynote today does not mean key in the sense of offering the key concept that is needed to unlock all subsequent deliberation on the important topic of forming Christians. In my case, key refers more frankly and realistically to the indispensable need to start somewhere by opening the door to subsequent work in the institute-with its speakers, its group sessions and workshops, and its liturgies. In a real sense what is needed at this year\u27s institute is not so much a key but a whole ring of keys. We who gather build on the two previous years\u27 work on worship, culture, and catholicity. Having explored the tensions between worship and culture in 1997, and the eschatological dimensions of those relationships in 1998, the institute this year turns to a moment of advocacy for the indispensable task of forming Christians

Transformed and Transforming: What God Effects through the Presence of Christians in the World

(Excerpt) On the model of mystagogy: remember your experience last night at the vigil, please, and recall these words from the eucharistic prayer: God and Father of our Lord Jesus Christ, open our eyes to see your hand at work in the world about us. Deliver us from the presumption of coming to this table for solace only and not for strength; for pardon only and not for renewal. Let the grace of this Holy Communion make us one body, one spirit in Christ that we may worthily serve the world in his name.\u27 And from Welcome to Christ: Dear Christian friends: Baptized into the priesthood of Christ, we are all called by the Holy Spirit to offer ourselves to the Lord of all creation in thanksgiving for all that God has done and continues to do fur us. It is our privilege to affirm those who are endeavoring to carry out their vocation as Christians in the world.

Equidistribution of cusp forms in the level aspect

Let f traverse a sequence of classical holomorphic newforms of fixed weight and increasing squarefree level q tending to infinity. We prove that the pushforward of the mass of f to the modular curve of level 1 equidistributes with respect to the Poincar\'{e} measure. Our result answers affirmatively the squarefree level case of a conjecture spelled out by Kowalski, Michel and Vanderkam (2002) in the spirit of a conjecture of Rudnick and Sarnak (1994). Our proof follows the strategy of Holowinsky and Soundararajan (2008) who show that newforms of level 1 and large weight have equidistributed mass. The new ingredients required to treat forms of fixed weight and large level are an adaptation of Holowinsky's reduction of the problem to one of bounding shifted sums of Fourier coefficients (which on the surface makes sense only in the large weight limit), an evaluation of the p-adic integral needed to extend Watson's formula to the case of three newforms where the level of one divides but need not equal the common squarefree level of the other two, and some additional technical work in the problematic case that the level has many small prime factors.Comment: 24 pages; slightly expanded, nearly accepted for