I work at the interface of arithmetic geometry and the analytic theory of automorphic forms. On the one hand, my work deals with the Iwasawa main conjectures for modular elliptic curves in dihedral extensions of CM fields, motivated by a novel strategy I proposed in "Some remarks on the two-variable main conjectures of Iwasawa theory for elliptic curves without complex multiplication" to address the two-variable main conjectures for elliptic curves using cyclic basechange.

Progress on this open problem is expected to reveal insights into the remaining cases of the conjecture of Birch and Swinnerton-Dyer. Here, I have published several theorems generalizing the required construction of p-adic L-functions and Euler systems to totally real fields to achieve part of this motivation — among them the papers "On the dihedral main conjectures of Iwasawa theory for Hilbert modular eigenforms", and "On the dihedral Euler characteristics of Selmer groups of abelian varieties".

This programme leads to giving a pure representation theoretic account of the p-adic L-functions constructed via toric period integral presentations via theorems of Waldspurger and others in my paper "p-adic interpolation of automorphic periods for GL(2)", which following recent advances on the Ichino-Ikeda conjecture I have generalized to higher-rank unitary groups in a new preprint. These works have now been used in various other constructions. A major hidden requirement for this programme is to know the nontriviality of the p-adic L-functions. This translates into a rich set of problems about moments of L-functions on higher-rank groups. Motivated again by the two-variable main conjectures, I have worked on a long-term project to derive nonvanishing estimates for families of central values of the corresponding (degree four) Rankin-Selberg L-functions in my papers "Rankin-Selberg L-functions in cyclotomic towers I, II, and III".

These have led me to consider other related problems with moments of GL(n)-automorphic L-functions, such as improvements on the Luo-Rudnick-Sarnak bounds towards the generalized Ramanujan conjecture for GL(n)-automorphic forms. Here, in preliminary work, I consider averages over primitive Dirichlet characters of prime-power modulus in "Dirichlet twists of GL(n)-automorphic L-functions and hyper-Kloosterman Dirichlet series", and derive special Voronoi summation formulae which I propose to develop to study Poincaré and Eisenstein series on GL(n). I have also reformulated open questions about moments of higher-rank L-functions in terms of certain L^2-automorphic forms on GL(2), which can be viewed as a stepping stone to progress on several major open problems. Finally, I have written a new work on integral presentations for central derivative values L-functions of elliptic curves over real quadratic fields à la Gross-Zagier, and look forward to pursuing more work in this direction.

Selected publications:

*J. Van Order, Dirichlet twists of GL(n)-automorphic L-functions and hyper-Kloosterman Dirichlet
series, Ann. Fac. Sci. Toulouse Math. (6) 30 no. 3 (2021), 633-703.

*J. Van Order, p-adic interpolation of automorphic periods for GL(2), Doc. Math. 22 (2017),
1467-1499.

*J. Van Order, On the dihedral Euler characteristics of Selmer groups of abelian varieties, in
"Arithmetic and Geometry", eds. L. Dieulefait et al., London Math Soc. Lecture Notes 420, Cambridge
University Press (2015).

*J. Van Order, Some remarks on the two-variable main conjecture of Iwasawa theory for elliptic
curves without complex multiplication, J. Algebra 350, (2012), 273 - 299.

*J. Van Order, On the dihedral main conjectures of Iwasawa theory for Hilbert modular eigenforms,
Canad. J. Math. 65 no. 2 (2013), 403-466.