Toric periods, modular forms, and number theory

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The research program which I propose over the next 5 years represents an evolution  of the themes I have explored for almost 20 years. In the early 2000s, I introduced an unexpected connection between ergodic theory and $p$-adic number theory; this fundamental insight is still the topic of active and fruitful research, and I now propose to extend the connections by introducing ideas from the $p$-adic Langlands programme, representation theory, and the Langlands programme in characteristic $p$. The overarching theme is that of toric periods, namely, the integrals of complex valued automorphic forms on a quaternion algebra along the orbits of embedded maximal tori. In my earlier work, the point was to show that these period integrals are nonzero, and this was accomplished by using Ratner's theorems  from ergodic theory on uniform distribution. This time, however, I propose to look at toric periods through completely different lenses: those of modular and p-adic representation theory, which is to say, representation theory in characteristic p and with coefficients in $p$-adic fields. The germs of this program are contained in the papers [Vat17] and [Vat18], but the new directions are unexpected and intriguing. The principal directions I propose to pursue are to develop the theory of toric periods for characteristic p and p-adic representations of GL_2(Q_p). I have already developed pieces of this theory in my work on test vectors, and completing the theory would lead to a theta correspondence in characteristic p, relating modular representations of PGL_2(Q_p) and the metaplectic cover of SL_2(Q_p). Such a correspondence has long been speculated, and it is exciting and energizing to feel that one is close to achieving it. Analogously, I propose to consider the geometry of p-adic representations of GL_2(Q_p), and to use the geometry of étale covers of Drinfeld's upper half plane to extend the period integral calculations made by Bertolini and Darmon for the special representations (which occur at the bottom of the tower) to the case of ramified supercuspidal representations.