I will introduce ‘canonical’ algebraic cycles for motives M enjoying a certain symmetry - for instance, some symmetric powers of elliptic curves. The construction is based on works of Kudla and Liu on some (conjecturally modular) theta series valued in Chow groups of Shimura varieties. The cycles have Heegner-point-like features that allow, under some assumptions, to establish an analogue of the BSD conjecture for M at an ordinary prime p. Namely, if the p-adic L-function of M vanishes at 0 to order exactly 1, then the Selmer group of M has rank 1 and it is generated by classes of algebraic cycles. Partly joint work with Yifeng Liu.