The Shadow Conjecture

The Batyrev-Manin conjecture predicts how many rational points of bounded height appear on algebraic varieties. It’s a precise asymptotic: the count grows like B^a (log B)^{b-1} with specific exponents determined by the geometry. For decades, verifying the conjecture has been a case-by-case affair, each variety requiring its own argument.

Bongiorno shows that for toric stacks, the Batyrev-Manin prediction follows as a consequence of understanding how rational points distribute across multiple height functions simultaneously. The multi-height distribution — how points are counted when you measure them with several heights at once — implies the single-height asymptotic.

The through-claim: the Batyrev-Manin conjecture is not an independent statement about point-counting. It is a shadow — a one-dimensional projection — of a richer multi-height structure. The conjecture holds because it is forced to hold by the finer distribution.

The method uses a generalized hyperbola method (from Pieropan and Schindler) to decompose the counting problem along multiple height axes simultaneously. Each axis gives a partial view; the full multi-height picture is overdetermined, and the single-height asymptotic falls out as a corollary.

If a conjecture seems hard to prove directly, it may be because it’s a projection of something easier to see in higher dimension.