We prove a general formula for the $p$-adic heights of Heegner points on modular abelian varieties with potentially ordinary (good or semistable) reduction at the primes above $p$. The formula is in terms of the cyclotomic derivative of a Rankin–Selberg $p$-adic $L$-function, which we construct. It generalises previous work of Perrin-Riou, Howard, and the author to the context of the work of Yuan–Zhang–Zhang on the archimedean Gross–Zagier formula and of Waldspurger on toric periods. We further construct analytic functions interpolating Heegner points in the anticyclotomic variables, and obtain a version of our formula for them. It is complemented, when the relevant root number is $+1$ rather than $-1$, by an anticyclotomic version of the Waldspurger formula. When combined with work of Fouquet, the anticyclotomic Gross–Zagier formula implies one divisibility in a $p$-adic Birch and Swinnerton-Dyer conjecture in anticyclotomic families. Other applications described in the text will appear separately.

Given $k\geqslant 2$ , we show that there are at most finitely many rational numbers $x$ and $y\neq 0$ and integers $\ell \geqslant 2$ (with $(k,\ell )\neq (2,2)$ ) for which

$$\begin{eqnarray}x(x+1)\cdots (x+k-1)=y^{\ell }.\end{eqnarray}$$

$\ell$

$(x,y,\ell )$

$y=0$

$\ell <\exp (3^{k})$

A certain sequence of weight $1/2$ modular forms arises in the theory of Borcherds products for modular forms for $\textrm{SL}_{2}(\Z)$. Zagier proved a family of identities between the coefficients of these weight $1/2$ forms and a similar sequence of weight $3/2$ modular forms, which interpolate traces of singular moduli. We obtain the analogous results for modular forms arising from Borcherds products for Hilbert modular forms.

let g be an algebraic group, $\gamma$ an arithmetic lattice of g and $x=\gamma\backslash g$. if h is an algebraic subgroup of g such that $h\cap \gamma$ is a lattice of h, then $\gamma\backslash \gamma h\subset x$ is endowed with a canonical h-invariant probability measure $\mu_h$. using ratner's theory, we give general examples where $\mu_{h_n}$ converges weakly to $\mu_g$ if hn is a strict sequence of algebraic subgroups of g. if $\gamma$ is a congruence subgroup of g, we define another probability measure $\mu_h^a$ on x by using the adelic description of the quotient. we conjecture that $\mu_{h_n}^a$ always converges weakly to $\mu_g$ if hn is a strict sequence. using automorphic forms and l-functions, we describe the case $g=\textit{sl}(2,f)$ for a number field f and a sequence of tori hn. the relation with similar problems on shimura varieties is explained.