By constructing suitable Borcherds forms on Shimura curves and using Schofer’s formula for norms of values of Borcherds forms at CM points, we determine all of the equations of hyperelliptic Shimura curves $X_{0}^{D}(N)$. As a byproduct, we also address the problem of whether a modular form on Shimura curves $X_{0}^{D}(N)/W_{D,N}$ with a divisor supported on CM divisors can be realized as a Borcherds form, where $X_{0}^{D}(N)/W_{D,N}$ denotes the quotient of $X_{0}^{D}(N)$ by all of the Atkin–Lehner involutions. The construction of Borcherds forms is done by solving certain integer programming problems.

We determine the additional structure which arises on the classical limit of a DQ-algebroid.

In this paper, we propose a conjectural identity between the Fourier–Jacobi periods on symplectic groups and the central value of certain Rankin–Selberg $L$-functions. This identity can be viewed as a refinement to the global Gan–Gross–Prasad conjecture for $\text{Sp}(2n)\times \text{Mp}(2m)$. To support this conjectural identity, we show that when $n=m$ and $n=m\pm 1$, it can be deduced from the Ichino–Ikeda conjecture in some cases via theta correspondences. As a corollary, the conjectural identity holds when $n=m=1$ or when $n=2$, $m=1$ and the automorphic representation on the bigger group is endoscopic.

We study the $p$-adic variation of triangulations over $p$-adic families of $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$-modules. In particular, we study certain canonical sub-filtrations of the pointwise triangulations and show that they extend to affinoid neighborhoods of crystalline points. This generalizes results of Kedlaya, Pottharst and Xiao and (independently) Liu in the case where one expects the entire triangulation to extend. We also study the ramification of weight parameters over natural $p$-adic families.

In this article, for nilpotent orbits of ramified quasi-split unitary groups with two Jordan blocks, we give closed formulas for their Shalika germs at certain equi-valued elements with half-integral depth previously studied by Hales. Associated with these elements are hyperelliptic curves defined over the residue field, and the numbers we obtain can be expressed in terms of Frobenius eigenvalues on the first $\ell$-adic cohomology of the curves, generalizing previous result of Hales on stable subregular Shalika germs. These Shalika germ formulas imply new results on stability and endoscopic transfer of nilpotent orbital integrals of ramified unitary groups. We also describe how the same numbers appear in the local character expansions of specific supercuspidal representations and consequently dimensions of degenerate Whittaker models.

We correct the proof of the main result of the paper, Theorem 5.7. Our corrected proof relies on weaker versions of a number of intermediate results from the paper. The original, more general, versions of these statements are not known to be true.

We correct an error in our paper ‘Connected components of affine Deligne–Lusztig varieties in mixed characteristic’.