ON CONSTANT COEFFICIENT PDE SYSTEMS AND INTERSECTION MULTIPLICITIES

In this paper we consider the concept of the multiplicity of intersection points of plane algebraic curves $p,q=0,$ based on partial differential operators. We evaluate the exact number of maximal linearly independent differential conditions of degree $k$ for all $k\ge 0.$ On the other hand, this gives the exact number of maximal linearly independent polynomial and polynomial-exponential solutions, of a given degree $k,$ for homogeneous PDE system $p(D)f=0,$ $q(D)f=0.$

Division by 1–ζ on Superelliptic Curves and Jacobians

Yuri Zarhin gave formulas for “dividing a point on a hyperelliptic curve by 2”. Given a point $P$ on a hyperelliptic curve $\mathcal{C}$ of genus $g$, Zarhin gives the Mumford representation of an effective degree $g$ divisor $D$ satisfying $2(D - g \infty ) \sim P - \infty $. The aim of this paper is to generalize Zarhin’s result to superelliptic curves; instead of dividing by 2, we divide by $1 - \zeta $. There is no Mumford representation for divisors on superelliptic curves, so instead we give formulas for functions that cut out a divisor $D$ satisfying $(1 - \zeta ) D \sim P - \infty $. Additionally, we study the intersection of $(1 - \zeta )^{-1} \mathcal{C}$ and the theta divisor $\Theta $ inside the Jacobian $\mathcal{J}$. We show that the intersection is contained in $\mathcal{J}[1 - \zeta ]$ and compute the intersection multiplicities.

Height pairings on orthogonal Shimura varieties

Let $M$ be the Shimura variety associated to the group of spinor similitudes of a quadratic space over $\mathbb{Q}$ of signature $(n,2)$. We prove a conjecture of Bruinier and Yang, relating the arithmetic intersection multiplicities of special divisors and complex multiplication points on $M$ to the central derivatives of certain $L$-functions. Each such $L$-function is the Rankin–Selberg convolution associated with a cusp form of half-integral weight $n/2+1$, and the weight $n/2$ theta series of a positive definite quadratic space of rank $n$. When $n=1$ the Shimura variety $M$ is a classical quaternionic Shimura curve, and our result is a variant of the Gross–Zagier theorem on heights of Heegner points.

The log-canonical threshold of a plane curve

We give an explicit formula for the log-canonical threshold of a reduced germ of plane curve. The formula depends only on the first two maximal contact values of the branches and their intersection multiplicities. We also improve the two branches formula given in [27].

LOCAL HEIGHTS ON ELLIPTIC CURVES AND INTERSECTION MULTIPLICITIES

In this short note we prove a formula for local heights on elliptic curves over number fields in terms of intersection theory on a regular model over the ring of integers.