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In [], the authors proved an explicit formula for the arithmetic intersection number $\left(CM\left(K\right).G_1\right)$ on the Siegel moduli space of abelian surfaces, under some assumptions on the quartic CM field $K$. These intersection numbers allow one to compute the denominators of Igusa class polynomials, which has important applications to the construction of genus $2$ curves for use in cryptography. One of the main tools in the proof was a previous result of the authors [] generalizing the singular moduli formula of Gross and...

One can define class invariants for a quartic primitive CM field $K$ as special values of certain Siegel (or Hilbert) modular functions at CM points corresponding to $K$. Such constructions were given by de Shalit-Goren and Lauter. We provide explicit bounds on the primes appearing in the denominators of these algebraic numbers. This allows us, in particular, to construct $S$-units in certain abelian extensions of a reflex field of $K$, where $S$ is effectively determined by $K$, and to bound the primes appearing...

We give new arguments that improve the known upper bounds on the maximal number $N_q\left(g\right)$ of rational points of a curve of genus $g$ over a finite field ${𝔽}_q$, for a number of pairs $(q,g)$. Given a pair $(q,g)$ and an integer $N$, we determine the possible zeta functions of genus-$g$ curves over ${𝔽}_q$ with $N$ points, and then deduce properties of the curves from their zeta functions. In many cases we can show that a genus-$g$ curve over ${𝔽}_q$ with $N$ points must have a low-degree map to another curve over ${𝔽}_q$, and often this is enough to...