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We compute equations for real multiplication on the divisor classes of genus two curves via algebraic correspondences. We do so by implementing van Wamelens method for computing equations for endomorphisms of Jacobians on examples drawn from the algebraic models for Hilbert modular surfaces computed by Elkies and Kumar. We also compute a correspondence over the universal family for the Hilbert modular surface of discriminant 5 and use our equations to prove a conjecture of A. Wright on dynamics over the moduli space of Riemann surfaces.
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We prove that the generic point of a Hilbert modular four-fold is not a Jacobian. The proof uses degeneration techniques and is independent of properties of the mapping class group used in preceding papers on locally symmetric subvarieties of the moduli space of abelian varieties contained in the Schottky locus.
Let $X$ be a quasi-projective variety and $fcolon Xto X$ a finite surjective endomorphism. We consider Zariski Dense Orbit Conjecture (ZDO), and Adelic Zariski Dense Orbit Conjecture (AZO). We consider also Kawaguchi-Silverman Conjecture (KSC) asserting that the (first) dynamical degree $d_1(f)$ of $f$ equals the arithmetic degree $alpha_f(P)$ at a point $P$ having Zariski dense $f$-forward orbit. Assuming $X$ is a smooth affine surface, such that the log Kodaira dimension $bar{kappa}(X)$ is non-negative (resp. the etale fundamental group $pi_1^{text{et}}(X)$ is infinite), we confirm AZO, (hence) ZDO, and KSC (when $operatorname{deg}(f)geq 2$) (resp. AZO and hence ZDO). We also prove ZDO (resp. AZO and hence ZDO) for every surjective endomorphism on any projective variety with larger first dynamical degree (resp. every dominant endomorphism of any semiabelian variety).
We consider the potential density of rational points on an algebraic variety defined over a number field $K$, i.e., the property that the set of rational points of $X$ becomes Zariski dense after a finite field extension of $K$. For a non-uniruled projective variety with an int-amplified endomorphism, we show that it always satisfies potential density. When a rationally connected variety admits an int-amplified endomorphism, we prove that there exists some rational curve with a Zariski dense forward orbit, assuming the Zariski dense orbit conjecture in lower dimensions. As an application, we prove the potential density for projective varieties with int-amplified endomorphisms in dimension $leq 3$. We also study the existence of densely many rational points with the maximal arithmetic degree over a sufficiently large number field.
A portrait is a combinatorial model for a discrete dynamical system on a finite set. We study the geometry of portrait moduli spaces, whose points correspond to equivalence classes of point configurations on the affine line for which there exist polynomials realizing the dynamics of a given portrait. We present results and pose questions inspired by a large-scale computational survey of intersections of portrait moduli spaces for polynomials in low degree.
We prove the Lefschetz hyperplane section theorem using a simpler machinery by making the observation that we can compose the Lefschetz Pencil with a Real Morse function to get a map from the variety to $mathbb{R}$ which is close to being a Real Morse function. The proof uses a new method unlike the conventional one which uses vanishing cycles, thimbles and monodromy. We prove the genus formula for plane curves using Morse theory, Lefschetz pencil and Bezouts theorem. And then we prove the Riemann Hurwitz formula for ramified maps between curves by employing techniques from deformation theory. Lastly, we prove the Lefschetz Hyperplane Section Theorem solely using Real Morse Theory and exact sequences.
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