# Workshop on Arithmetic and Complex Dynamics (17w5081)

Arriving in Oaxaca, Mexico Sunday, November 12 and departing Friday November 17, 2017

## Organizers

Holly Krieger (University of Cambridge)

Laura DeMarco (Northwestern University)

Mattias Jonsson (University of Michigan)

## Objectives

This workshop will bring together leading researchers from complex dynamics, non-Archimedean analysis and geometry, and algebraic and arithmetic geometry, with the goal of making progress on current problems in arithmetic dynamics. Recent breakthroughs have come from groups of mathematicians whose backgrounds span these varied disciplines. We will focus on sharing ideas and tools among researchers from diverse specialties in hopes of inspiring new questions and collaborations in arithmetic dynamics.

Arithmetic dynamics is an exciting and relatively new field, with many significant recent developments, so we plan to include a considerable number of young researchers. Our intended list of participants also includes a number of experts in complex dynamics and arithmetic geometry, since much of arithmetic dynamics concerns the connections between these two fields. For instance, the recent work on unlikely intersections in complex dynamics originated with a collaboration between non-Archimedean analyst Baker and complex dynamicist DeMarco, inspired by questions of arithmetic geometers Poonen, Masser, and Zannier. The workshop will sustain these extant collaborations, and found new cross-discipline research groups. To encourage this, the workshop will include casual open problem sessions on selected evenings during the week, and a speaker schedule that allows for interaction and discussion between talks.

We believe that the diverse group of researchers at the workshop will inspire many new questions in arithmetic dynamics and related fields; however, the workshop will focus on three main areas of research to guide the talks and open problem sessions.

Arithmetic dynamics is an exciting and relatively new field, with many significant recent developments, so we plan to include a considerable number of young researchers. Our intended list of participants also includes a number of experts in complex dynamics and arithmetic geometry, since much of arithmetic dynamics concerns the connections between these two fields. For instance, the recent work on unlikely intersections in complex dynamics originated with a collaboration between non-Archimedean analyst Baker and complex dynamicist DeMarco, inspired by questions of arithmetic geometers Poonen, Masser, and Zannier. The workshop will sustain these extant collaborations, and found new cross-discipline research groups. To encourage this, the workshop will include casual open problem sessions on selected evenings during the week, and a speaker schedule that allows for interaction and discussion between talks.

We believe that the diverse group of researchers at the workshop will inspire many new questions in arithmetic dynamics and related fields; however, the workshop will focus on three main areas of research to guide the talks and open problem sessions.

**Objective 1 (Unlikely intersections).} Bring participants up to date on recent progress in unlikely intersections in complex dynamics and in Diophantine geometry, and discuss the technical obstacles which must be overcome for future research, for example, towards developing a clean, well-formulated dynamical Andr\'e-Oort conjecture. Also of primary interest will be possibilities towards proving higher-dimensional versions of this conjecture, since all proved cases to date concern 1-dimensional varieties. Current results in this direction include progress on the dynamical analogs of well-known conjectures in arithmetic geometry, such as Mordell-Lang, Manin-Mumford, and Andr\'e-Oort [BD, BGT,Xie:DML, DF, GTZ, DWY, GKN, GKNY, GHT}. The dynamical proofs use a rich collection of techniques which include the deep equidistribution theorems of~\cite{BR, CL, FRL, YZ, Zhang:ICM], classical techniques of complex analysis and potential theory, and Ritt's theory of decomposition of polynomials, and are all illustrative of the general principle of unlikely intersections in arithmetic geometry, as in \cite{Andre, BMZ, O**.**Objective 2 (non-Archimedean geometry/analysis).} Discuss the status of equidistribution theorems in various contexts, building on work of \cite{FRL, BR, CL, YZ**, and the earlier ideas of Szpiro-Ullmo-Zhang, used to study abelian varieties. We now that we understand that weaker hypotheses are needed for various applications, and also that equidstribution does not always hold, even for "nice" height functions. As examples, there is the recent (non-dynamical) work of Rivera-Letelier, Burgos Gil, Philippon, and Sombra, studying the equidistribution on toric varieties, and the dynamical example of DeMarco, Wang and Ye showing that a desired ``adelic metrized line bundle" in the sense of Zhang is not always adelic. The existing equidistribution theorems have been used in many dynamical applications recently.**Objective 3 (Heights in arithmetic dynamics).} The concept of height plays a key role in arithmetic geometry, for example in Falting's proof of the Mordell conjecture and the proof of the Bogomolov conjecture by Szpiro-Ullmo-Zhang. In arithmetic dynamics, they are everywhere. Given a rational self-map of a projective variety defined over a number field, Silverman has formulated several conjectures that relate the asymptotic growth of the height along the orbit to quantities such as the dynamical degrees of the map. Special cases of these conjectures were recently proved in~\cite{Silverman:canheights, KS13,KS14,JW,JR**. The workshop will feature new developments in this area, as well as related topics such as heights for finitely generated extensions of the rational numbers as studied by Moriwaki or Yuan-Zhang.#### Bibliography

- [Andre} \au{Y.~Andr\'e}

{Finitude des couples d'invariants modulaires singuliers sur une courbe alg\'ebrique plane non modulaire.]

*J. Reine Angew. Math.*505 (1998), 203--208. - [BD] M.~Baker and L.~DeMarco.

{Special curves and postcritically-finite polynomials}.

*Forum of Mathematics Pi***1**(2013), 35 pages. - [BR} \au{M.~Baker, R.~Rumely]

{Potential theory on the Berkovich projective line}.

*Mathematical Surveys and Monographs, 159.*

American Mathematical Society, Providence, RI, 2010. - [BGT} \au{J.~Bell, D.~Ghioca, T.~Tucker]

{The dynamical Mordell-Lang problem for \'etale maps}.

*Amer. J. Math.***132**(2010), 1655--1675. - [BMZ} \au{E.~Bombieri, D.~Masser, U.~Zannier]

{Finiteness results for multiplicatively dependent points on complex curves}.

*Michigan Math J.***51**(2003), no. 3, 451--466. - [CL} \au{A.~Chambert-Loir]

{Mesures et \'equidistribution sur des espaces de Berkovich}.

*J. Reine Angew. Math.***595**(2006), 215--235, - [DWY} \au{L.~DeMarco, X.~Wang, H.~Ye]

{Torsion points and the Latt\`es family}.

{To appear,*Amer. J. Math.}* - [DF} \au{R.~Dujardin, C.~Favre]

{The dynamical Manin-Mumford problem for plane polynomial automorphisms}.

\texttt{arXiv:1405.1377}. - [FRL} \au{C.~Favre, J.~Rivera-Letelier]

{Equidistribution quantitative des points de petite hauteur sur la droite projective}.

*Math. Ann.***335**(2006), 311--361. - [GHT} \au{D.~Ghioca, L.-C.~Hsia,T.~Tucker]

{Preperiodic points for families of polynomials}.

*Algebra \& Number Theory***7**(2012), 701--732. - [GKN} \au{D.~Ghioca, H.~Krieger, K.~Nguyen]

{A case of the dynamical Andr\'e-Oort conjecture}.

to appear in*Intern. Math. Res. Not.*, 2015. - [GKNY} \au{D.~Ghioca, H.~Krieger, K.~Nguyen, H.~Ye]

{The dynamical Andr\'e-Oort conjecture for unicritical polynomials}.

\texttt{arXiv:1505.01447}. - [GTZ} \au{D.~Ghioca, T.~Tucker, S.~Zhang]

{Towards a dynamical Manin-Mumford conjecture}.

*Int. Math. Res. Not.*2011, no. 22, 5109--5122. - [JR] M. Jonsson, P. Reschke.

{On the complex dynamics of birational surface maps defined over number fields}.

\texttt{arXiv:1505.03559}. - [JW} \au{M.~Jonsson, E.~Wulcan]

{Canonical heights for plane polynomial maps of small topological degree}.

*Math. Res. Lett.***19**(2012), 1207--1217. - [KS13] S.~Kawaguchi and J.~H.~Silverman.

{Dynamical canonical heights for Jordan blocks, arithmetic degrees of orbits, and nef canonical heights on abelian varieties}.

To appear in*Trans. Amer. Math. Soc.*

\texttt{arXiv:1301.4964}. - [KS14] S.~Kawaguchi and J.~H.~Silverman.

{On the dynamical and arithmetic degrees of rational selfmans of algebraic varieties}.

*J. reine angew. Math.*

Published online 2014. - [MZ] D.~Masser and U.~Zannier.

{Torsion anomalous points and families of elliptic curves}.

*Amer. J. Math.***132**(2010), 1677--1691. - [Silverman:moduli] J.~H.~Silverman.

{Moduli spaces and arithmetic dynamics}, volume~30 of {\em CRM Monograph Series}.

American Mathematical Society, Providence, RI, 2012. - [Silverman:canheights] J.~H.~Silverman.

{Dynamical degree, arithmetic entropy and canonical heights for dominant rational selfmaps of projective space}.

*Ergodic Th. Dyn. Systems.***34**(2014), 647--678. - [Xie:DML] J.~Xie.

{ Dynamical Mordell-Lang Conjecture for polynomial endomorphisms of the affine plane}.

\texttt{arXiv:1503.00773}. - [YZ] X.~Yuan and S.-W.~Zhang.

{The arithmetic Hodge Theorem for adelic line bundles I,II}.

\texttt{arXiv:1304.3538,arXiv:1304.3539}. - [Zhang:ICM} S.-W. Zhang.

{Small points and {A}rakelov theory}.

In*Proceedings of the {I}nternational {C*ongress of {M}athematicians, {V}ol. {II] ({B}erlin, 1998)}, number Extra Vol. II, pages 217--225 (electronic), 1998. - [Zhang:questions} \au{S.-W.~Zhang]

{Distributions in algebraic dynamics}.

*Surveys in differential geometry.*Vol. X, 381--430.

Int. Press, Somerville, MA, 2006.