Curves and L-functions List of Participantsmatyd/Trieste2017/Booklet.pdf · L-functions and Galois...

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Curves and L-functionsList of Participants

Suman AhmedMohali

I work in Iwasawa theory of elliptic curves. So far Ihave been interested in comparing the lambda inva-riants, p-Selmer rank, global and local root numbersof two rational elliptic curves with good reduction atan odd prime p and equivalent mod p Galois repre-sentations.

Ambreen AhmedLahore

Mohamed AlaaeldinCairo

My area of interest is the arithmetic of elliptic cur-ves defined over number fields. I have been workingon exploring the rank of elliptic curves in familiesof quadratic twists. More precisely, I am concernedabout the construction of families of tuples of ellipticcurves for which simultaneous quadratic twists areof high rank.

ICTP Workshop & Participants

Brandon AlbertsUniversity of Wisconsin-Madison

I am a fifth year graduate student interested in al-gebraic number theory and arithmetic statistics. Cur-rently I am studying nonabelian Cohen-Lenstra heu-ristics, or more specifically the distribution of unra-mified nonabelian extensions over quadratic fields.

Samuele AnniHeidelberg

The main focus of my research is the study of Galoisrepresentations and automorphic forms. In particular,I am interested in the interplay between arithmeticgeometry and representation theory, also consideringthe related algorithmic aspects. Currently, I am stu-dying congruences between modular forms and ques-tions regarding the inverse Galois problem for repre-sentations attached to torsion subgroups of abelianvarieties.

Jennifer BalakrishnanBoston

My research is motivated by various aspects of theclassical and p-adic Birch and Swinnerton-Dyer con-jectures, as well as the problem of algorithmicallyfinding rational points on curves. I am particularlyinterested in explicit methods involving Coleman in-tegration and p-adic heights.

Gregorio BaldiUCL London

I am mainly interested in Arithmetic Geometry overnumber fields, in particular Galois representations at-tached to abelian varieties and K3 surfaces. In bothcases their moduli spaces have the structure of a Shi-mura variety, whose geometry affects the arithmetic.This led me also to more general questions aboutShimura varieties.


Alexander BestBoston

Alexander BettsKCL

I am primarily interested in applications of thetechniques of Minhyong Kim’s unipotent motivicanabelian geometry to Diophantine and arithmeticproblems, most recently including local and globalheights on abelian varieties and curves. Outside thisrather general area, I am interested in explicit pro-blems in number theory, mainly focusing on Tama-gawa numbers of abelian varieties and their behavi-

our under changing the variety or base field.

Francesca BianchiOxford

My work mostly concerns the arithmetic of ellipticcurves. In particular, my focus has been on analyticp-adic L-functions and their mu-invariants and thebehaviour of p-adic heights in families. I am also in-terested in Coleman integration on hyperelliptic cur-ves. I enjoy both the theoretical and computationalaspects of the area.


Matthew BisattKing’s College London

I am primarily interested in consequences ofthe Birch–Swinnerton-Dyer conjecture; in par-ticular studying root numbers of abelian va-rieties and their twists by Artin represesen-tations. Twisting can conjecturally force highorders of vanishing of the corresponding L-functions so I am also interested in howcompatible this theory is with current conjectu-res.

Jorge Cely

I work on motivic integration (Cluckers - Loeser the-ory) and applications to representation theory of p-adic groups (e.g. the fundamental lemma). I also haveinterest in applied model theory.

Stephanie ChanUCL London

I am a PhD student interested in the arithmeticof elliptic curves. My current research concerns thearithmetic statistics of certain families, in particularthe congruent number curves, via methods involvinga combination of algebraic and analytic techniques.I am also interested in the computation of rank dis-tributions.

Nirvana CoppolaPisa

I am a MSc student and I am currently studying mo-dular forms and their properties. In particular, I amworking on a theorem by Deligne and Serre, whoseaim is to attach a certain Galois representation to amodular form of weight 1, and on the converse pro-blem to associate a modular form to a given Galoisrepresentation, that leads to Serre’s Conjecture.


Daniele Cozzo

Marco D’addezioBerlin

Pranabesh DasNew Delhi

I am interested in the explicit solutions of exponen-tial Diophantine equations over integers and moregeneral totally real number fields. I have used mo-dular method, linear forms in logarithms, Chabautymethod during my PhD to obtain integral solutionsof a class of super elliptic curves. I am also interested

in Mahler’s method in transcendence and explicit ABC conjecture.

Julie DesjardinsGrenoble

I work on diophantine and analytic number theoryquestions. I am particularly interested in the densityof rational points on algebraic varieties such as ellip-tic surfaces and del Pezzo surfaces of degree 1. I usevarious methods, including the study of the variationof the root number, and I’m interested in problemssuch as the Squarefree conjecture, Chowla’s conjec-ture and the parity conjecture. I am also interestedin K3 surfaces and 3-dimensional Fano varieties.


Netan DograICL London

I am interested in rational points on algebraic varie-ties. My main focus has been on determining the setof rational points on curves of genus bigger than 1,using a method due to Minhyong Kim which invol-ves a combination of fundamental groups and p-adicHodge theory. I am also interested in various related

questions concerning the arithmetic information contained in the Galois action onfundamental groups of curves.

Tim DokchitserUniversity of Bristol

Most of my work concerns arithmetic of curves, theirL-functions and Galois representations. I am cur-rently very interested in various invariants of hig-her genus curves that are relevant to the Birch-Swinnerton-Dyer conjecture: regular and semistablemodels, conductors, root numbers, regular differenti-als and so on. I do quite a lot of algorithmic numbertheory, especially in Magma. I also have a recent petproject on names of finite groups, groupnames.org.

Vladimir DokchitserKing’s College London

The main focus of my research is the arithmetic of el-liptic curves and abelian varieties, especially the vari-ous phenomena that are predicted by the BSD and ot-her L-value conjectures. In particular, I’m interestedin the parity conjecture and root numbers, Galois re-presentations and various local invariants. RecentlyI’ve been working a fair bit on hyperelliptic curvesand their Jacobians over local fields (regular models,Galois representations, conductors, Tamagawa num-

bers...).


Thomas FisherCambridge

I work on the arithmetic of elliptic curves, specifi-cally descent calculations. I have contributed severalof the programs in Magma for carrying out higherdescents. I am also interested in applications of in-variant theory to arithmetic geometry, for example instudying congruences of elliptic curves and visibilityof Tate-Shafarevich groups.

Stevan GajovicOldenburg

Hui GaoHelsinki

Jedrzej GarnekPoznan

I’m a PhD student interested in arithmetic geome-try and algebraic number theory. So far my researchfocused mostly on p-torsion of elliptic curves over p-adic fields. Recently I’ve been working also on boundsfor the class number of division fields coming fromabelian varieties.


Francesca GattiBarcelona

I’m a PhD student and I’m working on generalizati-ons of the Elliptic-Stark conjecture. This conjecture,made by H. Darmon, A. Lauder and V. Rotger, rela-tes the central value of a p-adic L-function attachedto an elliptic curve (twisted by an Artin representa-tion) to a regulator attached to points on the ellipticcurve. More precisely, I’m studying a version of thisconjecture in which the role of the elliptic curve is

played by a cuspform of weight greater or equal than 2.

Neslihan GirginBogazici Universityl

I am a PhD student and I have just started to workon Arithmetic of Function Fields over Finite Fields.My research is motivated by some results of Cohen,McNay and Chapman which are about an iterativeconstruction of irreducible polynomials of 2-powerdegree over finite fields of odd order. I am trying tointerpret these results on the extensions of functionfields over finite fields. After that, I would like tounderstand finite fields elements of high order arisingfrom modular curves. Also before, I was interested inModular Forms, Modular Curves, Hecke Operatorsand Eichler-Shimura Relation for my master thesis.

Bayarmagnai GombodorjUlaanbaatar


Nadav GropperWeizmann Institute of Science

My main research focus is the arithmetic of ellip-tic curves over number fields. I spent most of mytime so far on the study of the Birch-Swinnerton-Dyer conjecture for elliptic curves with complex mul-tiplication and the Tate-Shafarevich groups of suchcurves. I am also interested in Berkovich’s theory ofnon-Archimedean analytic spaces.

Jeroen HanselmanUlm

I’m a Phd student, and I’m mostly interested in thearithmetic of curves of low genus. My research goalis to be able to explicitly glue curves of low genusalong their n-torsion. This will allow us to explicitlyconstruct higher genus curves with a specific endo-morphism ring.

Sachi HashimotoBoston

Daniel HastUniversity of Wisconsin-Madison

The main focus of my research is the non-abelianChabauty-Kim method for studying rational pointson higher genus curves. I am currently interested inseveral related problems in this area, including Sel-mer varieties for curves over number fields, genera-lizations to the bad reduction case, and the relationbetween Selmer varieties and unramified correspon-dences in the sense of Bogomolov-Tschinkel.


Richard HattonNottingham

I am a PhD student who is interested in the arithme-tic of elliptic curves. My current research involveslooking at modular points on elliptic curves, speci-fically self-points, and seeing if I can use them tobound certain Selmer groups with methods similar toKolyvagin’s use of creating derivative classes comingfrom Heegner points.

Wei HoMichigan

I’m interested in a wide range of questions inarithmetic geometry, but most of my work has beenrelated to using algebraic geometry, representationtheory, and analytic number theory to attack arithme-tic statistics problems. For example, I’ve spent a lot oftime thinking about explicit constructions of modulispaces for curves (elliptic and otherwise) and other(usually low-dimensional) varieties.

David HolmesLeiden

I spend most of my time thinking about thegeometry and arithmetic of families of no-dal curves and of their jacobians. My fa-vourite topics include existence and proper-ties of Néron models, behaviour of heig-hts in families (especially height-jumping),orders of torsion points, cycles on modulistacks (especially the double ramification cy-cle), moduli of stable maps, and log geome-try.


Elena IkonnikovaStPetersburg

I am now a PhD student at Saint Petersburg StateUniversity, and the area of my research is algebraicnumber theory and arithmetic geometry. I am cur-rently studying arithmetic of higher dimensional for-mal modules, including norm pairings. I also findthe topic of elliptic curves, and especially their cryp-tographic applications, very exciting.

Nikoleta KalaydzhievaUCL London

I am currently a first year PhD student at UCL un-der the supervision of Andrew Granville. Currently,I am working on continued fractions over Q[x] andtheir connection to hyperelliptic curves and quadraticforms.

Mohammed KamelCairo

I work on certain arithmetic questions related to al-gebraic curves. More precisely, I am interested in stu-dying different progressions on elliptic curves and therank of elliptic curves with prescribed torsion groups.Moreover, I have spent some time studying Diophan-tine m-tuples and their links with elliptic curves.

Matija KazalickiZagreb

In general, I study various problems related to thearithmetic of elliptic curves and modular forms. Re-cently, I’ve been working on the construction of rati-onal Diophantine sextuples, on counting Diophantinequadruples over finite fields, and on the connectionbetween the rank of an elliptic curve of prime conduc-tor p, and mod p zeros of the modular form attachedto this elliptic curve by the modularity theorem.


Kiran KedlayaUCSD

Pinar KiliçerOldenburg

Christian KlevdalUtah

Daniel KohenBuenos Aires

My main research interest is the study of rationalelliptic curves. During my PhD thesis I studied ge-neralizations of Heegner point constructions. Now Iam trying to understand some topics regarding p-adicL-series and Iwasawa theory.


Roman KohlsUlm

I am a PhD student interested in algebraic numbertheory and L-functions. Currently I am studying L-functions associated to curves and their local con-stants, such as conductors and root numbers. In par-ticular, I am very interested in the explicit computa-tion of local constants for a given curve.

Zunaira KosarLahore

Sabrina KunzweilerUlm

I have just started doing research in Arithmetic Geo-metry. Currently, my focus is on elliptic curves overlocal fields. Motivated by Ogg’s formula, I study thebehaviour of certain invariants of a curve under ra-mified extensions of its base field. In particular, Iam looking at fields of small residue characteristic(p=2,3).

Emmanuel LecouturierParis 7


Junxian Li

Wanlin LiUniversity of Wisconsin-MadisonI’m interested in various topics in arithmetic geome-try, like ranks of elliptic curves over function fields,maps between hyperelliptic curves, Jacobian varietiesand Galois representations.

Xiaobin LiSwjtu

My current research focus is on computationof higher genus orbifold Gromov-Witten invari-ants and its possible arithmetic property such asmodularity(quasi-modularity). Furthmore, I will in-vestigate its possible relation to L-function.

Guido LidoRome


Qing LiuBordeaux

My main research interests are arithmetic propertiesof algebraic curves and abelian varieties, especiallythose related to their degenerations (stable reductionof curves, Néron models of abelian varieties etc).

Davide LombardoUniversità di Pisa

My research revolves around the arithmetic of abe-lian varieties over number fields. I am especially inte-rested in the Galois representations arising from thenatural Galois action on the torsion points of abe-lian varieties, and in the interrelations between thearithmetic of curves of low genus and of their corre-sponding Jacobians. The questions I’m interested inoften have strong computational aspects, concerningfor example the effective determination of propertiesof concrete Jacobians, or the algorithmic resolutionof diophantine equations.

Céline MaistretUniversity of Bristol

I’m currently interested in the parity of ranks of abe-lian surfaces as well as the arithmetic of hyperel-liptic curves and their Jacobians. An application ofthe latter that I particularly care about is the com-putation of several invariants involved in the Birch-Swinnerton-Dyer conjecture.

Mariagiulia De MariaUniversity of Luxembourg,Université de Lille 1The main topic of my research is modularity forweight one modular forms for modulo prime po-wers rings and Hilbert modular forms of non-parallelweight one over finite fields. In particular, I am inte-rested in the properties of the Galois representationsassociated to these forms. I am currently workingon q-expansions of Hilbert modular forms of non-parallel weight 1.


Lukas MelninkasStrasbourg

My current research concerns various problems re-garding Weil–Deligne representations associated toabelian varieties defined over local fields, most no-tably the computation of the root number. I work inboth the classical l-adic setting and the p-adic one,the latter requiring tools from p-adic Hodge theory.

Hartmut MonienBonn

Adam MorganKCL London

My main research interests are centred around theBirch–Swinnerton-Dyer conjecture and its conse-quences, with a particular focus on the parity con-jecture. Usually, I work with abelian varieties of di-mension greater than 1 over number fields and try tounderstand how phenomena not exhibited for ellipticcurves (such as the possibility that the Shafarevich–Tate group may not have square order) influence thebehaviour of ranks, Selmer groups and other BSD-related invariants. I’m also interested in the behavi-our of ranks and Selmer groups in families, and the

various statistical questions one can ask in such situations.


Jan Steffen MullerOldenburg

I am mainly interested in explicit methods inarithmetic geometry. Most of my research involvesusing archimedean and p-adic height functions tostudy rational points on curves and abelian varieties.

Shoichi NakajimaGakushuin University, Tokyo

Formerly I worked on Galois coverings of algebraiccurves over fields of positive characteristic, and onclass numbers of cyclotomic fields of prime powerorder. Recently I want to go into the arithmetic ofcurves over number fields, and I regard this confe-rence a very good occasion for learning the subject.I am especially interested in BSD conjecture for CMelliptic curves (or Jacobians).

Sarah NakatoGraz

My research area is algebraic number theory, fo-cusing on non-unique factorizations in the ring ofinteger-valued polynomials over integral domains.My recent work was on sets of lengths in the ring ofinteger-valued polynomials over the ring of integersof an algebraic number field, and I’m currently inves-tigating factorizations in the ring of integer-valuedpolynomials over Krull domains.

Bartosz NaskreckiUniversity of Bristol,Adam Mickiewicz University

I am interested in arithmetic aspects of algebraic geo-metry. In particular, families of elliptic curves, ellipticsurfaces and Galois representations attached to etalecohomology groups and problems related to modula-rity and Generalised Fermat Conjecture. Currently Ialso study hypergeometric motives and explicit reali-sations of those. Typically algorithmic number theoryinvolves a lot of computations which I usually per-form in Magma.


Carlo PaganoLeiden

I am interested in Arithmetic statistic, Unit equa-tions, Arithmetic of local fields. I worked on fin-ding the joint distribution of 4-ranks of ray classgroups and class groups and a Cohen-Lenstra heu-ristics for ray class groups (with E.Sofos), on establis-hing bounds for unit equations in positive characte-

ristic depending only on the rank of the group (with P.Koymans), and on determininghow a new combinatorial invariant of local fields distributes over the space of Ei-senstein polynomials of given degree and relates to ramification invariants.

Jennifer ParkMichigan

Soohyun ParkMIT

I am interested in arithmetic aspects of certain pro-blems in algebraic geometry. One problem that I’mthinking about is related to decomposability of Ja-cobians over C and number fields. While one majorapplication is to moduli problems in algebraic geo-metry, there are also some interesting connections toranks of elliptic curves. In particular, I would like tofurther explore related work on understanding distri-butions of ranks of elliptic curves. Some other topics

that I like to think about in this general direction are those involving rational pointson higher dimensional varieties such as the density of rational points or rationalcurves on algebraic varieties.


Cihan PehlivanAnkara

My research area is number theory, more specifically,analytic number theory. In particular, I am interestedin Artin’s primitive root conjecture and its generali-zations, arithmetic statistics, sieves methods, analyticproblems for elliptic curves.

Gal PoratJerusalem

I am an M.Sc student, interested in Algebraic Num-ber Theory and Arithmetic Geometry. I am currentlyworking on questions related to Local Class Fieldtheory, in particular to Phi-Gamma modules obtai-ned using Lubin-Tate theory, due to Kisin and Ren.In the near future I am planning to study possibleapplications to the Iwasawa Theory of elliptic curves.

Gustavo Daniel Rama MoralesMontevideo

David RobertsMinnesota

My research centers around computational investiga-tions in situations where Galois theory and ramifi-cation play important roles. Particular topics includetabulating number fields, tabulating local fields, con-structing extreme number fields, Belyi covers, Hur-witz covers, hypergeometric motives, and connections

with modular forms.


Óscar Rivero SalgadoBarcelona

I am a PhD student working on several aspects ofarithmetic geometry, mainly those related with Eulersystems and p-adic L-functions. On the one hand,I am concerned about the relations between the the-ory of Beilinson-Flach elements and the Elliptic Starkconjecture. On the other, I am starting to deal with

some aspects of the theory of Stark-Heegner points and Darmon-Dasgupta units.

Soumya SankarUniversity of Wisconsin-Madison

My research interests revolve around the arithme-tic of curves and abelian varieties. I am interestedin questions about p-torsion in characteristic p andabout asymptotics of p-ranks and a-numbers. I amalso interested in questions about reduction of ellip-tic Q-curves and associated Galois representations.

Ciaran SchembriSheffield

Samuel SchiavoneDartmouth

I am a PhD student interested in algebraic geometryand algebraic number theory. I am especially inte-rested in the computational aspects of these areas,and often use SageMath or Magma to perform calcu-lations as a part of my work. I am currently workingon two projects: one on the computation of explicitequations for Belyi maps, and the other on classi-fying generalizations of quaternion algebras.


Jeroen SijslingUniversität Ulm

The leading subject of my research is that of alge-braic curves over number fields and their moduli. Acurrent main theme is the explicit calculation of en-domorphisms and decompositions of Jacobians. Thishelps in particular to find the modular form associa-ted to a given curve of low genus. Another theme isthat of reconstructing curves from their invariants,and more abstractly the question when the field ofmoduli is a field of definition. Roughly, this questionis as follows: if a curve is isomorphic to all its conju-gates over a given field, then does it allow a defining

equation over that field? A lot is known on this question, yet strange and interestingphenomena continue to appear when studying it.

Raffael Singer

Hironori ShigaChiba university

Current focus of the research: (1) To construct newmodular functions and modular forms based on theperiods of the families of algebraic curves (includingfamilies of hyper-elliptic curves), K3 surfaces andSchwarz maps of hypergeometric differential equati-ons. (2) Arithmetic applications of the above functi-ons. Especially, arithmetic properties of the specialvalues at CM points (including Hilbert’s 12th pro-blem). (3) Mirror properties coming from the above

modular functions.


Michael StollUniversität Bayreuth

The focus of my research is on rational points:Given a "nice" (smooth, projective, geometricallyintegral) variety V defined over Q, what can we sayabout its set V(Q) of rational points and how canwe determine it explicitly? I am mostly consideringthe case when V is a curve of higher genus. Inthis case, the set in question is finite and can bespecified by listing the points. Natural questions are:(how) can we check that all points have been found?,or what can we say about the number of points?My work covers both theoretical and computationalaspects.

Ersin SuerIstanbul

Andrew SutherlandMIT

My research focuses on computational aspectsof number theory and arithmetic geometry. Iam particularly interested in algorithms to ex-plicitly compute zeta functions, L-functions,and Galois representations in ways that allowone to investigate questions related to mo-dularity, BSD, the Sato-Tate conjecture, andother explicit aspects of the Langlands pro-gram.


Jack ThorneCambridge

I use Galois deformation theory to find provable in-stances of the conjectural Langlands correspondencebetween automorphic forms and Galois representati-ons. Currently I am especially interested in the pro-blem of modularity of elliptic curves over imaginaryCM number fields (the case of a totally real numberfield now being relatively well understood).

Gonzalo Tornaría LópezMontevideo

Nicholas George TriantafillouMIT

My research is currently directed towards statisticsof Selmer groups of (Jacobians of) curves in families.I also enjoy computational aspects of arithmetic geo-metry, especially algorithms (theoretical or practical)to determine sets of rational points or to computezeta functions. I am especially fond of hyperellipticand superelliptic curves.

Ling Sang TseToronto


Raymond Van BommelUniversiteit Leiden

My research interests are within algebraic geome-try and arithmetic geometry in particular. In particu-lar, I am very interested about computational aspectsof arithmetic geometry. For my PhD, I have beenworking on numerical verification of the Birch andSwinnerton-Dyer conjecture for hyperelliptic curve ofgenus 2 and 3 over Q.

John VoightDartmouth

In the spirit of the Langlands philosophy, I seek tomake effective the link between modular forms andgeometric objects, with particular attention to the set-ting of curves of higher genus as well as varieties ofhigher dimension (such as K3 surfaces). In particular,I like to design and implement algorithms to compute

automorphic forms and then to match the resulting automorphic representations withtheir realizations in the cohomology of varieties over number fields.

Ariel WeissSheffield

I am interested in the connections between automor-phic representations and Galois representations. Myresearch is focused on low weight Siegel modularforms and their associated Galois representations. Inparticular, for my PhD project, I am studying theimages of the Galois representations attached to lowweight Siegel modular forms. I’m also interested inthe conjectural relationship between weight 2 Siegelmodular forms and abelian surfaces.


Stefan WewersUlm

Most of my past research has been on reduction andlifting of covers of curves over local fields. My mainproject at the moment is the development of a Sagepackage for computation with semistable and regularmodels of curves (including the mathematical back-ground, of course). I am very interested in arithmeticapplications of these techniques, to study L-functionsand Galois representations, for instance. I have alsodone some work on the connection between Tate mo-

tives and diophantine equations, using the philosophy of Minhyong Kim.

Jeff Yelton

Di ZhangUniversity of Sheffield

I am a PhD student interested in algebraic numbertheory, Bianchi modular forms and L-functions. Mo-tivated by results of Kohnen and Zagier for the Shin-tani lifting, my research focuses on the analogoustheta lifting of Bianchi modular forms to Siegel mo-dular forms.


L U : T A, :

Abstracts

Samuele AnniThe inverse Galois problem for symplectic groupsThe inverse Galois problem is one of the greatest open problems in group theory andalso one of the easiest to state: is every finite group a Galois group? My interestaround this problem is connected to the realization of linear and symplectic groupsover finite fields as Galois groups over Q and over number fields. In this talk, I willbriefly describe "uniform realizations" using elliptic curves, genus 2 and 3 curves.After this introduction, I will show how to extend these results using Jacobians ofhigher genus (joint work with Vladimir Dokchitser). Here, Goldbach’s conjecture willplay a central role.

Jennifer BalakrishnanExplicit Coleman integration for curvesI will describe an algorithm for explicit Coleman integration on curves, starting fromthe case of hyperelliptic curves. I will also present a selection of computed examples.This is joint work with Jan Tuitman.

Alexander BettsComputing Tamagawa numbers of hyperelliptic curvesTBA.

Matthew BisattRoot Numbers of Abelian VarietiesThe expected sign of the L-function of an abelian variety is known as the root numberand this (conjecturally) determines the parity of the Mordell-Weil rank. In this talk,I will give explicit formulae to compute the root number and use my results to finda rational genus 2 hyperelliptic curve with a special property.


Julie DesjardinsDensity of rational points on rational elliptic surfacesFor an algebraic variety over Q, we are interested in X(Q) the set of rational points.Is it non-empty ? Is it infinite ? Is it Zariski-dense ? We look at those questionsin the case where X is a rational elliptic surface. This class of elliptic surfaces hasthe special property that the minimal model is either a conic bundle or a Del Pezzosurface.

Netan DograRational points on hyperelliptic curves via the Chabauty-KimmethodThis talk will discuss some joint work with Jennifer Balakrishnan and Steffen Mullerin which Kim’s generalisation of Chabauty’s method is used to determine the set ofrational points on certain hyperelliptic curves.

Tim DokchitserModels of curvesI will explain an approach to constructing regular and semistable models of curvesover DVRs. It is based on the ideas that are well-known for varieties over the complexnumbers in toric and tropical geometry, but can be imported to curves over arithmeticschemes as well. This often allows to compute important invariants of the curve ñconductor, semistable type, etale cohomology etc. in an elementary way.

Vladimir DokchitserArithmetic of hyperelliptic curves over local fieldsLet C : y2 = f(x) be a hyperelliptic curve over a local field K of odd residue characte-ristic. I will explain how several arithmetic invariants of the curve and its Jacobian,including its potential stable reduction, Galois representation and (in the semistablecase) Tamagawa numbers, can be simply extracted from combinatorial data comingfrom the roots of f(x). This is joint work with Tim Dokchitser, Celine Maistret andAdam Morgan.

Tom FisherVisualizing elements of order 7 in the Tate-Shafarevich group ofan elliptic curveMazur observed that elements of the Tate-Shafarevich group of one abelian variety(say E) can sometimes be explained by elements of the Mordell-Weil group of anotherabelian variety (say F). I will describe some examples I found where E is an ellipticcurve and F is a genus 2 Jacobian.


Wei HoOdd degree number fields with odd class numberTBA.

David HolmesArakelov geometry of hyperelliptic curvesArakelov geometry provides elegant machinery for working with canonical (Néron-Tate) heights on abelian varieties. We will begin with a gentle introduction to thistheory, with particular emphasis on how it can be used to compute these heightson the jacobians of hyperelliptic curves. We will then explain how the theory of‘height jumping’ (first noticed by R. Hain) puts rather strong constraints on how wella naive height can approximate the Néron-Tate height. If time allows we will discussconnections to existence of small points and the strong torsion conjecture.

Kiran KedlayaL-functions via deformations: from hyperelliptic curves tohypergeometric motivesThe “deformation method” is an approach due to Alan Lauder for computing zetafunctions of varieties over finite fields using the Frobenius structures on Picard–Fuchs equations (a/k/a Gauss–Manin connections). We first recall how this is donefor hyperelliptic curves, as implemented in Magma. We next propose an analogouscomputation for hypergeometric motives and explain the advantages it has over otherp-adic analytic methods.


Emmanuel LecouturierHigher Eisenstein elements in weight 2 and prime levelIn his classical work, Mazur considers the Eisenstein ideal I of the Hecke algebra Tacting on cusp forms of weight 2 and level Γ0(N) where N is prime. When p is anEisenstein prime, i.e. p divides the numerator of N−1

12 , denote by T the completionof T at the maximal ideal generated by I and p. This is a Zp-algebra of finite rankgp ≥ 1 as a Zp-module.Mazur asked what can be said about gp. Merel was the first to study gp. Assume forsimplicity that p ≥ 5. Let log : (Z/NZ)× → Fp be a surjective morphism. Then Merelproved that

gp ≥ 2

if and only ifN−1

2∑k=1

k · log(k) ≡ 0 (modulo p).

We prove that we have gp ≥ 3 if and only if

N−12∑k=1

k · log(k) ≡N−1

2∑k=1

k · log(k)2 ≡ 0 (modulo p).

We also give a more complicated criterion to know when gp ≥ 4. Moreover, we provehigher Eichler formulas. More precisely, let

H(X) =

N−12∑k=0

(N−12k

)2

· Xk ∈ FN[X]

be the classical Hasse polynomial. It is well-known that the roots of H are simpleand in F×

N2 . Let L be this set of roots. We prove that

∑λ∈L

log(H ′(λ)) ≡ 4 ·N−1

2∑k=1

k · log(k) (modulo p)

and, if gp ≥ 2,

∑λ∈L

log(H ′(λ))2 ≡ 4 ·N−1

2∑k=1

k · log(k)2 (modulo p) .

Qing LiuNew points on algebraic curves (Joint work with Dino Lorenzini)

Let K be a field and let L/K be a finite extension. Let X/K be a scheme of finite type.A point of X(L) is said to be new if it does not belong to ∪FX(F), where F runs over allproper subfields K ⊆ F ⊂ L. Fix now a separable extension L/K of degree d := [L : K].We investigate whether there exists a smooth proper geometrically connected curve ofgenus g > 0 with a new point in X(L). We show that if K is infinite with char(K) 6= 2and g ≥ bd/4c, then there exist infinitely many hyperelliptic curves X/K of genus g,pairwise non-isomorphic over K, and with a new point in X(L). When 1 ≤ d ≤ 10,we show that there exist infinitely many elliptic curves X/K with pairwise distinctj-invariants and with a new point in X(L).


Davide LombardoComputing twists of hyperelliptic curvesTwo curves over a field K are said to be twists of each other if there exists anextension of K over which they become isomorphic. The twists of a given curvecorrespond to elements of a suitable cohomology group, and there is a general recipeto describe a twist in terms of the corresponding cohomology class. However, writingdown explicit equations for a given twist can be challenging in practice: in this talk Iwill explain why hyperelliptic curves are particularly interesting in this context, andhow to quickly compute models for the twists of any such curve. This is joint workwith Elisa Lorenzo-GarcÌa.

Celine MaistretParity of ranks of abelian surfacesLet K be a number field and A/K an abelian surface. By the Mordell-Weil theorem,the group of K-rational points on A is finitely generated and as for elliptic curves, itsrank is predicted by the Birch and Swinnerton-Dyer conjecture. A basic consequenceof this conjecture is the parity conjecture: the sign of the functional equation of the L-series determines the parity of the rank of A/K. Under suitable local constraints andfiniteness of the Shafarevich-Tate group, we prove the parity conjecture for principallypolarized abelian surfaces. We also prove analogous unconditional results for Selmergroups.

Adam MorganParity of Selmer ranks in quadratic twist familiesWe study the parity of 2-Selmer ranks in the family of quadratic twists of a fixedprincipally polarised abelian variety over a number field. Specifically, we prove resultsabout the proportion of twists having odd (resp. even) 2-Selmer rank. This generaliseswork of Klagsbrun-Mazur-Rubin for elliptic curves and Yu for Jacobians of hyperel-liptic curves. Several differences in the statistics arise due to the possibility that theShafarevich-Tate group (if finite) may have order twice a square.

Jan Steffen MuellerCanonical heights on Jacobians of curves of genus twoTo find explicit generators for the Mordell-Weil group of an abelian variety over aglobal field, and to compute its regulator, one needs algorithms to compute canonicalheights of rational points and to enumerate all rational points of bounded canonicalheight. In my talk, I will discuss how this can be done efficiently for Jacobians ofcurves of genus 2. This is joint work with Michael Stoll.


Bartosz NaskreckiHypergeometric motives of low degrees and their connection toelliptic and hyperelliptic curvesIn this talk we will discuss the construction of hypergeometric motives as Chowmotives in explicitly given algebraic varieties. The class of hypergeometric motivescorresponds to Picard-Fuchs equations of hypergeometric type and forms a rich familyof pure motives with nice L-functions. Following recent work of Beukers-Cohen-Mellit we will show how to realise certain hypergeometric motives of weights 0 and2 as submotives in elliptic and hyperelliptic fibrations. An application of this work is acomputation of minimal polynomials of hypergeometric series with finite monodromygroups and proof of identities between certain hypergeometric finite sums, whichmimic well-known identities for classical hypergeometric series.

Jennifer ParkA heuristic for boundedness of elliptic curvesI will discuss a heuristic that predicts that the ranks of all but finitely many ellipticcurves defined over Q are bounded above by 21. This is joint work with Bjorn Poonen,John Voight, and Melanie Matchett Wood.

David RobertsAn inverse Hodge problem and some solutions from hypergeo-metric motivesWe pose a general "inverse Hodge problem" whose purpose is to provide a frameworkfor exploring the category of motives defined over Q with coefficients in Q. The pro-blem asks to find a full such motive for any given Hodge vector h = (hw,0, . . . , h0,w).Here “full” means that the motivic Galois group is as large as possible, and fullnessis imposed in order to avoid cheap solutions. For h = (g, g), a generic genus g curveprovides a positive solution. For more exotic Hodge vectors like h = (1, 2,0,0, 2, 1),there are no simple varietal sources. We show that the motives underlying classicalhypergeometric functions provide positive solutions for a wide range of h.

Jeroen SijslingDatabases of curvesWe discuss databases of curves, the information that they contain, and the relevantinterconnections with other mathematical objects. The genus 2 case can be exploredon the LMFDB; we will also see some preliminary results in genus 3.


Michael StollArithmetic of hyperelliptic curves in MagmaI will give a live demonstration of some of the available functionality for hyperellipticcurves and their Jacobians in Magma, with a focus on rational points.

Andrew SutherlandComputing L-series of hyperelliptic curvesLet X/Q be a hyperelliptic curve of genus g whose L-function L(X, s) satisfies itsconjectured functional equation. I will present an algorithm to compute the first nDirichlet coefficients of L(X, s) with a running time that is quasi-linear in n. This isjoint work with Andy Booker, David Harvey, and David Platt.

Gonzalo TornariaExplicit modularity for typical genus 2 curvesI will describe the computations we used to prove the (para)modularity of the genus2 curve of conductor 277. Joint work with Armand Brumer, Ariel Pacetti, Cris Poor,John Voight and David Yuen.

John VoightExplicit modularity for atypical genus 2 curvesWe discuss what it means for a genus 2 curve over the rationals to be modular whenit acquires extra endomorphisms over an extension field. More precisely, to everygenus 2 curve X we discuss conjectures and theorems that attach to X a modularform with a matching L-function: the description depends on the Galois modulestructure of the geometric endomorphism algebra of the Jacobian of X. This is jointwork with Andrew Booker, Jeroen Sijsling, Drew Sutherland, and Dan Yasaki.

Raymond Van BommelNumerical verification of the Birch and Swinnerton-Dyer con-jectureWe discuss recent work numerically verifying the BSD conjecture for several hundredhyperelliptic curves of genus 2 and genus 3 (up to squares), with particular emphasison the computation of a Néron differential (which is necessary for the real period). Iftime allows, we will discuss some particular examples and ideas for future research.


Stefan WewersMCLF - a Sage toolbox for computation with models of curvesIn my talk I will present a first and very premature version of the Sage toolbox MCLF(Models of Curves over Local Fields). It is based on the ’maclane’ infrastructurewritten by Julian Rueth. Future versions will hopefully be able to compute regularand semistable models of arbitrary curves. For the moment, it is able to computesemistable models of superelliptic curves of degree p, given by an equation yp = f(x),over the p-adic numbers.


Curves and L-functions List of Participantsmatyd/Trieste2017/Booklet.pdf · L-functions and Galois...

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