Computing neronseveri groups and cycle class groups volume 151 issue 4 bjorn poonen, damiano testa, ronald van luijk skip to main content we use cookies to distinguish you from other users and to provide you with a better experience on our websites. I would like to thank john and filip, and also my other coauthors jeroen demeyer. We find explicit equations for twocoverings of jacobians of genus two curves over an arbitrary ground field of characteristic different from two. Epsrc symposium on number theory warwick university 2012 20. In section 8, we work over number elds and discuss several arithmetic applications of the theory of balanced line bundles in this context. The authors would like to thank brendan hassett, damiano testa, and anthony v arillyalvarado for useful sugges. This version includes supplementary material for derivation of an equation. Ams proceedings of the american mathematical society. Alberta number theory days vith meeting april 19 april. This unpublished note was written probably around the time i was in princeton 19982000 since it resulted from discussions with nick katz. Algebraic geometry authorstitles recent submissions.
Ma246 number theory by damiano testa tests friday 11am, room ms. Classical invariant theory of quartic forms in this section, we identify the locus of binary quartic forms with a triple root. Conics on the fermat quintic threefold damiano testa junior number theory seminar university of oxford october 11, 2010 damiano testa conics on the fermat quintic threefold. Samir siksek, michael stoll, damiano testa summer school. I would also like to thank my examiners tom fisher and david mond for a large number of helpful comments and suggestions. The unimodality of a polynomial coming from a rational integral. The bounds are uniform in the curve and involve the rank of. My research interests include algebraic number theory, algebraic curves, abelian varieties, special values of modular functions, cryptography, divisibility sequences, and computational number theory. Alexander polynomials of alternating knots of genus two jong, in dae, osaka journal of mathematics, 2009. Counting rational points on cubic curves springerlink. Bjorn poonen mit department of mathematics 32 vassar st. Victor flynn, damiano testa submitted on 3 jan 2014 abstract.
Thank you to my parents and to annie for their care and suppport. Number theory for cryptography 2428 june 20 organizers and instructors. See sal73, especially chapter vi, for more details and complements. Reconstructing general plane quartics from their inflection lines. In this section we will describe a few typical number theoretic problems. Roger heathbrown, john cremona, samir siksek, michael stoll, damiano testa. Computing neronseveri groups and cycle class groups. Epsrc symposium on number theory university of warwick. It conjectures that, for a constant anot a perfect square and not 1, ais a. In this series of lectures, i will talk about del pezzo surfaces with an emphasis on arithmetic properties. On the existence and uniqueness of solution of boundarydomain integral equations for the dirichlet problem for the nonhomogeneous heat transfer equation defined on a 2d u. We apply this to several examples, where it can shown that nonreducible jacobians have nontrivial 3part of the tateshafarevich group. Im asking for a big list of not especially famous, long open problems that anyone can understand. Added a complete discussion of picard lattices of kummer surfaces of genus two curves with a nonhyperelliptic involutio.
Alberta number theory days vith meeting april 19 april 20, 2014 meals breakfast and lunches are complimentary, at vistas main dining room on the 4th. The bounds are uniform in the curve and involve the rank of the corresponding jacobian. Log fano structures and cox rings of blowups of products of projective spaces. Assuming the tate conjecture and the computability of etale cohomology with nite coe cients, we give an algorithm that computes the n eronseveri group of any smooth projective geometrically integral variety, and also the rank of the group of numerical equiv. Victor flynn, damiano testa, ronald van luijk submitted on 8 may 2009 abstract. We study the geometry of b\uchi s k3 surface showing that the rational points of this surface are zariskidense. Cox rings of degree one del pezzo surfaces, joint with tony varillyalvarado and mauricio velasco, algebra number theory 3 2009, no. Damiano testa mathematics institute zeeman building university of warwick coventry cv4 7al uk d. Number theory informed by computation, park city mathematics institute, july 525. Ams transactions of the american mathematical society. Roger heathbrown, john cremona, samir siksek, michael stoll, damiano testa summer school. In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the uniqueprimefactorization theorem, states that every integer greater than 1 ei ther is prime itself or is the product of prime numbers, and that this product is unique, up to the. Damiano testa junior number theory seminar university of oxford october 11, 2010 damiano testa conics on the fermat quintic threefold.
Diane fenton mathematics and statistics, university of calgary title. The unimodality of a polynomial coming from a rational. Back to the original proof article in journal of mathematical analysis and applications 4202 april 20 with 41 reads. Dan bernstein, john cremona, andreas enge, tanja lange, francois morain, samir siksek number theory, geometry and cryptography 15 july 20.
Some typical number theoretic questions the main goal of number theory is to discover interesting and unexpected relationships between different sorts of numbers and to prove that these relationships are true. This chapter lays the foundations for our study of the theory of numbers by weaving together the themes of prime numbers, integer factorization, and the distribution of primes. Each week you will have a workbook covering the material that will be the subject of the next inclass test. Algorithms and number theory, may 2429, 2009, at schloss dagstuhl diophantine equations at the hausdorff research institute for mathematics in bonn, januaryapril 2009 group theory, number theory and geometry in oxford, march 30 april 3, 2009 conference on the occasion of fritz grunewalds 60th birthday. Classically, this ties in with the invariant theory of binary quartic forms and is our starting point. Nils bruin, tom fisher, visibility of 4covers of elliptic curves, research in number theory 4. This project brings together three areas of mathematics. I am grateful to the epsrc for funding this research.
Gioia the theory of numbers markham publishing company 1970 acrobat 7 pdf 6. Algebraic number theory, analytic number theory and. Introduction to analytic number theory tu chemnitz. Bjorn poonen, damiano testa, and ronald van luijk abstract. We would like to thank brendan hassett and marta pieropan for comments on an early draft of this paper, and in particular brendan for suggesting ks04 as a reference for low degree rational curves on the degree 22 fano 3fold. I will touch upon the combinatorial structure of the set of 1curves, the existence of rational points, the construction of unirational parameterisations.
We settle a conjecture of batyrev and popov on the ideal of relations of the cox ring of a del pezzo surface in the. The rank of this group is called the picard number of the variety. Cox rings of degree one del pezzo surfaces, algebra number theory 3 2009, no. Chapter 6 makes use of the properties and according to theorem. We give parametrisation of curves c of genus 2 with a maximal isotropic zz32 in j3, where j is the jacobian variety of c, and develop the theory required to perform descent via 3,3isogeny. Accepted to appear in canadian conference on artificial intelligence canadian ai conference 2020, springer. We prove upper bounds for the number of rational points on nonsingular cubic curves defined over the rationals. We make use of kummer theory which we have explained earlier in chapter 2.
Artins primitive root conjecture is an old and wellstudied problem in number theory. Department of mathematics the department of mathematics at mit seeks to improve upon its top ranking in both research and teaching by aggressively hiring the very best faculty, with special a ention to the recruitment of top women and underrepresented minority candidates and by. Dan bernstein, john cremona, andreas enge, tanja lange, francois morain, samir siksek. Damiano testa conics on the fermat quintic threefold. Home research algorithmic number theory algorithmic number theory the frontiers between computable objects, algorithms above section, computational number theory and applications to security of cryptographic systems are very porous. Number theory, geometry and cryptography 15 july 20. Rational curves to study a variety we are going to search for parameterized curves inside it. In chapter 5 we prove the second half of theorem 1. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. The authors would like to thank brendan hassett, damiano testa, and anthony v arillyalvarado for useful suggestions and mihai fulger for providing an argument for lemma 4. Nils bruin, professor of mathematics at simon fraser university.
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