## Joerg Jahnel's Research

[back to home page]## Some papers

PreprintsJ. Jahnel:

On the distribution of small points on abelian and toric varieties[dvi] [ps] [pdf]

A.-S. Elsenhans and J. Jahnel:The Diophantine Equation x[dvi] [ps] [pdf]^{4}+ 2 y^{4}= z^{4}+ 4 w^{4}---A number of improvements

A.-S. Elsenhans and J. Jahnel:The Fibonacci sequence modulo p[dvi] [ps] [pdf]^{2}---An investigation by computer for p < 10^{14}

Habilitation ThesisJ. Jahnel:

Brauer groups, Tamagawa measures, and rational points on algebraic varieties, Göttingen 2008

Revised version:Brauer groups, Tamagawa measures, and rational points on algebraic varieties, Mathematical Surveys and Monographs 198, AMS, Providence 2014

ArticlesA.-S. Elsenhans and J. Jahnel:

On the component group of the algebraic monodromy group of a K3 surface, Journal of Algebra 646(2024)294-325 [dvi] [ps] [pdf]

A.-S. Elsenhans and J. Jahnel:Real and complex multiplication on K3 surfaces via period integration, Experimental Mathematics 33(2024)193-224 [dvi] [ps] [pdf]

A.-S. Elsenhans and J. Jahnel:Explicit families of K3 surfaces having real multiplication, Michigan Mathematical Journal 73(2023)3-32 [dvi] [ps] [pdf]

A.-S. Elsenhans and J. Jahnel:Frobenius trace distributions for K3 surfaces, Journal of the Ramanujan Mathematical Society 37(2022)385-409 [dvi] [ps] [pdf]

The histograms related to this project are available here at full resolution.

A.-S. Elsenhans and J. Jahnel:2-adic point counting on K3 surfaces, in: Proceedings of the ANTS XV conference (Bristol 2022), Research in Number Theory 8(2022)art.85 [dvi] [ps] [pdf]

A.-S. Elsenhans and J. Jahnel:Computing invariants of cubic surfaces, Le Matematiche 75(2020)457-470 [dvi] [ps] [pdf]

A.-S. Elsenhans and J. Jahnel:Computations with algebraic surfaces, in: Mathematical Software - ICMS 2020, Lecture Notes in Computer Science 12097, Springer, Berlin 2020, 87-93 [dvi] [ps] [pdf]

E. Costa, A.-S. Elsenhans, and J. Jahnel:On the distribution of the Picard ranks of the reductions of a K3 surface, Research in Number Theory 6(2020)art.27 [dvi] [ps] [pdf]

A.-S. Elsenhans and J. Jahnel:On plane quartics with a Galois invariant Cayley octad, European Journal of Mathematics 5(2019)1156-1172 [dvi] [ps] [pdf]

J. Jahnel and D. Schindler:On the frequency of algebraic 2-torsion Brauer classes on certain log K3 surfaces, Canadian Mathematical Bulletin 62(2019)551-563 [dvi] [ps] [pdf]

J. Jahnel and D. Schindler:On the algebraic Brauer classes on open degree four del Pezzo surfaces, Journal of Number Theory 203(2019)376-427 [dvi] [ps] [pdf]

A.-S. Elsenhans and J. Jahnel:On plane quartics with a Galois invariant Steiner hexad, International Journal of Number Theory 15(2019)1075-1109 [dvi] [ps] [pdf]

J. Jahnel and D. Schindler:On integral points on degree four del Pezzo surfaces, Israel Journal of Mathematics 222(2017)21-62 [dvi] [ps] [pdf]

J. Jahnel and D. Schindler:Del Pezzo surfaces of degree four violating the Hasse principle are Zariski dense in the moduli scheme, Annales de l'Institut Fourier 67(2017)1783-1807 [dvi] [ps] [pdf]

J. Jahnel and D. Schindler:On the Brauer-Manin obstruction for degree four del Pezzo surfaces, Acta Arithmetica 176(2016)301-319 [dvi] [ps] [pdf]

A.-S. Elsenhans and J. Jahnel:Point counting on K3 surfaces and an application concerning real and complex multiplication[dvi] [ps] [pdf], in: Proceedings of the ANTS XII conference (Kaiserslautern 2016), LMS Journal of Computation and Mathematics 19(2016)12-28

J. Jahnel and D. Loughran:The Hasse principle for lines on diagonal surfaces, Mathematical Proceedings of the Cambridge Philosophical Society 160(2016)107-119 [pdf]

J. Jahnel and D. Schindler:On the number of certain del Pezzo surfaces of degree four violating the Hasse principle, Journal of Number Theory 162(2016)224-254 [pdf]

J. Jahnel and D. Loughran:The Hasse principle for lines on del Pezzo surfaces, International Mathematical Research Notices 23(2015)12877-12919 [pdf]

Related to this project, there is some magma code.

A.-S. Elsenhans and J. Jahnel:Moduli spaces and the inverse Galois problem for cubic surfaces, Transactions of the AMS 367(2015)7837-7861 [dvi] [ps] [pdf]

A.-S. Elsenhans and J. Jahnel:On the characteristic polynomial of the Frobenius on étale cohomology, Duke Mathematical Journal 164(2015)2161-2184 [dvi] [ps] [pdf]

A.-S. Elsenhans and J. Jahnel:Cubic surfaces violating the Hasse principle are Zariski dense in the moduli scheme, Advances in Mathematics 280(2015)360-378 [dvi] [ps] [pdf]

A.-S. Elsenhans and J. Jahnel:Examples of K3 surfaces with real multiplication, in: Proceedings of the ANTS XI conference (Gyeongju 2014), LMS Journal of Computation and Mathematics 17(2014)14-35 [dvi] [ps] [pdf]

U. Derenthal, A.-S. Elsenhans, and J. Jahnel:On the factor alpha in Peyre's constant, Mathematics of Computation 83(2014)965-977 [dvi] [ps] [pdf]

A.-S. Elsenhans and J. Jahnel:Experiments with the transcendental Brauer-Manin obstruction, in: Proceedings of the ANTS X conference (San Diego 2012), MSP, Berkeley 2013, 369-394 [dvi] [ps] [pdf]

A.-S. Elsenhans and J. Jahnel:On the computation of the Picard group for certain singular quartic surfaces, Mathematica Slovaca 63(2013)215-228 [dvi] [ps] [pdf]

A.-S. Elsenhans and J. Jahnel:On the arithmetic of the discriminant for cubic surfaces, Journal of the Ramanujan Mathematical Society 27(2012)355-373 [dvi] [ps] [pdf]

A.-S. Elsenhans and J. Jahnel:The Picard group of a K3 surface and its reduction modulo p, Algebra & Number Theory 5(2011)1027-1040 [dvi] [ps] [pdf]

A.-S. Elsenhans and J. Jahnel:The discriminant of a cubic surface, Geometriae dedicata 159(2012)29-40 [dvi] [ps] [pdf]

A.-S. Elsenhans and J. Jahnel:Kummer surfaces and the computation of the Picard group, LMS Journal of Computation and Mathematics 15(2012)84-100 [dvi] [ps] [pdf].

Here are the raw data to this article.

A.-S. Elsenhans and J. Jahnel:On the order three Brauer classes for cubic surfaces, Central European Journal of Mathematics 10(2012)903-926 [dvi] [ps] [pdf]

A.-S. Elsenhans and J. Jahnel:On cubic surfaces with a rational line, Archiv der Mathematik 98(2012)229-234 BM_Trie [dvi] [ps] [pdf]

A.-S. Elsenhans and J. Jahnel:On the quasi group of a cubic surface over a finite field, Journal of Number Theory 132(2012)1554-1571 [dvi] [ps] [pdf]

A.-S. Elsenhans and J. Jahnel:On the computation of the Picard group for K3 surfaces, Mathematical Proceedings of the Cambridge Philosophical Society 151(2011)263-270 [dvi] [ps] [pdf]

J. Jahnel:More cubic surfaces violating the Hasse principle, Journal de Théorie des Nombres de Bordeaux 23(2011)471-477 [dvi] [ps] [pdf]

A.-S. Elsenhans and J. Jahnel:Cubic surfaces with a Galois invariant pair of Steiner trihedra, International Journal of Number Theory 7(2011)947-970 [dvi] [ps] [pdf]

A.-S. Elsenhans and J. Jahnel:On the Brauer-Manin obstruction for cubic surfaces, Journal of Combinatorics and Number Theory 2(2010)107-128 [dvi] [ps] [pdf]

A.-S. Elsenhans and J. Jahnel:On Weil polynomials of K3 surfaces, in: Algorithmic number theory, Lecture Notes in Computer Science 6197, Springer, Berlin 2010, 126-141 [dvi] [ps] [pdf]

A.-S. Elsenhans and J. Jahnel:On the smallest point on a diagonal cubic surface, Experimental Mathematics 19(2010)181-193 [dvi] [ps] [pdf]

A.-S. Elsenhans and J. Jahnel:Cubic surfaces with a Galois invariant double-six, Central European Journal of Mathematics 8(2010)646-661 [dvi] [ps] [pdf]

A.-S. Elsenhans and J. Jahnel:Estimates for Tamagawa numbers of diagonal cubic surfaces, Journal of Number Theory 130(2010)1835-1853 [dvi] [ps] [pdf]

A.-S. Elsenhans and J. Jahnel:New sums of three cubes, Math. Comp. 78(2009)1227-1230 [dvi] [ps] [pdf]

A.-S. Elsenhans and J. Jahnel:K3 surfaces of Picard rank one and degree two, in: Algorithmic number theory, Lecture Notes in Computer Science 5011, Springer, Berlin 2008, 212-225 [dvi] [ps] [pdf]

A.-S. Elsenhans and J. Jahnel:K3 surfaces of Picard rank one which are double covers of the projective plane, in: The Higher-dimensional geometry over finite fields, IOS Press, Amsterdam 2008, 63-77 [dvi] [ps] [pdf]

A.-S. Elsenhans and J. Jahnel:On the Smallest Point on a Diagonal Quartic Threefold, Journal of the Ramanujan Mathematical Society 22(2007)189-204 [dvi] [ps] [pdf],

A.-S. Elsenhans and J. Jahnel:Experiments with general cubic surfaces, in: Tschinkel, Y. and Zarhin, Y. (Eds.): Algebra, Arithmetic, and Geometry, In Honor of Yu. I. Manin, Volume I, Progress in Mathematics 269, Birkhäuser, Boston 2007, 637-654 [dvi] [ps] [pdf]

A.-S. Elsenhans and J. Jahnel:The Asymptotics of Points of Bounded Height on Diagonal Cubic and Quartic Threefolds, Algorithmic number theory, Lecture Notes in Computer Science 4076, Springer, Berlin 2006, 317-332 [dvi] [ps] [pdf]

A.-S. Elsenhans and J. Jahnel:The Diophantine Equation x, Math. Comp. 75(2006)935-940 [dvi] [ps] [pdf]^{4}+ 2 y^{4}= z^{4}+ 4 w^{4}---An investigation by computer for |x|, |y|, |z|, |w| < 2.5 10^{6}

J. Jahnel:The Brauer-Severi variety associated with a central simple algebra, Linear Algebraic Groups and Related Structures 52(2000)1-60 [dvi] [ps] [pdf]

J. Jahnel:Local singularities, filtrations and tangential flatness, Communications in Algebra 27(1999)2785-2808 [dvi] [ps] [pdf]

J. Jahnel:A height function on the Picard group of singular Arakelov varieties, in: Algebraic K-Theory and Its Applications, Proceedings of the Workshop and Symposium held at ICTP Trieste, September 1997, edited by H. Bass, A. O. Kuku and C. Pedrini, World Scientific 1999 [dvi] [ps] [pdf]

J. Jahnel:Heights for line bundles on arithmetic varieties, manuscripta mathematica 96(1998)421-442 [dvi] [ps] [pdf]

N. Hoffmann, J. Jahnel, and U. Stuhler:Generalized vector bundles on curves, Journal fuer die Reine und Angewandte Mathematik (Crelle's Journal) 495(1998)35-60 [dvi] [ps] [pdf]

J. Jahnel:Line bundles on arithmetic surfaces and intersection theory, manuscripta mathematica 91(1996)103-119 [dvi] [ps] [pdf]

J. Jahnel:Lech's conjecture on deformations of singularities and second Harrison cohomology, Journal of the London Mathematical Society 51(1995)27-40 [dvi] [ps] [pdf]

A funny note

Ph.D. Thesis

Diploma ThesisJ. Jahnel:

Zur Konvergenz regulierter Bewegungen(does not exist in electronic form)

## Some talks

Rational Points on Hypersurfaces in Projective Space, Clay Summer School at Göttingen, July 2006 [pdf]Rational Points on Hypersurfaces in Projective Space, ANTS VII at Berlin, July 2006 [pdf]Zur Geometrie der K3-Flächen, Habilitation talk at Göttingen, June 2009 [pdf]Rationale Punkte auf Hyperflächen im projektiven Raum, University of Siegen, July 2009 [pdf]K3 surfaces and their Picard groups, Mathematische Gesellschaft zu Göttingen, December 2011 [pdf]K3 surfaces and their Picard groups, École Normale Supérieure Paris, January 2012 [pdf]Experiments with the transcendental Brauer-Manin obstruction, ANTS X at San Diego, July 2012 [pdf]K3 surfaces and their Picard groups, Rational Points, MSRI Berkeley, October 2012 [pdf]Experiments with the transcendental Brauer-Manin obstruction, Rational Points, MSRI Berkeley, October 2012 [pdf]A solution to the inverse Galois problem for cubic surfaces, Rational Points, MSRI Berkeley, October 2012 [pdf]Moduli spaces and the inverse Galois problem for cubic surfaces, University of Bristol, November 2012 [pdf]Experiments with the Brauer-Manin obstruction, University of Sydney, March 2013 [pdf]On cubic surfaces violating the Hasse principle, University of Sydney, February 2014 [pdf]K3 surfaces with real multiplication, ANTS XI at GyeongJu, South Korea, August 2014 [pdf]On the Hasse principle for cubic surfaces, Leibniz University Hannover, December 2014 [pdf]On the Hasse principle for lines on del Pezzo surfaces, University of Göttingen, July 2015 [pdf]K3 surfaces with real multiplication, Brown University, Providence RI, October 2015 [pdf]Moduli spaces and the inverse Galois problem for cubic surfaces, DIAMANT Seminar, Soest/Netherlands, November 2016 [pdf]On the distribution of the Picard ranks of the reductions of a K3 surface, Banff Centre for Arts and Creativity, March 2017 [pdf]Real multiplication on K3 surfaces via period integration, Shepperton, May 2018 [pdf]On cubic surfaces violating the Hasse principle, Hannover, August 2018 [pdf]On integral points on open degree four del Pezzo surfaces, Chalmers, Göteborg, March 2019 [pdf]On integral points on open degree four del Pezzo surfaces, IST Austria, Klosterneuburg (via world wide web), Januar 2021 [pdf]

## The Hasse principle for lines on del Pezzo surfaces

The Hasse principle for lines on del Pezzo surfaces is not always satisfied. There are counterexamples in degree 1, 2, 3, 5, and 8. These form Zariski dense but thin subsets of the respective Hilbert schemes.

Related to these results, there is some code in magma. The results themselves may be found inThe Hasse principle for lines on del Pezzo surfaces.

## Number-theoretic software

hashingThe Hashing package searching for solutions of Diophantine equations of the form

f(x_{1}, ... ,x_{k}) = g(y_{1}, ... ,y_{l}),

Version 1.0.

Contains example programs for the equations

[demo] x ^{3}+ y^{3}= z^{3}+ w^{3},search limits: 0 < x, y, z, w < 5000, [kub] a x ^{3}= b y^{3}+ z^{3}+ v^{3}+ w^{3},search limits: a, b < 100, |x|, |y|, |z|, |v|, |w| < 5000, [quart] a x ^{4}= b y^{4}+ z^{4}+ v^{4}+ w^{4},search limits: a, b < 100, x, y, z, v, w < 100 000.

Source code: [tar.gz]swdCode searching for solutions of Sir P. Swinnerton-Dyer's equation

making use of Hashing.

[swd] x ^{4}+ 2 y^{4}= z^{4}+ 4 w^{4},search limits: x, y, z, w < 100 000 000,

Source code: [tar.gz]

## Cubic surfaces

The 27 lines on a smooth cubic surface form a highly symmetric configuration. Its group of symmetries is the Weyl group W(E_{6}) of order 51840. For a cubic surface over Q, a subgroup of W(E_{6}) operates on the 27 lines.

W(E_{6}) has exactly 350 conjugacy classes of subgroups.

For each of the subgroups, we constructed explicit examples of cubic surfaces over Q. They are spread over several lists.All subgroups stabilizing a sixer, all other subgroups stabilizing a double-six, all remaining subgroups stabilizing a pair of Steiner trihedra, part 1, part 2, all remaining subgroups stabilizing a line, all subgroups that still remain.

There is an example on the construction of such surfaces in the general case. (This is the magma code for Algorithm 5.1 ofA solution to the inverse Galois problem for cubic surfaces, working at the subgroup with gap-Nummer 73.)

Finally, I offer an example computation of the Brauer-Manin obstruction on a non-diagonal cubic surface. (This is the magma code for Example 4.30 ofOn the order three Brauer classes for cubic surfaces.)

## Sums of three cubes

Which integers may be written as a sum of three cubes?

By our calculations from the years 2006/07, we know 14288 essentially different integral vectors (a,b,c) such that 0 < a^{3}+ b^{3}+ c^{3}< 1000 where a^{3}+ b^{3}+ c^{3}is neither a cube nor twice a cube. This is our list threecubes_20070419.

We implemented a version of Elkies' method. Our source code is available here.

For the following 14 numbers below 1000, the question remained open.

33? 42? 74? 114? 165? 390? 579? 627? 633? 732? 795? 906? 921? 975?

The history of the problem and older lists may be found at Daniel Bernstein's homepage.

A solution for the number 33 has been found meanwhile by Andrew Booker.

## Experiments with the transcendental Brauer-Manin obstruction

For Kummer surfaces, associated to products of two elliptic curves, we made experiments on the transcendental Brauer-Manin obstruction.

The raw data of the experiments are offered here.

## Algebraic Brauer classes on open degree 4 del Pezzo surfaces

Two quadratic forms in five variables generically determine a del Pezzo surface of degree 4. We call the complement of a geometrically irreducible hyperplane section anopen del Pezzo surfaceof degree 4.

We systematically investigated the algebraic Brauer classes on open degree 4 del Pezzo surfaces. The result is that the algebraic Brauer group is generated by elements of four types.

For the evaluation of the elements of one of the four types, the4-torsion classes of type II,only a generic algorithm helps. Our implementation of a generic algorithm is to be found here, together with the concrete calculations related to Example 6.16 from the paper.

## 2-adic point counting

We implemented a 2-adic point counting algorithm for a particular class of K3 surfaces. The idea is to describe the transcendental part of étale cohomology with 4 torsion coefficients explicitly, as a Galois module. We show that this already suffices in order to determine the point count modulo 16.

Together with the point count modulop, for the determination of which, there are well established methods, the count may be determined exactly.

Our implementation is offered here as a demo file that immediately runs the generic example from the article.