## Joerg Jahnel's Research

[back to home page]## Some papers

PreprintsE. Costa, A.-S. Elsenhans, and J. Jahnel:

On the distribution of the Picard ranks of the reductions of a K3 surface[dvi] [ps] [pdf]

J. Jahnel and D. Schindler:On integral points on degree four del Pezzo surfaces[dvi] [ps] [pdf], to appear in: Israel Journal of Mathematics

J. Jahnel and D. Schindler:On the Brauer-Manin obstruction for degree four del Pezzo surfaces[dvi] [ps] [pdf], to appear in: Acta Arithmetica

J. Jahnel and D. Schindler:Del Pezzo surfaces of degree four violating the Hasse principle are Zariski dense in the moduli scheme[dvi] [ps] [pdf], to appear in: Annales de l'Institut Fourier

J. 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:

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 [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 lectures

Clay summer school 2006 at Göttingen [pdf] ANTS VII at Berlin [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 4.34 ofOn the order three Brauer classes for cubic surfaces.)

## Sums of three cubes

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

For the following 14 numbers below 1000, this is still not clear.

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

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.

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

## 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.