Joerg Jahnel's Research

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Some papers

  • My Habilitation Thesis
      J. Jahnel: Brauer groups, Tamagawa measures, and rational points on algebraic varieties [dvi] [ps] [pdf]

  • Preprints
      A.-S. Elsenhans and J. Jahnel: The Picard group of a K3 surface and its reduction modulo p [dvi] [ps] [pdf]

      A.-S. Elsenhans and J. Jahnel: On the computation of the Picard group for K3 surfaces [dvi] [ps] [pdf]

      A.-S. Elsenhans and J. Jahnel: Cubic surfaces with a Galois invariant pair of Steiner trihedra, to appear in: International Journal of Number Theory [dvi] [ps] [pdf]

      J. Jahnel: More cubic surfaces violating the Hasse principle [dvi] [ps] [pdf]

      A.-S. Elsenhans and J. Jahnel: On the Brauer-Manin obstruction for cubic surfaces, to appear in: Journal of Combinatorics and Number Theory [dvi] [ps] [pdf]

      A.-S. Elsenhans and J. Jahnel: The discriminant of a cubic surface [dvi] [ps] [pdf]

      A.-S. Elsenhans and J. Jahnel: The Diophantine Equation x4 + 2 y4 = z4 + 4 w4---A number of improvements [dvi] [ps] [pdf]

      A.-S. Elsenhans and J. Jahnel: The Fibonacci sequence modulo p2---An investigation by computer for p < 1014 [dvi] [ps] [pdf]

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

  • A funny note
      J. Jahnel: When is the (co)sine of a rational angle equal to a rational number? [dvi] [ps] [pdf]

  • Articles
      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 x4 + 2 y4 = z4 + 4 w4---An investigation by computer for |x|, |y|, |z|, |w| < 2.5 106, Math. Comp. 75(2006)935-940 [dvi] [ps] [pdf]

      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]

  • Ph.D. Thesis
      J. Jahnel: Tangential Flatness and Lech-Hironaka Inequalities [ps] [pdf]

  • Diploma Thesis
      J. 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]

    Number-theoretic software

  • hashing
      The Hashing package searching for solutions of Diophantine equations of the form
                  f(x1, ... ,xk) = g(y1, ... ,yl),
      Version 1.0.

      Contains example programs for the equations
         [demo]   x3 + y3 = z3 + w3,   search limits: 0 < x, y, z, w < 5000, 
         [kub]   a x3 = b y3 + z3 + v3 + w3,   search limits: a, b < 100, |x|, |y|, |z|, |v|, |w| < 5000, 
         [quart]   a x4 = b y4 + z4 + v4 + w4,   search limits: a, b < 100, x, y, z, v, w < 100 000. 

      Source code: [tar.gz]
  • swd
      Code searching for solutions of Sir P. Swinnerton-Dyer's equation
         [swd]      x4 + 2 y4 = z4 + 4 w4,           search limits: x, y, z, w < 100 000 000, 
      making use of Hashing.

      Source code: [tar.gz]

    Cubic surfaces with a Galois invariant double-six

    The 27 lines on a smooth cubic surface form a highly symmetric configuration. Its group of symmetries is the Weyl group W(E6), of order 51840. For a cubic surface over Q, a subgroup of W(E6) operates on the 27 lines.

    W(E6) has exactly 350 conjugacy classes of subgroups.

    For all subgroups stabilizing a double-six, we constructed explicit examples of cubic surfaces over Q.

    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 < a3 + b3 + c3 < 1000 where a3 + b3 + c3 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.