# Alan Lauder's Web Page

I am a university lecturer in the Mathematical Institute, and the tutorial fellow in pure mathematics at Hertford College.

My interests are in computational number theory, *p*-adic cohomology, elliptic
curves
and modular forms.
I can be contacted at "lauder" at "maths.ox.ac.uk".

I am a member of the Editorial Board of the Quarterly Journal of Mathematics.

I co-organised a conference on Effective methods in *p*-adic cohomology
and also a conference on Rational points on curves.

## Recent Papers:

- (with Kevin Buzzard) A computation of modulars forms of weight one and small level PDF. There is an accompanying webpage containing data on weight one modular forms. To appear in Annales Mathématiques du Québec.

- (with Henri Darmon and Victor Rotger) Overconvergent generalised eigenforms of weight one and class fields of real quadratic fields, Advances in Mathematics 283 (2015), 130-142. PDF (Erratum: the field on the second line of Example 1.1 should be K(√(3 + √21)/2).)

- (with Henri Darmon and Victor Rotger) Gross-Stark units and p-adic iterated integrals attached to modular forms of weight one, Annales Mathématiques du Québec, (2016), 1 - 30. PDF

- (with Henri Darmon and Victor Rotger) Stark points and p-adic iterated integrals attached to modular forms of weight one, Forum of Mathematics, Pi (2015), Vol. 3, e8, 95 pages. PDF Journal Version

- Efficient computation of Rankin p-adic L-functions, in "Computations with Modular Forms, Proceedings of a Summer School and Conference, Heidelberg, August/September 2011", Boeckle G. and Wiese G. (eds), Springer Verlag, 181-200, 2014. PDF

- Computations with classical and p-adic modular forms, LMS J. Comput. Math. 14, (2011), 214-231 PDF. (Freely available via JCM website.) Code for Algorithms 1 and 2 in SAGE: Alg1and2.sage. A fully documented python version of this sage file: Alg1and2.py. Code for Algorithms 1 and 2 in MAGMA: Alg1and2.m. (The algorithms are now included in the distributed versions of MAGMA and SAGE.)

## Older Publications:

- Degenerations and limit Frobenius structures in rigid cohomology, LMS J. Comp. Math. Vol. 14, 1-33, 2011. Freely available via JCM website. PDF
- Ranks of elliptic curves over function fields. LMS J. Comp. Math. Vol. 11, 172-212, 2008. Freely available at JCM website. PDF (Erratum: the deduction in the first sentence of the proof of Corollary 3.5 is incorrect. A more refined argument is required, taking into account the steps in the filtration, and so the exponent in the power of p in the statement should be multiplied by 2.)
- A recursive method for computing zeta functions of varieties, LMS J. Comp. Math. Vol. 9, 222-267, 2006. Freely available at JCM website. PDF
- (with Fatima Abu Salem and Shuhong Gao) Factoring polynomials via polytopes, Proceedings of ISSAC (International Symposium on Symbolic and Algebraic Computation) 2004, Gutierrez (Ed), 4-11. PDF
- Counting solutions to equations in many variables over finite fields, Foundations of Computational Mathematics Vol. 4 No. 3, (2004), 221-267. PDF
- Deformation theory and the computation of zeta functions, Proceedings of the London Mathematical Society, Vol. 88 Part 3, (2004), 565-602. PDF
- (with Daqing Wan) Computing zeta functions of Artin-Schreier curves over finite fields II, Journal of Complexity, Vol. 20 (2004), 331-349. PDF
- Computing zeta functions of Kummer curves via multiplicative characters, Foundations of Computational Mathematics, Vol 3 No. 3, pages 273-295, 2003. PDF
- Zero-patterns of polynomials and convex polytopes, J. of Combinatorial Theory Series A, Vol. 102 (No.1), (2003), 10-15. PDF
- (with Daqing Wan) Computing zeta functions of Artin-Schreier curves over finite fields, London Mathematical Society JCM Volume 5, (2002), pp 34-55. PDF. Freely available at JCM website.
- (with Richard Brent and Shuhong Gao) Random Krylov spaces over finite fields, SIAM J. on Discrete Mathematics, Vol. 16 No. 2, (2003), 276-287. PDF
- (with Daqing Wan) Counting points on varieties over finite fields of small characteristic, in "Algorithmic Number Theory: Lattices, Number Fields, Curves and Cryptography" (Mathematical Sciences Research Institute Publications), J. P. Buhler and P. Stevenhagen (eds.), Cambridge University Press, 579-612, 2008. (Version Oct 2002) PDF
- (with Shuhong Gao and Erich Kaltofen) Deterministic distinct-degree factorisation of polynomials over finite fields, Journal of Symbolic Computation, Vol. 38 No. 6, (2004), 1461-1470. PDF
- (with Kenny Paterson) Computing the error linear complexity spectrum of a binary sequence of period 2^n, IEEE Trans. Info. Theory Vol 49 No 1, (2003), 273-280. PDF
- (with Shuhong Gao) Hensel lifting and bivariate polynomial factorisation over finite fields, Math. Comp. 71, No. 240, (2002), 1663-1676. PDF
- (with Shuhong Gao) Decomposition of polytopes and polynomials, Discrete Comput. Geom 26, 89-104 (2001) PDF
- Continued fractions of Laurent series with partial quotients from a given set, Acta Arithmetica XC.3, 252-271 (1999) PDF
- Polynomials with odd orthogonal multiplicity, Finite Fields and Their Applications 4, 453-464 (1998) PDF
- Continued fractions and sequences, Ph.D. London University 1999 PDF

## Expository Papers:

- Rationality and meromorphy of zeta functions, Finite Fields and Their Applications 11 (2005), 491-510. PDF
- Rigid cohomology and p-adic point counting, J. Theor. Nombres Bordeaux 17 (2005), no. 1, 169-180. PDF
- Homotopy methods for equations over finite fields, in "Applied Algebra, Algebraic Algorithms and Error-Correcting Codes", Fossorier, Hoholdt and Poli (eds), LNCS 2643,18-24, 2003. PDF