Research interests

My main research interests are in the area of modular forms, a recurring theme being modular Calabi-Yau varieties.

Modular forms are beautiful functions, which I have tried to give an indication of visually in programs written in magma and java. These only really say as much about modular forms as the division of a line into segments of length 2 pi would say about the sine function, so the pictures only give a pale reflection of the theory of modular forms. Nevertheless, I hope that the images give the layman some indication of the symmetry properties involved. I gave some more details in a recent course on modular forms at LSU.

Research articles

In the following list I have tried to divide my papers into categories, though there is some overlap.
Preprints and papers are in the same list, but published papers include a reference to the published version.
A link is given to an electronic version, if available, though this may not be identical to the published version. I have added links to journals, in cases where you will be able to find the final version there, if you have a subscription to the journal. Links are also given to coauthors web pages, who may also have alternative electronic versions.

I have tried to be as complete as possible in this page, so I've also included a list of "fun", non research articles, since I don't have enough of these to give them a separate page. I've also listed here a book translation.

Contents: • Calabi-Yau varieties • fundamental domains • modular symbols • mod l Galois representations • Picard-Fuchs differential equations • Computational/Expository • For fun • Students' work

On Calabi-Yau varieties • Arithmetic of a certain Calabi-Yau threefold, by H. A. Verrill. In Number theory (Ottawa, ON, 1996), volume 19 of CRM Proc. Lecture Notes, pages 333--340. Amer. Math. Soc., Providence, RI, 1999. • The L-series of certain rigid Calabi-Yau threefolds, by H. A. Verrill. J. Number Theory, 81(2):310--334, 2000. electronic version (ps), Max-Planck preprint MPIM1999-17 • Update on the modularity of Calabi-Yau varieties, by Noriko Yui. In Calabi-Yau varieties and mirror symmetry (Toronto, ON, 2001), volume 38 of Fields Inst. Commun., pages 307--362. Amer. Math. Soc., Providence, RI, 200.3 With an appendix by Helena Verrill. • On modularity of rigid and nonrigid Calabi-Yau varieties associated to the root lattice A4. by K. Hulek and H. Verrill, To appear in the Nagoya Journal of Mathematics, September 2005. electronic version, math.AG/0304169 • On the modularity of Calabi-Yau threefolds containing elliptic ruled surfaces, by K. Hulek and H. Verrill, To appear in the proceedings of a conference on the arithmetic of Calabi-Yau threefolds, held in Banff, November 2003. electronic version, math.AG/0502158 • On the motive of Kummer varieties associated to Gamma1(7) - Supplement to the paper: The modularity of certain non-rigid Calabi-Yau threefolds (by R. Livne and N. Yui) by K. Hulek and H. Verrill, electronic version, math.AG/0506388 On fundamental domains • Fundamental domains for Shimura curves, by David R. Kohel and Helena A. Verrill. J. Théor. Nombres Bordeaux, 15(1):205--222, 2003. Les XXIIèmes Journées Arithmetiques (Lille, 2001). electronic version (pdf) . On modular symbols • Cuspidal modular symbols are transportable. by William A. Stein and Helena A. Verrill. LMS J. Comput. Math., 4:170--181 (electronic), 2001. older electronic version (ps), Max Planck preprint MPIM2001-7 • Transportable modular symbols and the intersection pairing, by H. A. Verrill. In Algorithmic number theory (Sydney, 2002), volume 2369 of Lecture Notes in Comput. Sci., pages 219--233. Springer, Berlin, 2002. On mod l Galois representations • On modular mod l Galois representations with exceptional images. by I. Kiming and H. A. Verrill. J. Number Theory, 110(2):236--266, 2005. electronic version (math.NT/0311082) . On Picard-Fuchs differential equations • Root lattices and pencils of varieties, by H. A. Verrill. J. Math. Kyoto Univ., 36(2):423--446, 1996. electronic version (ps) as a Kyoto University preprint. • Picard-Fuchs equations of some families of elliptic curves, by H. A. Verrill. In Proceedings on Moonshine and related topics (Montréal, QC, 1999), volume 30 of CRM Proc. Lecture Notes, pages 253--268, Providence, RI, 2001. Amer. Math. Soc. • Some congruences related to modular forms, by H. A. Verrill electronic version (ps), Max Planck preprint MPIM1999-26 • Thompson series, and the mirror maps of pencils of K3 surfaces, by Helena Verrill and Noriko Yui. In The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), volume 24 of CRM Proc. Lecture Notes, pages 399--432. Amer. Math. Soc., Providence, RI, 2000. • Sums of squares of binomial coefficients, with applications to Picard-Fuchs equations. by H. Verrill, electronic version, math.CO/0407327 Computational/Expository/Translation • Notes on toric varieties by D. Joyner and H. Verrill, electronic version, math.AG/0208065 • Elementary algebraic geometry, by Klaus Hulek, volume 20 of Student Mathematical Library. American Mathematical Society, Providence, RI, 2003. Translated from the 2000 German original by Helena Verrill. For fun This section includes papers which really have nothing to do with my research, but are included here for completeness. • Coin-moving puzzles, by Erik D. Demaine, Martin L. Demaine, and Helena A. Verrill. In More games of no chance (Berkeley, CA, 2000), volume 42 of Math. Sci. Res. Inst. Publ., pages 405--431. Cambridge Univ. Press, Cambridge, 2002. • Origami Tessellations by H. A. Verrill, Bridges 1998 Conference Proceedings Proceedings of the 1998 Bridges Conference on Mathematical Connections in Art, Music, and Science. Edited by Reza Sarhangi • Neutron, by H. A. Verrill Eureka 49, (March 1989), 62-66, journal of the Archimedeans, the Cambridge University student math society. (This was the first "article" I ever wrote; not serious mathematics, just for fun. It's about strategies in a simple game called "neutron", that we used to play a lot at the "puzzles and games ring". It's interesting to remember what I was playing with when I was a second year undergraduate, and compare with the far more advanced things undergraduates do in REUs these days!)

Students