Statement of Purpose November 13, 1999

Helena A. Verrill  verrill@math.ku.dk

Much of my recent work has involved making explicit number theoretical computations, which has included the application and development of algorithms. These computations arise from, and are connected with, many different areas of mathematics such as the study of Calabi-Yau threefolds, the Bloch-Kato conjectures and modular Galois representations. I am particularly interested in computations and algorithms related to the theory of modular forms. Despite recent major advances in this area, there are many difficult outstanding problems, such as the Birch–Swinnerton-Dyer conjecture and its generalization, due to Bloch and Kato, and the Lang–Trotter conjecture. Computation has allowed me to verify conjectures in specific examples, for example, my computations in [V1] and [V2] verify certain instances of the modularity conjecture for rigid Calabi-Yau threefolds, formulated by Noriko Yui [Y]. Verification can lead to better understanding, and refinement of the conjectures. By computation of examples I have also found new phenomena, such as some interesting relations between the L-series of certain Calabi-Yau threefolds and certain elliptic curves in [V2]. The algorithms I have recently been working on are an algorithm for determining the image of exceptional ℓ-adic modular Galois representations, and algorithms for finding the periods and the period lattice of modular forms of arbitrary weight. These projects are described in more detail below.

The MSRI Algorithmic Number Theory program presents a great opportunity for me to continue collaborative work with people such as Noriko Yui, and William Stein, who is also applying to participate in this program, and to begin new work with other participants. For example, it would be useful to have the opportunity to talk about the arithmetic of K3 surfaces with Hiro Goto, who is applying for general membership, or about supersingular primes of elliptic curves. with Noam Elikies, or with Joe Buhler about the image of modular Galois representations. Since I am collaborating with Noriko Yui, working on a monograph about Thompson series and mirror maps for families of K3 surfaces, it would be very useful to have enough time at the MSRI to work together on this project. This work is an extension of work in [VY].

Since I am interested in a wide range of computational problems related to number theory and arithmetic geometry, as described below, it is very likely that there will be other people interested in similar topics, with whom I expect to be able to embark on new projects. Also, since I have a large collection of examples of various phenomena, I expect that I will be able to interest other people in working with me on constructing theory to explain and generalize these examples. At Copenhagen University, where with Ian Kiming we are building on my computational work, which has motivated the production of more general theorems and methods.

I will now discuss some of my computational interests in L-series, periods of modular forms, and images of modular representations. I am also enclosing a list of abstracts of papers giving more detail of some of the problems that I have worked on.

1) L-series of varieties: I have explicitly computed the L-series of the middle cohomology of several rigid Calabi-Yau threefolds in [V1] and [V2], obtaining weight-4 modular forms. I am interested in using the methods to produce an algorithm for giving the L-series

Research Statement: page 1/3

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cover letter abstracts

Helena A. Verrill  verrill@math.ku.dk

of rigid Calabi-Yau threefolds more generally. I believe an algorithmic approach should be possible, given the defining equations over ℚ for an affine piece of the Calabi-Yau variety in ℙ^{4}, even though there is probably not a simple way to construct a completion of X in each case. This is because the cohomology of a Calabi-Yau threefold is very simple, and although the number of points on a Calabi-Yau threefold over 𝔽_{p} does not give the coefficient a_{p} of the L-series of H^{3}(X, ℚ_{ℓ}) in such as simple way as one obtains coefficients of the L-series of H^{1}(E, ℚ_{ℓ}) for an elliptic curve E, the difference is typically given by a polynomial in p of degree 3, with coefficients involving certain Kronecker symbols, and possibly also coefficients of modular forms of lower weight. I intend to study this in more detail to find out exactly what the possibilities are. One result that would be very useful in an algorithm based on this approach would be to extend Livné’s work, [L], which was based on work of Serre and Faltings. Livné’s result tells you how many coefficients of the L-series of two Galois representations one has to compute in order to show equality of the L-series. This theorem has become an important result in the area of computing L-series. I have been looking at the possibility of extending Livne’s result in various ways, such as, if one knows that the L-series of two varieties differ by a polynomial in p for enough p, can one conclude they are equal?

Also I am interested in computing the L-series of other varieties. One approach towards computing L-series, in the case of fibered varieties, is to use the relation with the Picard-Fuchs equation of the fibration. In [V1], [V2], and [V3], a fibration of a Calabi-Yau threefold is given, X → ℙ^{1}, with unifomizing parameter t of ℙ^{1} a weight-0 modular function t(τ). Then in certain cases, for q = exp(2πiτ), one has

\frac{q}{t(τ)} \frac{df}{dq} = g(τ) + Ag(B(τ),

for some integers A, B. Here f(τ), a solution of the Picard-Fuchs differential equation, is a weight-2 modular form.

I intend to pursue the study of computation of L-series further, in particular with regard to fibrations over other spaces than ℙ^{1}, and other varieties, such certain non rigid Calabi-Yau fibered threefolds. For example, for what I call the A_{1} ×A_{2} case in the notation of [V0], it is difficult to obtain either the L-series or the solutions of the Picard-Fuchs equation, but perhaps, combining partial results about the L-series and the Picard-Fuchs equation, one might be able to compute both simultaneously.

2) Periods of modular forms: I have worked with modular symbols to compute periods of modular forms. In [SV] we introduce the idea of “transportable modular symbols”, and give an algorithm to compute the period lattice of any modular form of weight k ≥ 2, generalizing work of Cremona [C], who computed period lattices of rational weight-2 modular forms. Our algorithms have applications to verification of parts of the Bloch-Kato conjectures, and also can be used to give a numerical approximation to the intermediate Jacobian of rigid Calabi-Yau threefolds as is partly explained in [V2] and [SV].

3) Images of modular representations: I have computed the images of several mod-p modular representations, starting from the data given by the coefficients of a cuspidal eigenform. Currently, with Ian Kiming, I am working on related matters, such as computation of

Research Statement: page 2/3

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cover letter abstracts

Helena A. Verrill  verrill@math.ku.dk

more examples, and generalizing the approach taken in the examples to give an algorithm, which, given an arbitrary cuspidal eigenform f(τ), and any prime p, will compute the image of the corresponding mod-p Galois representation. Some of our results are in [KV1] and [KV2]. These results are obtained by applying a generalization of results obtained by Kiming [K].

Currently my main interest is in understanding modular mod-p Galois representations and their images. Given a cuspidal eigenform f(τ), with associated Galois representation ρ : Gal(\overline{ℚ}/ℚ) → GL_{2}(ℚ_{l}), let ρ_{p} : Gal(\overline{ℚ}/ℚ) → GL_{2}(\overline{𝔽_{p}}) be the semisimplification of a mod-p reduction of ρ. By work of Ribet [R], there are only finitely many p for which the image of ρ_{p} is not as large as possible, but there is still no algorithm to give a list of all the p for which this happens. I am hoping to obtain such an algorithm, and understand the exceptional cases. The work I am currently undertaking with Ian Kiming is a first step in this direction.

I am also interested in understanding the supersingular primes of a cuspidal eigen form f(τ). A prime p is supersingular for a rational cuspidal eigenform f(τ) = ∑ a_{n} exp(2πiτ) if p divides a_{p}. The Lang-Trotter conjecture ([LT]) gives a conjectures for the distribution of supersingular primes, and Elkies [E] proved that there are infinitely many supersingular primes for weight-2 modular forms with rational coefficients. However, what happens in higher weight is not yet known. I hope my computations will give a better understanding of this problem, and will help me in working towards concrete results.

I believe that the environment of the MSRI Algorithmic number theory program will be excellent for work on the above projects. I fully expect to contribute to the program, with further advances in connections with the above circle of ideas, in particular contributing in the area of arithmetic geometry computations.

References

[C] Cremona, J.: Algorithms for modular elliptic curves, 2nd edition, Cambridge University Press, (1997).

[E] Elkies, N.: The existence of infinitely many supersingular primes for every elliptic curve over Q. Invent. Math. 89 (1987), no. 3, 561–567.

[K] Kiming, I.: On the liftings of 2-dimensional projective Galois representations over Q. J. Number Theory 56 (1996), no. 1, 12–35.

[KV1] Kiming, I and Verrill, H.: Computing the image of modular Galois representations, report, in preparation.

[KV2] Kiming, I and Verrill, H.: Twists of modular mod-ℓ representations with exceptional images, preprint, in preparation.

Research Statement: page 3/3

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cover letter abstracts

Helena A. Verrill  verrill@math.ku.dk

[LT] Lang, S. and Trotter, H.: Frobenius distributions in GL_{2}-extensions. Lecture Notes in Mathematics, Vol. 504.

[R] Ribet, K.: On l-adic representations attached to modular forms. Invent. Math. 28 (1975), 245–275.

[SV] Stein, W., and Verrill, H.: Computing period lattices of newforms, in preparation.

[V1] Verrill, H.: Root lattices and pencils of varieties, Kyoto Journal of mathematics, Volume 36. Number 2, July 1996, 421–446.

[V1] Verrill, H.: The L-series of some Calabi-Yau threefolds, to appear in the Journal of number theory.

[V2] Verrill, H.: The L-series of rigid Calabi-Yau threefolds from fiber products of elliptic curves, in preparation.

[V3] Verrill, H.: Some congruences related to modular forms, Max-Planck-Institut für Mathematik preprint 1999 (26).

[VY] Verrill, H. and Yui, N: Thompson series, and the mirror maps of pencils of K3 surfaces, to appear in the Proceedings of the NATO ASI/CRM Summer School “The Arithmetic and Geometry of Algebraic Cycles” Banff Alberta Canada 1998.

[Y] Yui, N.: The arithmetic of Calabi–Yau varieties and mirror symmetry to appear from the Institute for Advanced Study.

Research Statement: page 4/3

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cover letter abstracts