An L-series or an L-function is an infinite sum L(s) = ∑n=1∞an/ns in the complex variable s with complex coefficients an. Such functions arise in many places in mathematics and they are usually naturally associated with some kind of mathematical object, for instance a character, a number field, a curve, a modular form or a cohomology group of an algebraic variety. The coefficients an are certain invariants associated with that object. For example, in the case of a character χ: (ℤ/mℤ)*→ℂ* they are simply its values an = χ(n) when gcd(n,m) = 1 and 0 otherwise.
Magma is able to associate an L-series to various types of object. The heart of the analytic number theory package in Magma is an L-series package of routines originally due to T. Dokchitser. In this context an L-series can be thought of as a Dirichlet series with specified coefficients an given by L(s) = ∑n an/ns for the complex variable s. Almost all applications require L(s) to have an Euler product — namely L(s) = ∏p Lp(1/ps) where each Lp is a finite degree polynomial.
Such an L-series must also satisfy a functional equation Λ(s) = Λ(w + 1 – s) for some (nonnegative, integral) weight w upon completing the L-series via Λ(s) = γ(s)L(s)ℚs, where γ(s) is an appropriate product of Γ-functions and the conductor ℚ is an arithmetically-defined integer. In many cases, the most difficult computations arise when determining the Euler factors Lp at primes p dividing ℚ; such primes are called bad primes. A prototypical example of an L-series is the Riemann zeta function.
The barebones structure of this implementation makes possible the definition of general L-series and subsequent calculations with them. These include checking their functional equation and determining any special values to a given precision. T. Dokchitser first implemented such a framework in PARI/GP, but his versions of Magma from 2014 onwards contain a greatly expanded spectrum of possible L-series as listed below:
Dirichlet characters, number fields, elliptic curves over ℚ, and classical modular forms are relatively standard. The remainder are basically unique to Magma. Elliptic curves, hyperelliptic curves, Artin representations, hypergeometric motives and Jacobi motives are discussed in other chapters while the remaining cases are discussed briefly in the following subsections.
An Artin representation is a continuous homomorphism ρ: Gal(\Q/ \Q)→GL(V) from the absolute Galois group of \Q to a finite dimensional complex vector space V. By continuity, this factors through the Galois group of some finite extension K/ \Q. The Euler factors Lp can be computed by taking completions of K, while the computations for the places of ramification are more difficult.
The Dokchitser brothers have also implemented functionality in Magma which makes it possible to twist an elliptic curve over ℚ by an Artin representation (for instance), allowing faster computation. For example, this makes it possible to fix an elliptic curve over ℚ and study how the arithmetic varies when extending the field, the computations being expedited by the Artin representation machinery.
A hyperelliptic curve can be defined as y2 = f(x) where f is a polynomial of degree at least 5. (When f has degree 2 or less one obtains a conic, while degrees 3 and 4 give a genus 1 curve and so elliptic curve methods are more relevant). The determination of the Euler factors at good primes can be accomplished via Kedlaya's algorithm. However, to determine the Euler factors at bad primes, Magma relies on the integral model package of S. Donnelly (Magma), whereas an analogous PARI/GP version uses only Liu's algorithm.
Magma also allows one to work with hyperelliptic curves over number fields. Here the functionality seems not to be available elsewhere.
A Dirichlet character over a number field modulo an ideal I is a homomorphism from OK/I to ℂ, the image necessarily consisting of roots of unity. This defines a map on field elements; correspondingly, a Hecke character is defined on ideals, which means in particular that it must be trivial on the units. A Hecke character can be thought of as a Dirichlet character that is trivial on the units and is then extended by the class group, corresponding to the non-principal ideals. This is indeed essentially how the implementation proceeds as it dualises the ray class group machinery due to F. Hess and C. Fieker (Magma). The Hecke characters have associated L-functions, which is not directly the case for Dirichlet characters.
A Hecke Grossencharacter additionally has a nontrivial size component, which arises from embeddings into the complex numbers. For instance over ℚ(i), the ideal (2 + i) can be mapped to the complex number 2 + i (or 2 – i), and similarly at other principal ideals. The embeddings are determined by requiring a specific residue class for a generator of the ideal. This makes the subject matter much richer, although also much more delicate to work with, as it is necessary to ensure that the embeddings for each prime ideal are globally coherent. The theory behind computing with Grossencharacters was worked out by M. Watkins (Magma) over a three month period in 2009.
The Magma Grossencharacters package has been used by students of F. Calegari in their construction of a half-integral weight Hilbert modular form; in particular, they computed the values of the L-series at negative integers.
The generality of Dokchitser's package makes it easy to implement L-series for Hilbert modular forms once information from the Hecke operator computations is available. As noted in the previous section, this was used to construct a half-integral weight Hilbert modular form.
Magma provides a number of ways of specifying hypergeometric data of which the product of cyclotomic polynomials is one. Functions are provided for computing the hypergeometric trace according to the p-adic Γ-function definition. The Euler factor function computes the p-th Euler factor of the hypergeometric motive at t, a rational number. Finally, an L-series function tries to construct the L-series of the associated motive.
Functions are also provided that try to determine the associated Artin representation and the associated elliptic curve, as catalogued by Cohen.
A topic related to hypergeometric motives is that of Jacobi sum motives. These are indeed simpler, and in fact the tame prime information for hypergeometric motives can be determined from Jacobi motives, possibly twisted by Kummer and Tate characters.
The classical Jacobi sums were indicated by Weil to come from Grössencharacters and this functionality is also included, with it indeed being the preferred method to compute Euler factors and the L-series, once the reciprocity correspondence has been established and the Grössencharacter identified.
The Euler factor of Jacobi motive can be computed at a good prime. Also given a Jacobi motive, it can be identified as a Grössencharacter. This uses the Weil bound on the conductor, and then tries enough good primes to distinguish the character. This is now the preferred way to compute the LSeries of a Jacobi motive (though the latter still exists).
The creation of L-series from smaller ones, via symmetric powers and tensor products, is well developed in Magma. In particular, the case of symmetric powers of L-series for elliptic curves over ℚ is completely handled. This is based on theoretical work due to Dummigan, Martin, and Watkins. The Dokchitser brothers have recently developed methods for handling many cases of bad Euler factors for tensor products.
The symmetric square L-function of an elliptic curve plays a major role in the computation of its degree of modular parametrisation (the existence of this being a result coming out of Andrew Wiles' work on proving Fermat's Last Theorem). Watkins (Magma), in joint work with Dummigan and also Martin, has produced extensive numerical data for higher symmetric powers, including analysis of central vanishings (for odd powers) and divisibility of special values.
One of the principal directions of inquiry in the area of L-functions concerns their order of vanishing at the central point. This is conjecturally related to the rank of a finitely generated abelian group in great generality; the simplest case is the Birch and Swinnerton-Dyer conjecture (a Clay Milliennial problem), which states that the rank of an elliptic curve should be equal to the order of vanishing of its L-function at the central point. Knowing how large the rank of an elliptic curve (or the order of vanishing) can be is also a significant open problem, having been hotly disputed over the years. M. Watkins has undertaken many experiments, both for elliptic curves and other types of L-functions, which have led to a refined understanding. Two specific problems studied include (a), predicting the growth rate for rank 3 curves in a quadratic twist family and (b), whether rank 7 is the maximal rank for quadratic twists of the congruent number curve. The congruent number curve is the elliptic curve y2 = x3 – n2x, where n is a positive integer.