For this purpose, we design a practical version of the method of complex approximations presented in [EC11]. For example, we use fast exponentiation in the modular Jacobian instead of analytic continuation, which greatly reduces the need to compute Abelian integrals, since most computations deal with divisors. We also present an efficient way to compute arithmetically well-behaved functions on Jacobians, a method for expanding peak shapes in quasilinear time, and a trick that makes computing the image of a Frobenius element with a modular Galois representation more efficient. .
In particular, we get rid of the sign ambiguity that results from the use of a projective representation as in [Bos07]. Indeed, our method relies on numerical calculations in C, so we need a bound on the height of the result to prove it. Namely, we checked that the discriminant of the polynomial F(X) that defines the representation (see the next section) is of the form vM2.
For example, we were able to prove the correctness of the projective version of the representation ρ∆,29 of level 29 attached to the new form f = ∆, which we calculated (cf. our results section): J. We thus now have two `-torsion divisors D1 and D2, whose images of the Abel-Jacobi map form the basis of the `-torsion subspace Vf,l. We first show in subsection 3.1 how to quickly compute a large number of terms of the q-expansion at infinity of the peak forms of weight 2 and level.
Finally, we explain in 3.6 how to construct a well-behaved function on the jacobian J1(`) and how to evaluate it at the `-torsion divisors, and we conclude by describing in 3.7 an efficient way to compute the image of the Frobenius. elements of the Galois representation.
Expanding the cuspforms of weight 2 to high pre- cision
More precisely, to compute disseq expansions to the precision O(qB), we first compute a generator of the Hecke algebra T2,`⊗ZQ, by choosing a Hecke operator and testing whether it is a Q-algebra generator . Ps should also be chosen large enough for the reduction modp of the coefficients to be credible. We can then calculate the q-expansions of the formsωwith trivial nebentypusε =1 inB1 as follows.
Indeed, its degree is the maximum number of zeros of the 1-form ωdqq plus the number of poles of the 1-formedj. Once this is done, we can compute the q-expansions of the forms ω with non-trivial nebentypusε as follows. Let ω0 ∈ B1 be one of g0 forms2 with trivial nebentypus whose q-expansion we have just calculated.
Finally, we apply to the q-expansions of the forms that we have just calculated, the change of the basis matrices from Bε to the basis of the eigenforms that we have calculated at the beginning, so that we obtain the q-expansions of the new forms. For a fixed primary level `, the number of bitwise operations required to compute the q-expansion of the new forms in S2 Γ1(`).
It is useful for our purpose because it can be used to map a point τ with a small imaginary part to −1`τ, which can have a much larger imaginary part. Indeed, we checked that p63 is very often sufficient for wp to span the rational homology of the modular curve over T2,' and p67 is enough for all but 661 levels.
Arithmetic in the jacobian J 1 (`)
This space is the direct sum of all cusp shapes of weight 2 and of the scalar multiples of Eisenstein series e1,2 and e1,3 of weight 2 that vanish at all cusps except c1 and c2 for 1,2 and except c1 and c3 for e1,3 , so we have. From now on, we will identify weight-6 modular shape spaces with the corresponding modular function spaces obtained by dividing by f03 without explicitly stating it. To calculate these q-expansions, we assume the q-expansions at ∞, and apply diamond operators and Fricke involutions to achieve all other cusps, as explained below.
We could also have represented shapes only by their q-expansions at ∞, but we think that using q-expansions at multiple cusps is better for numerical stability. The knot points above 0 are all rational, while the knot points above ∞ constitute a single Galois orbit. We know how the Fricke operator behaves on new forms of weight 2 (cf. subsection 3.2 on the periods), and on Eisenstein series (cf. next subsection 3.4).
In addition, all the forms we are dealing with have a nebentypus, so that the action of each diamond operator hdion their q expansions is easy to compute: it amounts to multiplying by the value of their character by d. Thus, using these two kinds of operators, we get the q expansions of the new forms and of the Eisenstein series on all cusps of their q expansions at.
Finding the appropriate Eisenstein series
Consequently, in the case where N =` is prime, only two cases remain, namely Gχ,12 and G1,χ2 , where χ is a non-trivial even Dirichlet character modulo. We construct Eisenstein series e1,2 and e1,3 as linear combinations of the E2χ,1's and the E21,χ's, because they have nicer q-expansions than their G counterparts. Now it easily follows from the orthogonality relations between Dirichlet characters that the Eisenstein sequence.
Computing an `-torsion divisor
However, this is not direct, since these algorithms only work with divisors of the form D − D0, where D is an effective divisor of degree d0, and D0 and d0 are defined in the beginning of Section 3.3. To get around this, we fix what we call a stuffing divisor, that is, an effective divisor C of degrees d0−g = g+ 1, we feed the divisors Pg. We do this by evaluating the q series in the basis of V at the points of D and doing linear algebra.
Because we will need to evaluate the q-series at C, it turns out to be convenient to choose a divisor C supported by vertices, hence the notation C. The ± sign is not a problem because we get a basis vector for Vf,l whatever the sign is, and that's all we really need.
Evaluating the torsion divisors
If we want this function Ξ to play the role of α, then we want it to be computationally doable, so ξ should be chosen as low as possible. Any point x ∈Jac(X) can be written as [Ex−gO], where Ex is an effective divisor of degree g on X that is generically unique, and O ∈ X is a fixed point. The divisor of tx has the form (tx) = −Π +Ex+Rx, where Rx is a residual effective divisor of degree g on X, which is the reflection's image of Ex.
This map is only well-defined on a Zariski-dense subset of Jac(X) due to the genericity assumptions, and it is defined over Q if X, Π, A, B and O are defined over Q. The divisor of poles of α is the sum of only two translations of the Θ divisor. It is even optimal in a certain sense, since by the Riemann-Roch theorem for abelian varieties (cf [HS00, theorem A.5.3.3]), no non-constant function on Jac(X) a single translation of Θ as divisor of poles has , while a generic curve X NS has Jac(X).
RentalsD ∈V such that sD WD,red spans C, we know that the divisor of sD is of the form. Again by the Riemann-Roch theorem, ED is generically alone in its linear equivalence class. This allows us to compute the map α, which will be defined over Q if C1,C2,A and B are.
As in the previous section, it is convenient to choose the divisors C1 and C2 to be supported by cusps so that the q series are easy to evaluate, hence the notation C1 and C2.
Finding the Frobenius elements
We then compute complex approximations of the resolvents ΓC(X) by listing matrices in the similarity classes of GL2(F`). This subgroup S is the subgroup consisting of the elements of odd order in F∗`, that is, the 20-subgroup of F∗`. It is clear that the image of the conjugation class of gbyπ is exactly the conjugation class of eg, so that πg is well-defined and surjective.
1 6∈ S, we can derive the similarity class of the image of the Frobenius element in GL2(F`) using our knowledge of its determinant. As a result, with this trick we can still calculate the full, non-quotient representation ρf,l and we have saved a factor |S|2 in the calculation of the roots of the solvent and a factor |S| in their expansion and in identification of their coefficients as rational numbers. Computation times were as follows: computation of the q-expansion of cuspforms and the Eisenstein series in O(q5000).
We found a polynomial F(X) ∈ Q[X] that defines the representation, of degree and with a common denominator of 142 decimal digits, and finally it took a little less than 20 minutes to calculate the solvents ΓCe(X) thanks to the quotient representation trick and for massive parallelization, after which it takes about 30 minutes to derive the similarity classes of the image of a Frobenius element at p≈101000. The prime ` = 23 is actually one of the finitely many primes we have to rule out for f = ∆, as we mentioned at the beginning of the introduction. Computing the period lattice took a little less than 2 hours, computing each of the two 23-torsion divisors took 5 and a half hours, and computing the F23 plane spanned by them took just over 11 hours.
Calculating the periods took just over 6 hours, calculating each of the two 29 torsion dividers took 120 hours, and calculating the F29 plane spanned by them took about 100 hours. Then calculating the resolutions ΓCe(X) thanks to the quotient representation trick took about 60 hours, and finally deriving the image of the Frobenius on the same prime number sp≈101000 as in level 19 took 2 hours. We used a precision of 15 kbits inC for the calculation of the defining polynomial F(X), and a precision of 18 Mbits for the calculation of the resolvents Γ.
The most time-consuming part of the computation of the polynomial F(X)∈ Q[X] that defines the representation is the arithmetic in Jacobian J1(`). Let H be the logarithm of the common denominator of F(X), so computing F(X) with our method requires a precision of O(H) bits in C. The precision in C we need to work with for this is O ( `2H), so that the total complexity of the computation of the solvents ΓC(X) is O(`e 8H) bit operations, which is the same as the rest of the computation.
For example, using this trick allows us to reduce the complexity of calculating the resolvents Γ. Note that limiting to such ` does not exacerbate the complexity of calculating coefficients by Chinese remainders.
