Computer Code

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Computer code and data for "A family of surfaces with p_g=q=1, K^2=2, and large Picard number"

  1. Verifying the A_5-singularity. This Mathematica file verifies that the divisor Z(mu(tau)) in Theorem 5.3 has A_5-singularities at the points indicated, by analyzing blow-ups.
  2. Obtaining the coefficients of the family (Mathematica). This Mathematica file takes the file above as a starting point, but uses general coefficients to show (as discussed in Remark 5.5) how one may use the same blow-up procedure to derive the coefficients of mu(tau) given in Theorem 5.3. At one point, it also refers to the calculations in the Magma file below.
  3. Obtaining the coefficients of the family (Magma). Accompanies the file above.
  4. Divisors Z(mu(tau)) with further singularities. This Magma code determines those (non-excluded) values of tau for which Z(mu(tau)) that have a further singularity outside of the expected A_5-singularity, the answer being given in Proposition 6.2.
  5. Computing the zeta function of Y(mu(3)). The Magma code in this file computes the zeta function of the surface Y(mu(3)) over F_q, for a prime power q. It is used in conjunction with the file below to prove Proposition 7.1.
  6. Counting points on X^+(mu(3)) and X^-(mu(3)). The Magma code analyzes orbits (under iota^+, iota^-, and the Galois group) of the F_{q^2}-points of Y(mu(3)) to determine the point counts of X^+(mu(3)) and X^-(mu(3)) over F_q, for a prime power q. It also contains the output when q=11, and is used in conjunction with the file above to prove Proposition 7.1.

 

Computer code and data for "A surface over Q with p_g=q=1, K^2=2, and minimal Picard number"

  1. Determining nonsingularity of B(a). This file contains Singular code to determine whether the divisor B(a) on E^(2) is nonsingular (by checking whether the corresponding double cover is nonsingular). The code here is useful in a few places in the paper for specific values of a.
  2. Determining the zeta function of a double cover. This file contains Magma code to determine the zeta function of the reduction of the double cover of X(a) to F_q. More precisely, it allows one to determine the nontrivial factor coming from the middle cohomology of the surface.
  3. Counting Frobenius-invariant iota-orbits. While it can be done by hand, this file discusses how one may use Magma (including some of the code in the previous file) to show that the surface X_1 in Theorem B satisfies #X_1(F_3)=9. This is what allows us to determine the zeta function from that of its double cover.
  4. Verifying the properties of the pencil $Pi$. This file discusses how one may use Singular to verify that the specific pencil $Pi$ has the properties claimed.

 

Computer code and data for "The Tate Conjecture for a family of surfaces of general type with p_g=q=1 and K^2=3"

  1. Singular codePDF File . The code in this file represents calculations carried out for the pencil J_1 using the computational commutative algebra software Singular.
  2. Numerical dataPDF File . The numerical data in this file was calculated using Singular via the code in the file above, and is too lengthy to include in that file. The data reflects the singular fibers and singular points of the pencil J_1.
  3. Rational map. This Mathematica file uses the numerical data above to check that the rational map eta is defined at the singular points of all singular divisors in the pencil J_1. Created with Mathematica 5.1.