Version 1.0

2022-03-08

Authors: Catherine Hsu, Preston Wake, Carl Wang-Erickson

This code is written for Sage, Version 9.2.

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   Copyright (C) 2022 Hsu--Wake--Wang-Erickson 

Distributed under the terms of the GNU General Public License (GPL)

This code is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
General Public License for more details.

The full text of the GPL is available at:

              http://www.gnu.org/licenses/

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This program implements the algorithms described in Explicit Non-Gorenstein R=T via Rank Bounds II: Computation. To use this program, load RR3_main.sage and then call the function RR3_computations() for each example in which you want to check the conditions of Theorem 4.5.1.

RR3_computations() requires the following inputs:

• values for p, ell0, and ell1 satisfying Assumption 2.1.1
• do_a1_p_check = True or False, True if you want the program to check if a1 is split at p, False otherwise
• mode = "Full" or "Ell1", both modes produce the same output, but Ell1 mode assumes that certain S-units in Qz have already been computed in Sage

When in Full mode, RR3_computations() does not assume any previously computed data and calls the following files:

1. initial_setup.sage: This implements the framework required to execute these computations, including fixing the relevant parts of the pinning data in Definition 2.2.1.
2. functions.sage: This implements the functions used throughout the rest of the program.
3. Qz_extensions.sage: This implements a computation of S-units in Qz for a0 and c1.
4. Kolyvagin_setup.sage: This computes a1_cand and b2_cand following Theorem 3.5.1.
5. a1_side.sage: This computes the local adjustments for a1 (Algorithms 1 and 2) and then checks if Condition (i) in Theorem 4.5.1 holds (Algorithm 5).
6. a1_p_splitting.sage: This optional computation checks if a1 splits at p, which can simplify the local adjustments of b2.
7. b2_prelim.sage: This implements the preliminary adjustment for b2_cand following Lemma 4.3.1.
8. b2_side.sage: This computes the local adjustments for b2 (Algorithms 3 and 4) and then checks if Condition (ii) in Theorem 4.5.1 holds (Algorithm 6).

When in Ell1 mode, RR3_computations() assumes that S-units in Qz have already been computed for a0 and c1. As such, the program calls initial_setup_ell1.sage, a slightly modified version of initial_setup.sage, and omits the file Qz_extensions.sage. Other than these modifications, Ell1 mode is identical to Full mode.