Contents

A list of tickets we're working on can be found here. If you work on a ticket, please add `sd87` to the list of keywords so that it appears!

Most of the code for working with p-adics can be found here and here if you want to browse.

## Big Sage projects

Add general extensions of p-adic fields in Sage #23218 (David Roe)

Add Julian's Mac Lane package which provides a general framework for discrete valuations to Sage #21869 (Julian)

Add Julian's Completion package, for general p-adic extension backed by number fields, to Sage #22956 (Julian)

Polynomial factorization, using Julian's Mac Lane package and/or Brian Sinclair's ticket #12561 (Ticket needs review) (Brian - meeting in UHS 115)

- make sure simpler factoring methods are in good shape, like Hensel lifting and Panayi's root finding.

- Lattice precision for p-adics (in particular p-adic matrices, polynomials) (Xavier)
Smith normal form #23450 (ready for review), determinant #23478 (ready for review)

Hermite normal form #23486 (ready for review)

Lattices in p-adic vector spaces: this is handled by the generic code for modules over PID (after the implementation of HNF above and this simple ticket #23503)

Design a framework for lattice precision #23505

- Power series via p-adic templates (Adriana)
Linkage files for p=2 and/or using longs for the case that p^{\text{prec}} < 2^{62}

Implementation of Gröbner bases and tropical Gröbner bases algorithm (F4, F5, FGLM), doctest and submission (Tristan). A ticket on F5 has been posted (#23461, needs review). A ticket on a Tropical F5 is in progress (#23501).

We might finish reviewing the inclusion of openf4 at #18749 and patch it to avoid going through strings all the time

- And also look at the performance of Singular, polybori, giac, ...

- Zeta function tickets
Roadmap for regular models in Sage using Mac Lane package, Suchandan Pal's code and Stefan Wewers' work. (Julian)

Etale algebras (maybe see also ticket #21413) (Ricky)

For an old list of possible projects, see padics

## Smallish Sage projects

- Norms, traces, frobenius, matrix mod pn for relative p-adic extensions (David)
- Add more black-box testing to p-adics, performance benchmarketing (Aly)
nth roots, square roots that create extensions (extend=True as for integers) #12567 (Marc, David)

Artin-Hasse exponentials #12560 (Xavier)

Gauss sums via the Gross-Koblitz formula, which uses code on p-adic gamma functions #23456 (Adriana and Ander)

- Better coercion/conversion to and from residue fields (Aly, Marc)
Optimized implementation of Frobenius automorphism #12657 (Ander)

p-adic polylogarithms #20260 (Alex) Ready for review!

bug in matrix of Frobenius when p = 3 #11960 (Jen)

Switching to exact defining polynomials for p-adic extensions #23331 (David)

Change p-adic constructors to not care about the base ring of a defining polynomial #18606 (David)

Investigate slowness in unramified extensions #23172 (Xavier)

Review Xavier's fast exponential code #23235 (Xavier)

Add Monge-reduction for Eisenstein polynomials (first over \mathbb{Q}_p, then over unramified extensions) (Sebastian)

Generic zeta function method for schemes #20308 (Edgar)

Elliptic curve point counting over F_q using PARI #16931 #16949 (J-P Flori, Kevin)

Expose PARI code for Frobenius matrix on hyperelliptic curves #20309 (Marc)

Raise coverage of schemes/hyperelliptic_curves/monsky_washnitzer.py to 100% #15645 (Edgar)

For number fields, extend the range of degrees where is_galois() works, add method is_abelian(), and for abelian number fields add method conductor() (JJ)

- Requested by Anna Haensch: A weak approximation function

## Non Beginner Sage tickets needing review

## Beginner Sage projects

Change root_field to return a p-adic field #14893, #20073,#20244 (Aly)

Add an

`exact_ring`method for p-adic rings and fields #23507 (Adele)- Update and improve the p-adic tutorial (Rob)
- Add more thematic tutorials in number theory (Rob)
- Here are some tickets that should be easy to review (feel free to add more!):
#23190 (possible dependencies for global tests -- Adele)

#23185 (Needs work, doctests are failing)

#23482 (Claire)

#23483 (Adele)

#23376 (Samuel)

#23473 (This tickets now needs a review ! -- David A.) (Freda)

#23456 (Adele)

#23495 (Adele)

#23499 (Adele)

#23193 (Freda)

#23194 (Edgar)

#23235 (Adele)

#23479 (Sara)

#23503 (Angie)

#12657 (Sara)

#20308 (David A.)

#23509 (Sara)

polylogarithms #20260

#23507 (Sara)

#23510 (Adele)

#23512 (really small) (Edgar)

## LMFDB projects

Fix polredabs related issues #2135 (JJ)

Finish up: https://github.com/LMFDB/lmfdb-collab/wiki//Warwick-workshop-June-2017, precisely:

- Doc-Testing utilities (David Lowry-Duda)
Hecke algebras: see https://github.com/sanni85/HeckeAlgebras and https://github.com/sanni85/lmfdb/tree/hecke_alg (Samuele) pull request #2189

Display Hecke eigenvalues in terms of an integral basis. See: #975, pull request #2197 (Edgar, Sam Schiavone, Michael Musty)

- Compute Galois splitting models (Ben, Angie)
Display local algebras (JJ)

- Better handling of character tables (JJ)
- Some finished work:
Sanitize API interface: https://github.com/LMFDB/lmfdb/issues/2053 (Edgar)

Prototype a Sage/LMFDB interface. See https://github.com/LMFDB/lmfdb/issues/1360 and https://github.com/LMFDB/lmfdb/issues/2169 (Edgar, Simon Brandhorst and David Lowry-Duda)

PR #2184

Moving import scripts: pull request #2186

Pull request #2198: Hilbert modular forms search CM and base change, solves issues #1975, #1972

Yoshida lifts of Hilbert modular forms: adding function to compute, working on displaying the data (Malcolm, Samuele) https://github.com/sanni85/lmfdb/tree/paramodular_lift and Pull Request https://github.com/LMFDB/lmfdb/pull/2201

- Compute shapes of cubic and quartic number fields (Rob)
Shapes of number fields (Rob, Samuele) pull request #2205

- Idle:
Work on the isogeny classes of abelian varieties over finite fields. See http://beta.lmfdb.org/Variety/Abelian/Fq/ (Christelle)

Work on Hypergeometric Motives over ℚ. See http://beta.lmfdb.org/Motive/Hypergeometric/Q/ (Dave Roberts)

- Use Monge-reduced polynomials (and ones related to them) for defining polynomials (JJ)
Dirichlet characters modulo l: see https://github.com/sanni85/Dirichlet_modL (Samuele)

Galois representations modulo l: see https://github.com/sanni85/Mod-l-galois-representations and http://beta.lmfdb.org/Representation/Galois/ModL/ (Samuele)