Fifteenth Algorithmic Number Theory Symposium, ANTS-XV,
University of Bristol
August 8 - 12, 2022

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Where the schedule says LUNCH BREAK or COFFEE BREAK, it means that we will be serving lunch or tea/coffee/snacks. At other times participants may have to forage for sustenance on their own.

Monday

• 9:30 - 10:45 Coffee and arrival
• 10:45 Welcome and opening remarks
• 11:00 Andrei Seymour-Howell. Rigorous computation of Maass cusp forms of squarefree level ( slides )
• 11:30 Aurel Page and Pascal Molin. Computing groups of Hecke characters ( slides )
• 12:00 — 13:30 LUNCH BREAK
• 13:30 Francois Morain. Modular curves over number fields and ECM ( slides )
• 14:00 Wouter Castryck, Marc Houben, Frederik Vercauteren and Benjamin Wesolowski. On the Decisional Diffie-Hellman Problem for Class Group Actions on Oriented Elliptic Curves ( slides )
• 14:30 (via Zoom) David Lubicz and Damien Robert. Fast change of level and applications to isogenies ( slides )
• 15:00 — 16:00 COFFEE BREAK
• 16:00 — 17:00 Kevin Buzzard

  Teaching number theory to computers.

  Computers have been used to great effect in number theory, to do computations far beyond what humans are able to do. My talk is about using computers to do mathematics in a different way, namely to reason about mathematics. As an example, we can use computers to compute class groups of global fields, but nowadays we can also use a computer to prove that the class group of a global field is finite. I'll give an introduction to the area, show you the software in action, and explain what I think the future has in store.

Tuesday

• 9:30 Tom Fisher. On binary quartics and the Cassels-Tate pairing ( slides )
• 10:00 Daniel Hast. Explicit two-cover descent for genus 2 curves
• 10:30 — 11:00 COFFEE BREAK
• 11:00 Antoine Comeau-Lapointe, Chantal David, Matilde Lalin and Wanlin Li. On the vanishing of twisted L-functions of elliptic curves over function fields ( slides )
• 11:30 Marc Houben and Marco Streng. Generalized Class Polynomials ( slides )
• 12:00 — 13:30 BREAK
• 13:30 Kiran S. Kedlaya. The relative class number one problem for function fields, I ( slides )
• 14:00 Madeleine Kyng. Computing zeta functions of algebraic curves using Harvey's trace formula ( slides )
• 14:30 Juanita Duque-Rosero and John Voight. Triangular modular curves of small genus ( slides )
• 15:00 — 16:00 COFFEE BREAK
• 16:00 — 17:00 Fredrik Johansson

  Real numbers: a computational perspective.

  Can we make computing with real and complex numbers as easy and reliable as arithmetic in "exact" domains like the rational numbers and finite fields? I will discuss open problems and recent work towards this goal. There are two fundamental, dual tasks: numerical evaluation (for establishing inequalities), and algebraic computation (for establishing equalities). On the numerical side, we want to obtain provably accurate approximations, often with very high precision (hundreds or thousands of digits). This requires tools to deal with slow convergence, numerical instability and degenerate cases like function singularities. On the algebraic side, the central challenge is to deal with algebraic and transcendental number fields of high degree and with non-canonical representations. Finally, an overarching problem is to integrate a multitude of specialized representations and algorithms in a useful form in mathematical software.

• 18:30 — 22:00 Conference Party / Dinner at Zero Degrees.

Wednesday

• 9:30 — 10:30 Alina Ostafe

  On some GCD, linear recurrence and unlikely intersection problems.

  Let \(a,b\) be multiplicatively independent positive integers. Bugeaud, Corvaja and Zannier (2003) proved that $$ \gcd(a^n-1,b^n-1)\le \exp(\varepsilon n) $$ for any fixed \(\varepsilon>0\) and sufficiently large \(n\). Ailon and Rudnick (2004) were the first to consider the function field analogue and proved a much stronger result in this setting. These results triggered a floodgate of various extensions and generalisations, from the number case, to function fields in both zero and positive characteristics. In this talk I will discuss some of these results in the function field case and their connections to unlikely intersection problems for parametric curves, as well as a generalisation to linear recurrence sequences. The latter is also motivated and connected to the the well-known Skolem Problem, for which I will discuss the issue of decidability for parametric families of linear recurrence sequences.

• 10:30 — 11:00 COFFEE BREAK
• 11:00 Andreas-Stephan Elsenhans and Jörg Jahnel. 2-adic point counting on K3 surfaces ( slides )
• 11:30 Edgar Costa, David Harvey and Andrew V. Sutherland. Counting points on smooth plane quartics
• 12:00 — 13:30 BREAK
• 13:30 Harald Helfgott and Lola Thompson. Summing mu(n): a faster elementary algorithm ( slides )
• 14:00 — 15:00 Psych Session*
  • Wouter Castryck. An efficient key recovery attack on SIDH ( slides )
  • Luciano Maino. An attack on SIDH with arbitrary starting curve ( slides )
• 15:00 — 16:00 COFFEE BREAK
• 16:00 (via zoom) Noam Elkies and Gaurav Goel. On powerful integers expressible as sums of two coprime fourth powers ( slides )
• 16:30 (via zoom) Nils Bruin and Eugene Filatov. Twists of the Burkhardt Quartic Threefold
• 17:00 — ~18:30 POSTERS AND DRINKS
• ~18:30 — ??? BUSINESS MEETING AND RUMP SESSION.

Thursday

• 9:30 — 10:30 (via zoom) Tanja Lange

  S-unit attacks.

  Lattice-based cryptography is a strong contender for post-quantum cryptography, having scored three out of four announced winners in the NIST post-quantum competition. All three of those winners use structured lattices derived from cyclotomic fields. A standard argument for the security of structured lattices is a "worst-case-to-average-case reduction" proving that an attack would imply an attack against (approximate) Ideal-SVP, the problem of finding short nonzero elements of a nonzero ideal of the ring of integers of a cyclotomic field. This raises the question of whether Ideal-SVP is in fact hard. This talk explains S-unit attacks against Ideal-SVP, including some recent results.

• 10:30 — 11:00 COFFEE BREAK
• 11:00 Catherine Hsu, Preston Wake and Carl Wang-Erickson. Explicit Non-Gorenstein R=T via rank bounds II: Computation ( slides )
• 11:30 Eran Assaf, Dan Fretwell, Colin Ingalls, Adam Logan, Spencer Secord and John Voight. Definite Orthogonal Modular Forms: Computations, Excursions and Discoveries ( slides )
• 12:00 — 12:30 BREAK
• 12:30 Tímea Csahók, Peter Kutas, Mickaël Montessinos and Gergely Zábrádi. Explicit isomorphisms of quaternion algebras over quadratic global fields ( slides )
• 13:00 Alex Cowan. Computing newforms using supersingular isogeny graphs ( slides )
• 13:30 — ??? CONFERENCE PHOTO
• 13:30 — ??? FREE AFTERNOON

Friday

• 10:00 Jonathan Webster and Andrew Shallue. Tabulating Carmichael Numbers n = Pqr with small $P$ ( slides )
• 10:30 Samuele Anni, Eran Assaf and Elisa Lorenzo García. On smooth plane models for modular curves of Shimura type ( slides )
• 11:00 Nikola Adžaga, Shiva Chidambaram, Timo Keller and Oana Padurariu. Rational points on hyperelliptic Atkin-Lehner quotients of modular curves and their coverings ( slides )
• 11:30 — 13:00 LUNCH BREAK
• 13:00 — 14:00 (via Zoom) Peter Sarnak

  An underdetermined moment problem for eigenvalues of matrices in classical groups and its application to computing root numbers and zeros of L functions.

  We describe thresholds for the recovery of the determinant and the exact count of eigenvalues in certain intervals, of random matrices in classical groups of dimension n which share the same traces of their powers up to k (less than n). Key to this is the study of the real algebraic geometry and shapes of semialgebraic sets that are associated with compact moment curves. This study is applied to give subexponential in the conductor, algorithms to compute the root numbers and exact counts of zeros of L-functions coming from arithmetical algebraic geometry. Joint work with Michael Rubinstein.

• 14:00 — 14:30 COFFEE BREAK
• 14:30 David Harvey and Markus Hittmeir. A deterministic algorithm for finding r-power divisors ( slides )
• 15:00 (via zoom) Daniel J. Bernstein. Fast norm computation in smooth-degree Abelian number fields

*These papers are not part of the ANTS proceedings.