The Thirteenth Algorithmic Number Theory Symposium took place at the University of Wisconsin, Madison on July 16-20, 2018.

The Invited speakers were:

- Jennifer Balakrishnan “Effective aspects of quadratic Chabauty”
Jennifer explained Minhyong Kim’s non-abelian Chabauty framework to obtain an effective methods for determining the rational points on a curve using Coleman integrals. She explained how p-adic heights provide a bilinear function that enables Kim’s approach to be performed in the “quadratic” case. She surveyed new results, including resolving the problem of rational points on the “cursed curve” that parameterises split Galois representations of level 13.

- Noam Elkies “Curves with many points over number fields”
Noam demonstrated constructions of infinite families of curves over Q of fixed genus with many rational points.

- Steven Galbraith “Current trends and challenges in post-quantum cryptography”
The talk gave an overview of post-quantum crypto and the NIST standardisation process, then listed a number of open problem in isogeny crypto. Slides are here: http://www.math.grinnell.edu/~paulhusj/ants2018/TalkSlides/Galbraith.pdf.

- Melanie Matchett Wood “Effective Chebotarev density theorems for families of number fields without GRH”
Melanie surveyed her joint work with Lillian Pierce and Caroline Turnage-Butterbaugh on effective Chebotarev density theorems for certain families of number fields (the fields in each family all have the same Galois group).

- Emmanuel Thomé “Computation of discrete logarithms in finite fields”
Emmanuel gave an overview of number field sieve algorithms for discrete logarithms in , including recent research on variants for different regions of interest in terms of He also mentioned the LOGJAM attack on TLS (from 2015) and the “hidden-SNFS” 1024-bit discrete logarithm (2016).

The contributed papers included the following of interest to ECC researchers:

- S. Abelard, P. Gaudry and P.-J. Spaenlehauer, Counting points on genus-3 hyperelliptic curves with explicit real multiplication. The paper is about Schoof-type algorithms for point counting on genus 3 hyperelliptic curves over finite fields.
- T. Kleinjung and B. Wesolowski, A new perspective on the powers of two descent for discrete logarithms in finite fields. The paper is about the descent step in quasi-polynomial-time algorithms for the DLP in for small and large .
- A. V. Sutherland, Fast Jacobian arithmetic for hyperelliptic curves of genus 3. This paper gives formulas for fast computation of divisor class groups of genus 3 curves with two points at infinity.
- B. Wesolowski, Generating subgroups of ray class groups with small prime ideals. The paper has results about isogeny graphs in higher dimensions.

The winners of the Selfridge Prize for best paper were Michael Musty, Sam Schiavone, Jeroen Sijsling and John Voight for their paper “A database of Belyĭ maps”.

The poster prize was awarded to Travis Scholl for his poster on “Isolated Abelian Varieties over Finite Fields”. Here is a list of all the posters.

The Rump session took place late on the Thursday afternoon. I put up an announcement for ECC 2018 at Osaka Japan on November 19-21 (preceded by a 2 day autumn school). The rump session also included by presentation by Benjamin Smith giving a general way to break tri-linear maps of the type proposed by Huang, by using the intersection pairing on the endomorphism ring.

— Steven Galbraith