DIAMANT Symposium Autumn 2023

Tight bounds for reconstructing graphs from distance queries

Suppose you are given the vertex set of a graph and you want to discover the edge set. An oracle can tell you, given two vertices, what the distance is between these vertices in the graph. (For example, in a computer network, this would represent the minimum number of communication links needed to send a message from one computer to another.) Based on the answer, you may select the next query. The (labelled) graph is reconstructed when there is a single edge set compatible with the answers. How many queries are needed, in the worst case? The question becomes interesting for bounded degree graphs. We provide tight bounds for various classes of graphs, improving both the lower and upper bound, in both the randomized and deterministic setting. Based on joint work with Paul Bastide (https://arxiv.org/abs/2306.05979).

Slopes of modular forms and the ghost conjecture

Modular forms are holomorphic functions with a wealth of symmetries. Even though these functions are borne out of complex analysis, their Fourier coefficients contain a wealth of arithmetic information. Even bounding the sizes of these coefficients involve very deep mathematics — the best bounds follow from Deligne’s proof of the Weil conjectures, for which he was awarded the Fields medal.

In this talk, rather than looking at complex absolute values, we will instead focus on the p-adic size of p-th Fourier coefficient for a prime number p. We will state a conjecture (the ghost conjecture) which gives an exact description of these sizes for all modular forms. This funnily named conjecture converts difficult automorphic questions into more accessible combinatorial ones. We will discuss the state of this conjecture and its applications to several open questions on slopes of modular forms.

On lightweight cryptography

In recent years research in symmetric cryptography has focused on lightweight ciphers, leading even to an open NIST competition dedicated to it. In this talk we take a step back and look at lightweight cryptography from a more historical perspective. We will philosophize about which cryptographic functionality is required and what the lightweight requirements really are in the real world.

Unified treatment of Artin-type problems

Since Hooley’s seminal 1967 resolution of Artin’s primitive root conjecture under the Generalized Riemann Hypothesis, numerous variations of the conjecture have been considered. We present a framework generalizing and unifying many previously considered variants, and prove results in this full generality (under GRH). This is joint work with Olli Järviniemi and Pietro Sgobba.

Weak Spatial Mixing for The anti-ferromagnetic Potts Model on The d-ary Tree

Given a q-coloring on a graph, the probability that a specific vertex v is colored i is 1/q. However, if we impose a boundary condition, i.e., we require that a set of vertices is pre-colored, then given this boundary condition, the probability that v is colored i doesn’t have to be 1/q anymore. If the distance between v and this set of pre-colored vertices goes to infinity, we might hope that the probability that v is colored i actually does converge to 1/q. If this behavior occurs, we speak of weak spatial mixing (WSM). This property depends on q and the maximum degree Delta of the graph. We generalize this problem for the anti-ferromagnetic Potts model instead of just proper colorings and share what is being conjectured regarding how WSM for the Potts Model on the d-ary tree is related to the parameters. We will also share some new results.

Based on a joint project with Ferenc Bencs and Guus Regts.
https://arxiv.org/abs/2310.04338

Experiments with Ceresa classes of cyclic Fermat quotients

Let C be a curve of genus g embedded in its Jacobian J. The Ceresa cycle C-[-1]*C is a homologically trivial algebraic cycle of codimension g-1 on J. When C is hyperelliptic, this cycle is trivial modulo algebraic equivalence, whereas for a general curve C it is non-trivial by work of Ceresa. The first example of a non-hyperelliptic curve with torsion Ceresa class modulo algebraic equivalence was found by Beauville and Schoen. We give two new examples of curves with such behavior. All three examples (including the one of Beauville and Schoen) are cyclic quotients of Fermat curves. We compute the central order of vanishing of the L-functions of the relevant motives and give evidence in one case for the conjecture of Beilinson-Bloch. This is joint work with Ari Shnidman.

Gross and Zagier’s Fascinating Formula

Some elliptic curves are more special than others, and we say that such curves “have CM”. The j-invariant of an elliptic curve determines its isomorphism class. Therefore, the j-invariants of CM-curves are of particular interest. In the 1980s, Gross and Zagier observed and proved fascinating factorisation phenomena for these values, paving the way for many more exciting and ground-breaking discoveries. In this talk, we rediscover and explore the fascinating factorisation formula by Gross and Zagier and conclude by outlining how these ideas can be generalised to the setting of Shimura curves, for which I proved similar factorisation formulae in recent work.

Finding orientations of supersingular elliptic curves and quaternion orders

Orientations of supersingular elliptic curves encode the information of an endomorphism of the curve. Computing the full endomorphism ring is a known hard problem, so one might consider how hard it is to find one such orientation. We prove that access to an oracle which tells if an elliptic curve is $\mathfrak{O}$-orientable for a fixed imaginary quadratic order $\mathfrak{O}$ provides non-trivial information towards computing an endomorphism corresponding to the $\mathfrak{O}$-orientation. We provide explicit algorithms and in-depth complexity analysis. We also consider the question in terms of quaternion algebras. We provide algorithms which compute an embedding of a fixed imaginary quadratic order into a maximal order of the quaternion algebra ramified at $p$ and $\infty$. We provide code implementations in Sagemath which is efficient for finding embeddings of imaginary quadratic orders of discriminants up to $O(p)$, even for cryptographically sized $p$. This is joint work with J. Clements, P. Dartois, J. Eriksen, P. Kuta’s, and B. Wesolowski.

Logarithmic Hochschild (co)homology of log smooth schemes

Hochschild (co)homology is an invariant that has a nice description for smooth schemes, known as HKR isomorphism and proved by Arinkin and Căldăraru. Can one extend this notion to logarithmic schemes encoding their combinatorics? We give a positive answer by setting up a framework where one can revisit the steps of the classical theory by looking at log smooth schemes.
Based on a joint work with Márton Hablicsek and Leo Herr.

The scheme of monogenic generators

Let $L$ be a number field, with $O_L$ its ring of integers. The theorem of the primitive element shows $L = Q(t)$ is generated by a single element over the rationals. Is the same true for the rings of integers $O_L = Z[t]$? Called the “Hasse problem,” this depends on the number field $L$. We describe a scheme $M$ parameterizing all possible choices of $t$ generating the ring of integers $O_L = Z[t]$. The rational points of $M$ govern the Hasse problem, whose hidden geometry offers another perspective. Joint work with S. Arpin, S. Bozlee, H. Smith.

Frobenius distributions in a vertical sense

Elliptic curves over a finite field F_q come in two flavours: ordinary and supersingular. As q varies over powers of a fixed prime p, the eigenvalues of Frobenius of an ordinary elliptic curve are uniformly distributed on a circle, while those of a supersingular elliptic curve are supported are finitely many places. In joint work with Santiago Arango-Piñeros and Deewang Bhamidipati, we study this phenomenon for abelian varieties in higher dimensions. In this talk, I will discuss some of our results and some open questions in this area.

Strong approximation, and beyond number fields

Strong approximation property of algebraic varieties generalize the Chinese Remainder Theorem in a natural way. One important tool to tackle such a problem is the Brauer-Manin obstruction, which is also used in the study of local-global principle. Going beyond number fields, we can study certain function fields of curves which behave in a similar way as number fields, and analogous notions of strong approximation and Brauer-Manin obstruction can be defined. We give results of certain varieties over these function fields where this method applies and successfully predicts the strong approximation property.