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# Algebra Seminar 2020/21

Due to the current Covid-19 pandemic, we will host online research talks until further notice. Send an email to steffen.muller@rug.nl if you want to participate.

We meet on Wednesdays, usually at 15:00.

## Schedule

• 14-07-21, 15:00 - 16:30: Master's thesis presentations:
15:00 Anne Tuijp (Groningen): McEliece variation under attack: cryptanalysis of a public key cryptosystem based on algebraic geometry codes [Abstract]
The McEliece cryptosystem is a public key cryptosystem that has (so far) not been broken by any quantum algorithms and may therefore be resistant to quantum computer attacks. McEliece makes use of error correcting codes in the process of encryption and decryption. The original McEliece cryptosystem uses classical Goppa codes, but these have the disadvantage that the public and private keys of the cryptosystem are very large matrices. Many variations with shorter keys have been proposed - and then been proven to be insecure. One example is the proposal to use algebraic geometry codes instead of classical Goppa codes. An attack on this variation, based on Schur products and error correcting pairs, is examined in detail in this project.
15:45 Floor Dogger (Groningen): Classifying abelian threefolds of p-rank 0, 1, 2 and 3 [Abstract]
The aim of the thesis is to find relations between the endomorphism algebra, the p-rank and the factorization of the rational prime p into prime ideals in the number field generated by the Frobenius endomorphism for abelian threefolds. For elliptic curves and abelian surfaces, these relations are discussed in Chapter 1 and Chapter 2. The main theorem follows in Chapter 3 and gives a complete classification of the p-rank in terms of the splitting behaviour of the rational prime p in the maximal order of the number field generated by the Frobenius endomorphism of an absolutely simple abelian threefold. In Chapter 4, the reduction of absolutely simple CM abelian threefolds is studied. Only abelian threefolds with CM by a CM field of which the Galois group of the normal closure is cyclic or isomorphic to the dihedral group D_6 are considered. For both options, the endomorphism algebra and the p-rank of the reduced CM abelian threefold is determined from the prime factorization of a rational prime p in the ring of integers of the CM field.
• 07-07-21, 15:00 Martin Lüdtke (Goethe-Universität Frankfurt): Refined Selmer equations for the thrice-punctured line [Abstract]
Betts-Dogra have proposed a refinement of the Chabauty-Kim method to find S-integral points of a curve by including in the definition of the Selmer scheme local conditions at the primes contained in S. We are applying the theory in the case of the thrice-punctured line. We determine explicit equations for the refined Chabauty-Kim sets in depth 2 for #S=2 and show that they are satisfied precisely by the solutions of the S-unit equation in many cases.

The results were obtained by last year's Arizona Winter School project group under the guidance of Alex Betts. This is joint work with him and Alex Best, Theresa Kumpitsch, Angus McAndrew, Lie Qian, Elie Studnia, and Yujie Xu.

• 30-06-21, 15:00 Vandita Patel (University of Manchester): Values of the Ramanujan tau-function [Abstract]
The infamous Ramanujan tau-function is the starting point for many mysterious conjectures and difficult open problems within the realm of modular forms. In this talk, I will discuss some of our recent results pertaining to odd values of the Ramanujan tau-function. We use a combination of tools which include the Primitive Divisor Theorem of Bilu, Hanrot and Voutier, bounds for solutions to Thue–Mahler equations due to Bugeaud and Gyory, and the modular approach via Galois representations of Frey-Hellegouarch elliptic curves. This is joint work with Mike Bennett (UBC), Adela Gherga (Warwick) and Samir Siksek (Warwick).
• 23-06-21, 15:00 Sebastian Falkensteiner (JKU Linz): Tropical Differential Algebra [Abstract]
Tropical Differential Algebraic Geometry considers difficult or even intractable problems in Differential Equations and tries to extract information on their solutions from a restricted structure of the input.
The fundamental theorem of Tropical Differential Algebraic Geometry and its extensions state that the support of power series solutions of systems of ordinary differential equations (with formal power series coefficients over an uncountable algebraically closed field of characteristic zero) can be obtained either, by solving a so-called tropicalized differential system, or by testing monomial-freeness of the associated initial ideals.
Tropicalized differential equations work on a completely different algebraic structure which may help in theoretical and computational questions, particularly on the existence of solutions.
• 16-06-21, 15:00 Cecília Salgado (Groningen): Mordell Weil rank jumps and the Hilbert property [Abstract]
In this talk I will discuss recent progress around the variation of the Mordell-Weil rank in families of elliptic curves. The first part will be dedicated to give an introduction to the theme, by motivating and discussing the state of the art. The second part will be dedicated to presenting a recent work joint with Dan Loughran (Bath) and work in progress with Renato Dias. More precisely, let X be surface endowed with a (non-constant) elliptic fibration with a section defined over a number field. Specialization theorems by Néron and Silverman imply that the rank of the Mordell-Weil group of special fibers is at least equal to the MW rank of the generic fiber. We say that the rank jumps when the former is strictly larger than the latter. I will discuss rank jumps for elliptic surfaces fibred over the projective line. If the surface admits a conic bundle we show that the subset of the line for which the rank jumps is not thin in the sense of Serre.
• 09-06-21, 15:00 Pim Spelier (Leiden): A geometric linear Chabauty comparison theorem [Abstract]
Let $C$ be a curve over the rationals of genus $g \geq 2$. Such a curve has finitely many rational points, but finding them all is difficult. Assuming the rank of the Mordell-Weil group of the Jacobian is less than $g$, the Chabauty-Coleman method is very effective in practice, returning a small set of $p$-adic points containing the set of rational points. In this talk I will discuss a variant of Chabauty, geometric linear Chabauty (based on geometric quadratic Chabauty as developed by Edixhoven and Lido), that can outperform Chabauty-Coleman in some cases; I will compare this method to the classical Chabauty-Coleman method, and characterise the differences. This talk is based on joint work with Sachi Hashimoto.
• 02-06-21, 15:00 Barinder Singh Banwait (Harish-Chandra Research Institute): Explicit isogenies of prime degree over quadratic fields [Abstract]
Let K be a quadratic field which is not an imaginary quadratic field of class number one. We describe an algorithm to compute a superset of the set of primes p for which there exists an elliptic curve over K admitting a K-rational p-isogeny. Combining this algorithm with recent work on the determination of quadratic points on low-genus modular curves, we determine - conditional upon the Generalised Riemann Hypothesis - the above set of isogeny primes for several quadratic fields, providing the first such examples after Mazur's 1978 determination for K = Q. We will give a live demo of the Sage and PARI/GP implementations of the algorithm.
• 26-05-21, 15:00 Bernd Sturmfels (MPI Leipzig & UC Berkeley): Linear PDE with Constant Coefficients [Abstract]
We discuss algebraic methods for solving systems of homogeneous linear partial differential equations with constant coefficients. The setting is the Fundamental Principle established by Ehrenpreis and Palamodov in the 1960’s. Our approach rests on recent advances in commutative algebra, and it offers new vistas on schemes and coherent sheaves in computational algebraic geometry. Please check out arXiv:2104.10146.
• 19-05-21, 15:00 Alice Garbagnati (Università Statale di Milano): Elliptic fibrations and Kodaira dimensions of Schreieder's varieties [Abstract]
We consider the $d$-dimensional Schreieder's varieties $X_{n,d}$, which are the quotient of the product of a certain curve $C_n$ by itself $d$ times by a diagonal action of the group $(\mathbb{Z}/n\mathbb{Z})^{d-1}$. They were originally constructed to obtain varieties with prescribed Hodge numbers; in particular if $n=3^c$ Schreieder also gave an explicit desingularization of $X_{3^c,d}$ and computed its full Hodge diamond. If $n=3,4$ these varieties were already studied by Cynk and Hulek as examples of modular Calabi--Yau varieties.

More recently Flapan proved that if $n=3^c$ and $d=2$, a smooth model of the surface $X_{3^c,2}$ is a modular elliptic surface and if $n=3^c$ and $d=3$ a smooth model of threefold $X_{3^c,3}$ admits a fibration in K3 surfaces. She also showed that for any dimension $d$, $X_{3^c,d}$ has Kodaira dimension at most 1.

In this talk we generalize these results describing some geometric properties of the quotient $X_{n,d}$, without any assumption on $n$ and $d$. In particular we describe fibrations on these varieties (with fibers genus 1 curves and with higher dimensional fibers) and we show that their Kodaira dimension is at most 1.

• 12-05-21, 15:00 Aline Zanardini (University of Pennsylvania): Stability of pencils of plane curves [Abstract]
In this talk I will discuss some recent results on the problem of classifying pencils of plane curves via geometric invariant theory. We will see how the stability of a pencil is related to the stability of its generators, to the log canonical threshold, and to the multiplicities of a base point. In particular, I will present some results on the stability of certain pencils of plane sextics called Halphen pencils of index two.
• 05-05-21, No seminar (bank holiday)
• 28-04-21, 15:00 Vladimir Mitankin (Hannover): Rational points on del Pezzo surfaces of degree 4 [Abstract]
In this talk I shall explain how often failures of local-to-global principles arise in a family of del Pezzo surfaces of degree four. This is addressed in terms of the Brauer group. More precisely, we give an explicit description of its generators modulo constants and incorporate in the Brauer-Manin obstruction the information obtained. This allows us to use tools from analytic number theory to get sharp upper and lower bounds for the number of surfaces in the family with a prescribed Brauer group as well as bounds for the number of Hasse and weak approximation failures. This talk is based on a joint work with Cecília Salgado.
• 21-04-21, 15:00 Christian Wuthrich (University of Nottingham): Integrality of twisted $L$-values of elliptic curves [Abstract]
The values at $s=1$ of the $L$-function of an elliptic curve $E/\mathbb{Q}$ twisted by a Dirichlet character $\chi$ enter the formulation of the generalised Birch and Swinnerton-Dyer conjecture. When normalised by a period, one obtains an algebraic number $\mathscr{L}(E,\chi)$. In joint work with Hanneke Wiersema, we determine under what conditions $\mathscr{L}(E,\chi)$ is an algebraic integer.
• 14-04-21, 15:00 Alp Bassa (Boğaziçi University): Rational points on curves over finite fields and their asymptotic behaviour [Abstract]
Curves over finite fields with many rational points have been of interest for both theoretical reasons and for applications. To obtain such curves with large genus various methods have been employed in the past. One such method is by means of explicit recursive equations and will be the emphasis of this talk. The recursive nature of these towers makes them very special and in fact all good examples have been shown to have a modular interpretation of some sort. In this talk I will try to give an overview of the landscape of explicit recursive towers and their modularity.
• 07-04-21, No seminar (NMC and Diamant symposium)
• 31-03-21, 15:00 Timo Keller (Universität Bayreuth): Exact verification of the strong BSD conjecture for some absolutely simple modular abelian surfaces [Abstract]
Let $X$ be a quotient of the modular curve $X_0(N)$ by a subgroup generated by Atkin-Lehner involutions such that its Jacobian $J$ is a $\mathbf{Q}$-simple modular abelian surface. We prove that for all but two such $J$, the Shafarevich-Tate group of $J$ is trivial and satisfies the strong Birch-Swinnerton-Dyer conjecture. (To prove this also for the remaining two abelian surfaces, we are currently performing descent.)

To achieve this, we compute the image and the cohomology of the mod-$\mathfrak{p}$ Galois representations of $J$, show effectively that almost all of them are irreducible and have maximal image, make Kolyvagin-Logachev effective, compute the Heegner points and Heegner indices, compute the $\mathfrak{p}$-adic $L$-function, and perform $\mathfrak{p}$-descents. Because many ingredients are involved in the proof, we will give an overview and focus on the computation of the mod-$\mathfrak{p}$ Galois representation in our talk.

• 24-03-21, 15:00 George Turcas (Babeș-Bolyai University): Irreducibility of mod p Galois representations of elliptic curves with multiplicative reduction over number fields [Abstract]
For every integer d ≥ 1, there exists an explicit constant B_d such that the following holds. Let K be a number field of degree d, let q > max{d − 1, 5} be any rational prime that is totally inert in K, and let E be any elliptic curve defined over K such that E has potentially multiplicative reduction at the prime above q. Then, for every rational prime p > B_d, E has an irreducible mod p Galois representation. In this talk, we will discuss the ingredients that go into the proof of the aforementioned result and we will present an application to Diophantine equations.

All the results presented are in joint work with Filip Najman.

• 17-03-21, 15:00 Nicholas Triantafillou (University of Georgia): Nonexistence of exceptional units via modified Skolem-Chabauty [Abstract]
An exceptional (S-)unit is a unit x in the ring in of (S-)integers of a number field K such that 1-x is also an (S-)unit. For fixed K and S, the set of exceptional S-units is finite by work of Siegel from the early 1900s. In the hundred years since, exceptional S-units have found wide-ranging applications, including to enumerating elliptic curves with good reduction outside a fixed set of primes and to proving "asymptotic" versions of Fermat's last theorem.

In this talk, we give an elementary p-adic proof of a new nonexistence result on exceptional units: there are no exceptional units in number fields of degree prime to 3 where 3 splits completely. We will also explain the geometric inspiration for the proof -- a version of Skolem-Chabauty's method for finding integral points on curves. Time permitting, we may discuss an application to periodic points of odd order in arithmetic dynamics.

• 10-03-21, 15:00 Laura Capuano (Politecnico di Torino): GCD results on semiabelian varieties and a conjecture of Silverman [Abstract]
A divisibility sequence is a sequence of integers d_n such that, if m divides n, then d_m divides d_n. Bugeaud, Corvaja, Zannier showed that pairs of divisibility sequences of the form a^n-1 have only limited common factors. From a geometric point of view, this divisibility sequence corresponds to a subgroup of the multiplicative group, and Silverman conjectured that a similar behaviour should appear in (a large class of) other algebraic groups.

Extending previous works of Silverman and of Ghioca-Hsia-Tucker on elliptic curves over function fields, we will show how to prove the analogue of Silverman’s conjecture over function fields in the case of abelian and split semiabelian varieties and some generalizations. The proof relies on some results of unlikely intersections. This is a joint work with F. Barroero and A. Turchet.

• 24-02-21, 15:00 Harry Justus Smit (MPIM Bonn): The 4-rank of class groups of biquadratic number fields [Abstract]
Class groups are among the most fundamental objects in number theory, yet they remain relatively inaccessible, with many problems concerning their behaviour still open. In 1801 Gauss formulated his genus theory, explicitly describing the elements of order 2 in quadratic number fields. Furthermore, the 4-rank and 8-rank of the class groups of certain families of quadratic number fields, indexed by the primes, are governed, that is, their 4-ranks and 8-ranks are determined completely by the splitting of their index in some (fixed) number field. About the p-parts for p an odd prime, very little is known explicitly.

Their average behaviour, though, is slightly better understood. Davenport and Heilbronn (1971) obtained the average size of the 3-torsion subgroups of class groups of quadratic number fields by counting binary cubic forms. In general, when considering all p-parts (p an odd prime) of the class groups of quadratic number fields, one should expect, according to the Cohen-Lenstra conjectures, that a finite abelian p-group occurs with probability proportional to the reciprocal of the size of its automorphism group. For the 2-part the conjectures have to be slightly modified to account for Gauss's genus theory. The prediction about the 4-rank has been proven by Fouvry and Klüners in 2000 and the full conjectures for the 2-part for imaginary quadratic fields have recently been proven by breakthrough work of Smith.

Collaborating with Koymans and Morgan, we extended the work of Fouvry and Klüners to obtain an average of the 4-rank of biquadratic fields. We show that in 100% of the cases, the 4-rank of the class group of a biquadratic number field can be explicitly related to the 2-rank of the class group of one of its quadratic subfields, along with some constants and some "noise". We then use the analytical techniques of Fouvry and Klüners to show that this noise is negligible in 100% of the cases.

• 17-02-21, 15:00 Filip Najman (Zagreb): Q-curves over odd-degree number fields [Abstract]

By reformulating and extending results of Elkies, we prove some results on Q-curves over number fields of odd degree. We show that, over such fields, the only prime isogeny degrees p which an elliptic curve without CM may have are those degrees which are already possible over Q, and show the existence of a bound on the degrees of cyclic isogenies between Q-curves depending only on the degree of the field. We also prove that the only possible torsion groups of Q-curves over number fields of degree prime degree q>7 are just the 15 groups that appear as torsion groups of elliptic curves over Q. This is joint work with John Cremona.

• 10-02-21, 15:00 Angeliki Mali (Groningen): Research studies on university mathematics education: The teaching practice of research mathematicians
• 10-02-21, 09:00 Levent Alpöge (Columbia University): Effectivity in Faltings' theorem II [Abstract]

I will show that in certain cases (i.e. after imposing conditions on K and C) there is an unconditional finite-time algorithm to compute (K,C)\mapsto C(K), using potential modularity theorems. Example: given K totally real of odd degree and a\in K^\times, one can effectively compute C_a(K) where C_a : x^6 + 4y^3 = a^2.

• 03-02-21, 17:00 Levent Alpöge (Columbia University): Effectivity in Faltings' theorem I [Abstract]

In joint work with Brian Lawrence, we show that, assuming standard motivic conjectures (Fontaine-Mazur, Hodge, Tate), there is a finite-time algorithm that, on input (K,C) with K a number field and C/K a smooth projective hyperbolic (i.e. genus > 1) curve, outputs C(K). The algorithm has the property that, if it terminates, the output is unconditionally correct --- one uses the conjectures to show that it always terminates in finite time.

• 27-01-21, 15:00 Damaris Schindler (Göttingen): On the distribution of Campana points on toric varieties [Abstract]

In this talk we discuss joint work with Marta Pieropan on the distribution of Campana points on toric varieties. We discuss how this problem leads us to studying a generalised version of the hyperbola method, which had first been developed by Blomer and Bruedern. We show how duality in linear programming is used to interpret the counting result in the context of a general conjecture of Pieropan-Smeets-Tanimoto-Varilly-Alvarado.

• 20-01-21, 13:00 Marc Masdeu (Universitat Autònoma de Barcelona): Quaternionic rigid meromorphic cocycles [Abstract]

Rigid meromorphic cocycles were introduced by Darmon and Vonk as a conjectural p-adic extension of the theory of singular moduli to real quadratic base fields. They are certain cohomology classes of SL2(Z[1/p]) which can be evaluated at real quadratic irrationalities and the values thus obtained are conjectured to lie in algebraic extensions of the base field.

I will present joint work with X.Guitart and X.Xarles, in which we generalize (and somewhat simplify) this construction to the setting where SL2(Z[1/p]) is replaced by an order in an indefinite quaternion algebra over a totally real number field F. These quaternionic cohomology classes can be evaluated at elements in almost totally complex extensions K of F, and we conjecture that the corresponding values lie in algebraic extensions of K. I will show some new numerical evidence for this conjecture, along with some interesting questions allowed by this flexibility.

• 06-01-21 Francesca Bianchi (Groningen): p-adic heights and p-adic sigma functions on Jacobians of genus 2 curves [Abstract]

Extending work of Mazur and Tate on elliptic curves, Blakestad recently constructed a p-adic analogue of the complex sigma function on Jacobians of genus 2 curves. We use Blakestad's function to define, compute and study p-adic heights on such Jacobians. P-adic heights are of much arithmetic interest. Just to cite one application, they figure prominently in the quadratic Chabauty method for the computation of integral and rational points on curves.

• 09-12-20 Valentijn Karemaker (Utrecht): Mass formulae for supersingular abelian threefolds [Abstract]

Using the theory of polarised flag type quotients, we determine mass formulae for all principally polarised supersingular abelian threefolds defined over an algebraically closed field k of characteristic p. We combine these results with computations of the automorphism groups to study Oort's conjecture; we prove that every generic principally polarised supersingular abelian threefold over k of characteristic >2 has automorphism group Z/2Z.

• 02-12-20 Netan Dogra (King's College): p-adic differential equations and rational points on semistable curves [Abstract]

If X is a curve over a p-adic field with semistable reduction, then its de Rham cohomology has the "hidden structure" of a filtered (phi,N)-module. This structure was explicitly described by Coleman and Iovita in terms of p-adic integration and combinatorics of the dual graph. In this talk I will explain a generalisation of this to the fundamental group of X, and applications to the algorithmic determination of the rational points on higher genus curves over number fields. This is joint work with Jan Vonk.

• 25-11-20 Ziyang Gao (IMJ-PRG): Bounding the number of rational points on curves [Abstract]

Mazur conjectured, after Faltings’s proof of the Mordell conjecture, that the number of rational points on a curve of genus g at least 2 defined over a number field of degree d is bounded in terms of g, d and the Mordell-Weil rank. In particular the height of the curve is not involved. In this talk I will explain how to prove this conjecture and some generalizations. I will focus on how functional transcendence and unlikely intersections are applied in the proof. If time permits, I will talk about how the dependence on d can be furthermore removed if we moreover assume the relative Bogomolov conjecture. This is joint work with Vesselin Dimitrov and Philipp Habegger.

• 11-11-20 Rosa Winter (Leiden / MPI Leipzig): Density of rational points on a family of del Pezzo surfaces of degree 1 [Abstract]

Del Pezzo surfaces are classified by their degree d, which is an integer between 1 and 9 (for d ≥ 3, these are the smooth surfaces of degree d in P^d). For del Pezzo surfaces of degree at least 2 over a field k, we know that the set of k-rational points is Zariski dense provided that the surface has one k- rational point to start with (that lies outside a specific subset of the surface for degree 2). However, for del Pezzo surfaces of degree 1 over a field k, even though we know that they always contain at least one k-rational point, we do not know if the set of k-rational points is Zariski dense in general. I will talk about a result that is joint work with Julie Desjardins, in which we give necessary and sufficient conditions for the set of k-rational points on a specific family of del Pezzo surfaces of degree 1 to be Zariski dense, where k is a number field. I will compare this to previous results.

• 04-11-20 Lazar Radicevic (Cambridge): Explicit realization of elements of the Tate-Shafarevich group constructed from Kolyvagin classes [Abstract]

We consider the Kolyvagin cohomology classes associated an elliptic curve over Q from a computational point of view. We explain how to go from a model of a class as an element of E(L)/pE(L), where p is prime and L is a dihedral extension of degree 2p, to a geometric model as a genus one curve embedded in P^{p-1}. We then give a method to compute Heegner points explicitly. As an application, we obtain explicit equations for genus one curves that represent non-trivial elements of Sha(E)[p], for p<=11, and hence are counterexamples to the Hasse priniple.

• 28-10-20 Fabien Pazuki (Copenhagen): Bertini and Northcott [Abstract]

I will report on joint work with Martin Widmer. Let X be a smooth projective variety over a number field K. We prove a Bertini-type theorem with explicit control of the genus, degree, height, and field of definition of the constructed curve on X. As a consequence we provide a general strategy to reduce certain height and rank estimates on abelian varieties over a number field K to the case of jacobian varieties defined over a suitable extension of K. We will give examples where the strategy works well!

• 21-10-20 Anna Somoza (Rennes): Reduction types of Ciani quartics
• 14-10-20 Berno Reitsma (Groningen): A complete algorithm that computes the rational torsion subgroup for Jacobians of hyperelliptic curves of genus 3 [Abstract]

In 1999, Stoll designed an algorithm to compute the rational torsion subgroup for Jacobians of hyperelliptic curves of genus 2. It uses p-adic lifting methods on the Kummer Variety and height bounds to find a method to determine whether an F_p-point lifts to J(Q) or not. This thesis generalizes the algorithm for any genus and gives an explicit proof. The (theoretical) algorithm itself always works, but in practice, you need to have algorithms for computing certain quantities. All the required procedures are explicitly described and implemented for Jacobians of hyperelliptic curves of genus 3. This results in a complete implementation for genus 3, based on explicit arithmetic on the Kummer variety of such Jacobians due to Stoll. We use this to compute the rational torsion structures for genus 3 hyperelliptic curves with low discriminant for the LMFDB.