In this talk we will apply 2-cover descent to compute the rational points on the family of hyperelliptic curves defined by y^2 = x(x^2-p^2)(x^2-4p^2), for a prime p > 3. The method itself is not new, and perfectly implemented in Magma, but doing the method for infinitely many curves as once complicates things and requires some modifications to make the method work. The method succesfully computes the rational points for all primes, except for 1/16'th of the primes of the form 1 mod 24, and for 1/4'th of the primes of the form -1 mod 24.

Given a hyperelliptic curve y^2 = f(x) defined over a number field, can one find simple conditions on f to determine whether its Jacobian is absolutely simple or not? Or, even better, obtain information on the structure of its (geometric) endomorphism ring?

Zarhin has shown that in many cases when the Galois group of f is "large" (insoluble, two-transitive, ...) the possibilities for the endomorphism ring are heavily restricted. In this talk, we will see that many restrictions persist when the Galois group of f is merely cyclic of large prime order. In fact, for certain base fields, we are able to give a finite explicit list.

The problem of finding rational points on curves is thousands of years old. Given a two variable polynomial f(x,y) = 0, how can we decide that we have a complete list of all of the rational solutions x and y?

After giving some background in the area of rational points and Diophantine problems, we will discuss recent advances in p-adic methods for attacking this problem, and compare the methods. In particular, we discuss Chabauty's method, quadratic Chabauty, and the geometric variants of these methods.

Given a generically étale morphism of compact quasi-smooth Berkovich analytic curves over a complete discretely valued field, one can define a logarithmic different function on the source measuring wild ramification. Such a logarithmic different function was introduced and studied earlier by Cohen-Temkin-Trushin, Temkin and Brezner-Temkin in the setting of an algebraically closed ground field. Similarly to the work of Temkin e.a., in the DVR setting the logarithmic different is a piecewise integral affine function that satisfies a balancing condition at divisorial points which can be viewed as a generalized Riemann-Hurwitz formula. For Galois degree p covers, the logarithmic different detects the topological ramification locus. In the setting of a finite seperable base change of an analytification of an algebraic curve, the logarithmic different explains some of the wild behaviour of base change of arithmetic surfaces. In particular, one can clarify some results and questions of Lorenzini on wildly ramified algebraic curves.

• 17.01.19: Jeroen Sijsling, Picard-Kurven: Gleichungen und Invarianten
• 24.01.19: Irene Bouw, Spezielle Picard-Kurven: Twists und Führerexponenten
• 31.01.19:  Duc Khoi Do, Rekonstruktion von Frobenius-halbeinfachen Weildarstellungen mithilfe ihrer lokalen Polynome
• 07.02.19: Paula Truöl,  Massey products and linking numbers
• 02.05.19: Irene Bouw, Der Parshin-Trick
• 09.05.19: Jeroen Hanselman, Semi-abelsche Varietäten und ihre Néron-Modelle
• 23.05.19: Duc Khoi Do, Galoisdarstellungen und Endlichkeitssätze
• 06.06.19: Sabrina Kunzweiler, Die Faltingshöhe, I
• 13.06.19: Stefan Wewers, Die Faltingshöhe, II
• 27.06.19:  Jeroen Sijsling, Die Tate-Vermutung
• 27.06.19: Angel Villanueva, Joint Distribution of Hecke and Casimir Eigenvalues for Automorphic Forms
• 04.07.19: Irene Bouw, p-dividierbare Gruppen und die Hodge-Tate-Zerlegung
• 27.06.19: Jeroen Sijsling, Beweis der Mordell-Vermutung
• 17.10.219: Dr. Sophie Schmieg (Google, USA), Cryptography at Google
• 24.10.19: Felix Göbler (Frankfurt), Das Zahlkörpersieb
• 31.10.19: Jeroen Sijsling, Kubische Erweiterung mit vorgeschriebener Verzweigung
• 14.11.19: Andreas Pieper, Die Leopoldt-Vermutung und die Geometrie der Zahlen
• 21.11.19: Andreas Pieper, Die Leopoldt-Vermutung und die Geometrie der Zahlen II: Gitterpunkte in Kreuzpolytopen und Kugeln
• 28.11.19: Sabrina Kunzweiler, Reduktionstypen von ebenen Quartiken
• 05.12.19: Stefan Wewers, Torische Flächensingularitäten
• 12.12.19: Bogdan Dina, Hyperelliptic curves with complex multiplication via Shimura reciprocity
• 23.01.20: Jeroen Hanselman, Algorithmen für das Verkleben von Jacobischen
• 24.01.20: Tim Evink (Universiteit Groningen), 2-Descent on hyperelliptic curves of genus 2
• 30.01.20: Duc Do, Galois-Darstellungen elliptischer Kurven mit potentiell guter Reduktion
• 06.02.20: Robert Slob (Universiteit Utrecht), Divisibility sequences of elliptic curves of characteristic 0
• 07.02.20: Mike Daas (Universiteit Amsterdam), The sympletic method
• 28.05.20: Sabrina Kunzweiler, Superelliptische Kurven und ganze Differentialformen
• 15.10.20: Jeroen Sijsling, Abstieg algebraischer Kurven
• 22.10.20: Jeroen Sijsling, Abstieg algebraischer Kurven returns
• 29.10.20: Stefan Wewers, The Tate curve: Illustrating the connection between reduction and non-archimedean uniformization

• 12.01.17: Stefan Wewers, Picard-Kurven mit kleinem Führer
• 26.01.17: Jeroen Sijsling, Die Rekonstruktion quartischer Kurven aus ihren Invarianten
• 02.02.17: Jeroen Hanselman, Non-hyperelliptic genus 3 covers of curves
• 09.02.17: Mohamed Barakat, Category theory is a programming language
• 23.02.17: Stefan Wewers, Rigid analytic spaces a la Berkovich and adic spaces: a short introduction
• 09.03.17: Irene Bouw, Dynamical Belyi maps
• 20.04.17 Tudor Micu, Etale morphisms
• 26.05.17 Pinar Kilicer (Oldenburg), On primes dividing the invariants of Picard curves. 
• 01.06.17 Angelos Koutsianas, The Chabauty-Coleman method and rational points on curves.
• 20.06.17 Roman Kohls, Spurformel und Gauss-Summen
• 29.06.17 Irene Bouw, Reduction of Picard curves, I
• 06.07.17  Jeroen Sijsling, Canonical models of arithmetic (1,∞)-curves
• 13.07.17 Stefan Wewers, Reduction of Picard curves, II
• 20.07.17 Jeroen Hanselman, Pairings and line bundles on abelian varieties
• 26.10.17: Stefan Wewers, "MCLF: ein Werkzeugkasten zur Berechnung von Modellen von Kurven über lokalen Körpern"
• 02.11.17: Christian Steck, Auflösen von zahmen zyklischen Quotientensingularitäten auf gefaserten Flächen
• 16.11.17: Jeroen Sijsling, Endomorphismen algebraischer Kurven
• 23.11.17: Stefan Wewers, Duality and canonical sheaf
• 30.11.17: Martin Djukanovic, Some remarks on split Jacobians
• 07.12.17: Martin Djukanovic, Split Jacobians--continued
• 25.01.18: Jeroen Hanselman, Gluing Curves along their 2-torsion
• 08.02.18: Sabrina Kunzweiler, Ogg's formula
• 15.02.18: Tudor Micu, Models, valuations, and Berkovich trees
• 22.02.18: Stefan Wewers, What is Intersection theory?
• 19.04.18: Jeroen Sijsling, Split Jacobians in genus 3: an inventory
• 03.05.18: Jeroen Sijsling, Endomorphisms of kind of special Picard curves and Richelot isogenies
• 24.05.18: Jeroen Sijsling, Endomorphisms and Divisors
• 07.06.18: Martin Djukanovic, Families of (3, 3)-split Jacobians
• 14.06.18: Jeroen Sijsling, Endomorphisms and Divisors II
• 21.06.18: Sabrina Kunzweiler, Discriminants of hyperelliptic curves
• 28.06.18: Jeroen Hanselman, Prym Varieties
• 12.07.18: Stefan Wewers, Resolution of wild arithmetic quotient singularities
• 25.10.18: Jeroen Sijsling, Eine Datenbank von Belyi-Morphismen
• 08.11.18: Andreas Pieper, Abelsche Varietäten mit komplexer Multiplikation
• 15.11.18: Sabrina Kunzweiler, Differentialformen auf hyperelliptischen Kurven mit semistabiler Reduktion
• 22.11.18: Roman Kohls, Eine obere Schranke für den Führerexponenten einer hyperelliptischen Kurve
• 29.11.18: Stefan Wewers, Rationale Singularitäten und die kanonische Garbe
• 06.12.18: Jeroen Hanselman, Ein Algorithmus zum Verkleben einer Kurve vom Geschlecht 1 und einer Kurve vom Geschlecht 2 entlang ihrer 2-Torsion
• 13.12.18: Matthew Bisatt, Root numbers of abelian varieties

• 12.05.15: Michel Börner, L-Reihen hyperelliptischer Kurven
• 26.05.15: Michael Eskin, Semistabile Reduktion von 3-Punkt-Überlagerungen
• 02.06.15: Stefan Wewers, Semistabile Reduktion einer gewissen Kurve vom Geschlecht 4 über Q_2
• 09.06.15: Irene Bouw, Konstruktion von Kurven mit schlechter Reduktion an vorgegebenen Stellen
• 16.06.15: Stefan Wewers, Was ist etale Kohomologie?
• 23.06.15: Christian Steck, Etale Morphismen
• 14.07.15: Roman Kohls, Lokale Konstanten und elliptische Kurven
• 03.08.15: Tudor Micu, A weak version of Beilinson's conjecture for Rankin-Selberg products of modular forms
• 22.10.15: Stefan Wewers, Reguläre und semistabile Modelle elliptischer Kurven: ein Miniprojekt
• 05.11.15 Jeroen Hanselman, Bounding the field extension necessary to find a semistable model of finite covers of curves
• 12.11.15: Stefan Wewers, Arithmetische Flächen
• 19.11.15: Christian Steck, Reguläre Modelle und Quotientensingularitäten
• 26.11.15: Christian Steck, ______, Teil 2
• 03.12.15: Tudor Micu, The Kodaira classification of  the special fiber of the minimal proper regular model of an elliptic curve
• 10.12.15: Roman Kohls, Classification of automorphism groups of elliptic curves
• 21.01.16: Christina Höhn, Faktorisieren mit elliptischen Kurven
• 28.01.16: Tudor Micu, MacLane valuations
• 04.02.16: Michel Börner, Picard curves and L-functions
• 11.02.16: Angelos Koutsianas, Computing elliptic curves with good reduction outside S
• 18.02.16: Christian Steck, Blow-ups
• 25.02.16: Roman Kohls, The sign of the functional equation of the L-function of an elliptic curve
• 27.10.16: Angelos Koutsianas, Lebesgue-Nagell equation and the modular Approach
• 10.11.16: Dimitris Xatzakos (Bristol), Hyperbolic lattice point counting in conjugacy classes 
• 17.11.16 November: Gunther Cornelissen (Utrecht), A combinatorial Li-Yau equality and rational points on curves
• 24.11.16 November: Tudor Micu: Sheaves, cohomology and local systems
• 01.12.16: Roman Kohls, Semistable reduction of curves and Weil-Deligne representations, I
• 08.12.16 Dezember: Roman Kohls, Semistable reduction and Weil-Deligne representations, II
• 12.12.16 Dezember: Davide Lombardo (Orsay), On division fields of CM abelian varieties