Number theory and arithmetic geometry invited talks at UAB-UB-UPC (2021--Present)


Speaker: Jordan Ellenberg (Wisconsin University).
Time and place: 10/7/2023, Aula T1 of Universitat de Barcelona.
Title: Cohomology of configuration spaces with coefficients and Malle’s conjecture over function fields.
Abstract: It turns out that a wide variety of questions in arithmetic statistics, when transposed from the more customary setting of the rational numbers to the setting of a global function field like F_q(t), turn out to be expressible in surprisingly topological terms, in particular involving the cohomology of configuration spaces with coefficients in various local systems. I will give an overview of known results and work in progress that fits into this program; I will try especially to say something about a recent preprint with TriThang Tran and Craig Westerland, or more accurately a recent revision to an old preprint. The content of this work, which is in spirit a sequel to the work of the first and third author with Venkatesh, is to prove the upper bound in the weak Malle’s conjecture for function fields, which holds that for each finite group G, the number of G-extensions of F_q(t) with discriminant at most X grows as X^a (log X)^b, for specified constants a and b depending on G and q.

Speaker: Pankaj Vishe (Durham University).
Time and place: 6/7/2023, Aula T1 of Universitat de Barcelona.
Title: A two dimensional delta method and applications to quadratic forms.
Abstract: We develop a two dimensional version of the delta symbol method and apply it to establish quantitative Hasse principle for a smooth pair of quadrics defined over Q defined over at least 10 variables. This is a joint work with Simon Myerson (warwick) and Junxian Li (Bonn).

Speaker: Alex Smith (Stanford University).
Time and place: 24/5/2023, Aula B7 of Universitat de Barcelona.
Title: Quadratic twist families of elliptic curves with unusual $2^\infty$-Selmer groups.
Abstract: Given any elliptic curve E over the rationals, we show that 50% of the quadratic twists of E have $2^\infty$-Selmer corank 0 and 50% have $2^\infty$-Selmer corank 1. As a result, we show that Goldfeld's conjecture follows from the Birch and Swinnerton-Dyer conjecture.

Speaker: Giuseppe Ancona (Université de Strasbourg).
Time and place: 29/11/2022, Aula B7 of Universitat de Barcelona.
Title: Quadratic forms arising from geometry.
Abstract: The cup product on topological manifolds or the intersection product on algebraic varieties induce quadratic forms which turn out to be a fine invariant of these geometric objects. We will discuss some old theorems on the signature of these quadratic forms and some applications both of geometric and arithmetic origins. Finally we will study an old conjecture of Grothendieck about those signatures and explain some new evidences.

Speaker: Sachi Hashimoto (Max Planck Institute for Mathematics in the Sciences).
Time and place: 12:30 of 11/11/2022, Aula B1 of Universitat de Barcelona.
Title: p-Adic Gross--Zagier and rational points on modular curves.
Abstract: Faltings' theorem states that there are finitely many rational points on a nice projective curve defined over the rationals of genus at least 2. The quadratic Chabauty method makes explicit some cases of Faltings' theorem. Quadratic Chabauty has recent notable success in determining the rational points of some modular curves. In this talk, I will explain how we can leverage information from p-adic Gross--Zagier formulas to give a new quadratic Chabauty method for certain modular curves. Gross--Zagier formulas relate analytic quantities (special values of p-adic L-functions) to invariants of algebraic cycles (the p-adic height and logarithm of Heegner points). By using p-adic Gross--Zagier formulas, this new quadratic Chabauty method makes essential use of modular forms to determine rational points. .

Speaker: Lennart Gehrmann (Universitat de Barcelona).
Time and place: 14:45 of 5/10/2022, 19/10/2022 and 2/11/2022, Aula B1 of Universitat de Barcelona.
Title: Rigid meromorphic cocycles I, II, III (minicourse).
Abstract: In the early 2000s, Darmon initiated a fruitful study of analogies between Hilbert modular surfaces and quotients Y := SL_2(Z[1/p]) \ H x H_p, where H is the complex upper half plane and H_p is Drinfeld's p-adic upper half plane. As Y mixes complex and p-adic topologies, making direct sense of Y as an analytic space seems difficult. Nonetheless, Darmon-Vonk have described an incarnation of meromorphic functions on Y, so called rigid meromorphic cocycles. In this mini course I will describe joint work with Henri Darmon and Michael Lipnowski, in which we study generalizations Y' of the space Y to orthogonal groups G for quadratic spaces over the rationals of arbitrary signature. We define explicit rigid meromorphic cocycles on for Y'; these RMCs are analogous to meromorphic functions on orthogonal Shimura varieties with prescribed special divisors first studied by Borcherds, and they generalize the RMCs constructed by Darmon-Vonk.

Speaker: Samuele Anni (Aix-Marseille Université).
Time and place: 16:45 of 9/9/2022, Aula T1 of Universitat de Barcelona.
Title: Counting modular forms mod p satisfying constraints at p
Abstract: The structure of the algebra of modular forms over finite fields has been widely studied, in part for its applications in establishing congruences. In this talk, after recalling classical geometric arguments of Ogg and Kenku, I will show how, for N prime with p, one can count the number of classical modular forms of level Np and weight k with both a residual Galois representation and an Atkin-Lehner sign at fixed p, generalizing Martin's recent results, and dimension formulas given by Jochnowitz and by Bergdall-Pollack. Most of these results can be stated as equivariant isomorphisms for the Hecke operators between certain modules, thanks to a p-adic refinement of the Brauer-Nesbitt theorem. A theoretical framework for proving such isomorphisms is given, using the Eichler-Selberg trace formula. This method applies in the case where the level is divisible by the residual characteristic, contrary to the pre-existing approaches. This is work in progress with Alexandru Ghitza (University of Melbourne) and Anna Medvedovsky (Boston University).

Speaker: Kiran Kedlaya (UCSD).
Time and place: 15:10 of 9/9/2022, Aula T1 of Universitat de Barcelona.
Title: The relative class number one problem for function fields.
Abstract: Building on my lecture from ANTS-XV, we classify extensions of function fields (of curves over finite fields) with relative class number 1. Many of the ingredients come from the study of the maximum number of points on a curve over a finite field, such as the function field analogue of Weil's explicit formulas (a/k/a the "linear programming method"). Additional tools include the classification of abelian varieties of order 1 and the geometry of moduli spaces of curves of genus up to 7.

Speaker: Andreas Mihatsch (Universität Bonn).
Time and place: 15:10 of 17/6/2022, Aula B6 of Universitat de Barcelona.
Title: Linear Arithmetic Fundamental Lemmas.
Abstract: Arithmetic Fundamental Lemmas (AFLs) are certain identities in arithmetic geometry that relate intersection numbers on moduli spaces of $p$-divisible groups with derivatives of orbital integrals. They form the local building blocks of global conjectures that relate cycles on Shimura varieties and derivatives of L-functions. In this talk, I will motivate and explain some new AFL identities for $GL_n$, which is joint work with Qirui Li.

Speaker: Heidi Goodson (Brooklyn College, CUNY).
Time and place: 17:00 of 27/4/2022, Aula B7 of Universitat de Barcelona.
Title: Sato-Tate Groups in Dimension Greater than 3.
Abstract: The focus of this talk is on Sato-Tate groups of abelian varieties -- compact groups predicted to determine the limiting distributions of local zeta functions. In recent years, complete classifications of Sato-Tate groups in dimensions 1, 2, and 3 have been given, but there are obstacles to providing classifications in higher dimensions. In this talk, I will describe my recent work on families of higher dimensional Jacobian varieties. This work is partly joint with Melissa Emory.

Speaker: Michele Fornea (Columbia University).
Time and place: 15:00 of 20/4/2022, Aula C3B/158 of Universitat Autònoma de Barcelona.
Title: On the algebraicity of polyquadratic plectic points.
Abstract: Gehrmann, Guitart, Masdeu and myself recently proposed and numerically computed plectic generalizations of Stark-Heegner points. The construction is p-adic, cohomological, and unfortunately lacking a satisfying geometric interpretation. Nevertheless, we formulated precise conjectures on the algebraicity of plectic points and their significance for the arithmetic of elliptic curves of large rank. In this talk I will report on joint work with Lennart Gehrmann where we establish direct evidence for our conjectures in special cases..

Speaker: Andrea Conti (Université de Luxembourg).
Time and place: 14:30 of 21/10/2021, Aula B3 of Universitat de Barcelona.
Title:Lifting trianguline Galois representations along isogenies.
Abstract: Starting with a $p$-adic family of automorphic forms for a reductive group, one can construct $p$-adic representations of global Galois groups that are “trianguline” locally at $p$. This property is roughly expected to characterize the representations constructed this way. We study how triangulinity behaves under functoriality: in particular, if $G \to H$ is an isogeny of reductive groups and $\rho$ is a global Galois representation into $H(\mathbb Q_p)$ that is trianguline at $p$, we show under mild conditions that any lift of $\rho$ to $G(\mathbb Q_p)$ is also trianguline at $p$. We expect this result to help in the study of congruences between $p$-adic families of automorphic forms for different groups.