## Upcoming seminars

Abstract: We attempt -- and almost entirely succeed -- to classify the automorphic representations \pi of reductive groups over number fields, for which the algebraicity of the Hecke eigenvalues (or Satake parameters) is reducible via Langlands functoriality to cases already known for algebro-geometric reasons (if \pi is singular, the only current option being via the coherent cohomology of Shimura varieties). Historically, an important case -- for which algebraicity is still wide open -- is that of Maass forms of Galois-type: non-holomorphic eigenfunctions of the hyperbolic Laplacian on the upper half-plane with eigenvalue 1/4. Also, using functoriality to reduce to the cohomology of Shimura varieties has been applied successfully by many people to the construction of automorphic Galois representations, in cases where the algebraicity of the Hecke eigenvalues was already known (usually via the Betti cohomology of a locally symmetric space). However, as far as we know, no new case of algebraicity of Hecke eigenvalues had previously been established by reducing via functoriality to Shimura varieties (where we transfer \pi

In the positive direction, we give several examples of non-holomorphic automorphic forms which are superficially similar to Maass forms, but whose algebraicity does reduce to the coherent cohomology of Shimura varieties via either known or open cases of functoriality; in the known cases of functoriality we are also able to attach Galois

representations. We also introduce new notions of "D" and "M-algebraic", which generalize/refine the "L" and "C" of Buzzard-Gee and the "W-algebraic" of Patrikis, and we give examples of cuspidal, M-algebraic \pi which are "farther" from L-algebraic than previously considered yet still have algebraic Hecke eigenvalues, answering a question of

Patrikis. In the negative direction, we give a conceptual, group-theoretic explanation for why Maass forms and many other forms are not reducible to known cases via functoriality (so the sign error found by Taylor in a previous attempt on Maass forms was not coincidental, but rather necessary). There remains only a sliver of cases where it is still perhaps unclear whether or not reduction via functoriality is possible.

## Past seminars

Abstract: Landau-Ginzburg models are a family of quantum field theories characterized by a polynomial (satisfying some conditions) usually called ‘potential’. Often appearing in mirror-symmetric phenomena, they can be collected in categories with nice properties that allow direct computations. In this context, it is possible to introduce an equivalence relation between two different potentials called `orbifold equivalence’. We will present some recent examples of this equivalence, and discuss the computational challenges posed by the search of new ones. Joint work with Timo Kluck.

Abstract: In Mazur's work proving the torsion theorem for rational elliptic curves, he studied congruences between cusp forms and Eisenstein series in weight two and prime level. One of his innovations was to measure such congruences using a residually Eisenstein Hecke algebra. He asked for generalizations of his theory to squarefree levels. The speaker made progress toward such generalizations in joint work with Preston Wake; however, a crucial condition in their work was that the Hecke algebra be Gorenstein, which is often but by no means always true. We present joint work with Catherine Hsu and Preston Wake in which we study the smallest possible non-Gorenstein cases and leverage this smallness to draw an explicit link between its size and an invariant from algebraic number theory.

Abstract: We show that the one-level density for L-functions associated with the cubic residue symbols χn, with n ∈ Z[ω] square-free, satisfies the Katz-Sarnak conjecture for all test functions whose Fourier transforms are supported in (−13/11, 13/11), under GRH. This is the first result extending the support outside the trivial range (−1, 1) for a family of cubic L-functions. This implies that a positive proportion of the L-functions associated with these characters do not vanish at the central point s = 1/2. A key ingredient is a bound on an average of generalized cubic Gauss sums at prime arguments, whose proof is based on the work of Heath-Brown and Patterson. Joint work with Ahmet M. Güloglu.

Abstract: The Bloch--Kato conjecture is a very general conjecture relating the properties of arithmetic objects over number fields to special values of L-functions, generalising a wealth of earlier theorems and conjectures such as the Birch--Swinnerton-Dyer conjecture for elliptic curves. I will explain a little about how this conjecture arises, and how it can be attacked using tools known as 'Euler systems'. In recent years there has been dramatic progress in the theory of Euler systems, and I'll explain recent work of Sarah Zerbes and myself from 2020, in which we use Euler systems to prove new cases of the Bloch--Kato conjecture for L-functions arising from the symplectic group GSp(4).

Abstract: We study a sequence of normalized counting functions on the space of 2d-dimensional symplectic lattices where d is at least 3. Using a combinatorial device introduced by Björklund and Gorodnik in order to estimate cumulants (alternating sums of moments), we prove that our sequence satisfies a central limit theorem. This is work in progress.

Abstract: Many problems, such as resolution of singularities, are solved by blowing up smooth varieties along smooth centers. This is a very useful and explicit construction but sometimes a little restrictive. If we replace (smooth) varieties with (smooth) stacks/orbifolds, we obtain several new similar operations (root stacks, stacky blowups, weighted stacky blowups) that we can add to our toolbox. I will describe these new operations and some of their applications such as easier and more efficient algorithms for resolution of singularities (Abramovich--Temkin--Wlodarczyk) and weak factorization, as well as other applications such as \'etalification and destackification. There are also close relations to toric and logarithmic geometry.

This is partly joint work with Daniel Bergh and Ming Hao Quek.

Abstract: In recent works with H. P. A. Gustafsson, A. Kleinschmidt, D. Persson, and S. Sahi, we found a way to express any automorphic form through its Fourier coefficients, using adelic integrals, period integrals and discrete summation – generalizing the Piatetski-Shapiro – Shalika decomposition for GL(n). I will explain the general idea behind our formulas, and illustrate it on examples. I will also show applications to vanishing and Eulerianity of Fourier coefficients.

Abstract: It is well known that the class group of a number field is of size bounded above by roughly the square root of its discriminant. But one expects by conjectures of Cohen-Lenstra that the

*n*-torsion part of this group should be much smaller and there have recently been several papers on this by prominent mathematicians. We present in our talk some new bounds which are sharper than those in the literature.

Abstract: We present an improvement to Roth's theorem on arithmetic progressions, implying the first non-trivial case of a conjecture of Erdős: if a subset A of {1,2,3,...} is not too sparse, in that the sum of its reciprocals diverges, then A must contain infinitely many three-term arithmetic progressions. Although a problem in number theory and combinatorics on the surface, it turns out to have fascinating links with geometry, harmonic analysis and probability, and we shall aim to give something of a flavour of this.

Abstract: Modularity conjectures lie at the heart of Kudla's program, which is a historic development of the earlier work of Hirzebruch--Zagier, Gross--Zagier, Gross--Keating, and Kudla--Millson.

The modularity conjecture for orthogonal Shimura varieties over Q was fully resolved by Bruinier--Raum in 2015, thanks to earlier work of Richard Borcherds and Wei Zhang. In this talk, I will explain how to use the method of Bruinier--Raum to show the modularity conjecture for certain unitary Shimura varieties based on a result of Yifeng Liu.

If time allows, I'll also talk about recent development of certain arithmetic applications, including the work of Li--Liu on arithmetic inner product formula, which is a natural analogue of the Gross--Zagier formula in higher dimension.

Abstract: Questions about the distribution of primes in an arithmetic progression are closely linked to the Generalized Riemann Hypothesis (GRH), which unfortunately appears out of reach. A very useful unconditional substitute for the GRH is the Bombieri-Vinogradov Theorem, which shows that the GRH is true 'on average'.

I'll talk about some recent results on primes in arithmetic progressions which goes beyond the Bombieri-Vinogradov Theorem, and corresponds to proving something stronger than the Riemann Hypothesis holds 'on average'.

Abstract: I will survey some old and new ideas on "geometrizing" the Langlands correspondence, by viewing it as a sheaf of representations over a moduli space of Galois representations (or other "Langlands parameters"). Towards the end, I hope to briefly discuss some work in progress on p-adic endoscopy for SL(2) over totally real fields utilizing this perspective, which is joint with Judith Ludwig.

Abstract: We establish the analytic Hasse principle for Diophantine systems consisting of one diagonal form of degree k and one general form of degree d, where d is smaller than k. By employing a hybrid method that combines ideas from the study of general forms with techniques adapted to the diagonal case, we are able to obtain bounds that grow exponentially in d but only quadratically in k, reflecting the growth rates typically obtained for both problems separately. Time permitting, we may also discuss some of the most interesting generalisations of our approach. This is joint work with Scott T. Parsell.

18.11.2020 - Tim Browning (IST Austria): Poor bounds for a rich problem

Abstract: Thanks to Néron heights and the Mordell-Weil theorem we can count rational points of bounded height on elliptic curves. Faltings's theorem tells us that there are only finitely many rational points (of any height) on hyperelliptic curves of genus at least 2. Unfortunately, these estimates depend intimately on the individual curve and it is hard to apply them to study the corresponding problem for surfaces that are fibred into elliptic or hyperelliptic curves. In this talk I shall discuss an alternative approach that uses sieves, though the bounds are far from optimal! This is joint work with Dante Bonolis.

04.11.2020 - John Christian Ottem (University of Oslo): Enriques surface fibrations of even index

Abstract: I will explain a geometric construction of an Enriques surface fibration over P1 of even index. This answers a question of Colliot-Thèlene and Voisin, and provides new counterexamples to the integral Hodge conjecture. This is joint work with Fumiaki Suzuki.

28.10.2020 - Scott Ahlgren (University of Illinois, Urbana-Champaign): OBS! 14.30-15.30

Title: Congruences for the partition function

21.10.2020 - Peter Sarnak (Princeton University/Institute for Advanced Study): Applications of points on subvarieties of tori

Abstract: The intersection of the division group of a finitely generated subgroup of a torus with an algebraic subvariety has been understood for some time (Lang, Laurent,...). After a brief review of some of the tools in the analysis and their recent extensions (André-Oort conjectures), we give some old and new applications; in particular to the additive structure of the spectra of metric graphs and crystalline measures.

Joint work with P. Kurasov.

07.10.2020 - Jack Shotton (University of Durham): Moduli of local Galois representations and representation theory

24.09.2020 - Jan Gerken (Chalmers/GU): Single-valued maps at genus zero and one

Abstract: The single-valued map for for multiple zeta values (MZVs) due to Francis Brown and Oliver Schnetz is an intriguing algebra homomorphism for MZVs. Surprisingly, it appears in the leading contribution to scattering amplitudes in string theory, relating open- and closed-string amplitudes. Since MZVs are periods of configuration spaces of punctured genus-zero Riemann surfaces, a natural next step is the generalization of the single-valued map to genus-one surfaces. In string theory, these correspond to the subleading contributions to the scattering amplitudes. Using the structures provided by string theory, we propose a genus-one generalization of the single-valued map for MZVs which acts on the level of generating functions of genus-one periods.

Abstract: Tensor hierarchy algebras constitute a new class of non-contragredient Lie superalgebras, whose finite-dimensional members are the simple Lie superalgebras of Cartan type in Kac’s classification. They have proven useful in describing gauge structures in physical models related to string theory. I will review their construction by generators and relations and some of the remarkable features they exhibit.

*L*-functions

Abstract: In an unpublished paper from 2007, Conrey discovered certain ‘reciprocity relations’ satisfied by twisted moments of Dirichlet L-functions, linking the arithmetics of the finite fields F_p, F_q for two different primes p,q (as is the case with quadratic reciprocity). In this talk I will discuss a generalization to twisted moments of twisted modular L-functions. This will lead to a discussion of the notion of quantum modular forms due to Zagier, and in particular we will explain that additive twists of modular L-functions define examples of quantum modular forms.

Abstract: Hurwitz class numbers, class numbers of imaginary quadratic fields, and partition counts are among the most classic quantities in number theory, and for each of them their factorizations, i.e. divisibilities, are celebrated open questions. In the case of class numbers the Cohen-Lehnstra Heuristics provides predictions of of statistical nature. In the case of partition counts, Ramanujan congruences opened the door to a whole new research area in~1920.

We survey recent progress on divisibilities of class numbers and partition counts on arithmetic progressions. These result rely on a two new methods exploiting the finer structure of Fourier coefficients of real-analytic and meromorphic modular forms.

The project on class numbers is partially based on joint work with Olivia Beckwith and Olav Richter. The project on partition counts is partially based on joint work with Olivia Beckwith and Scott Ahlgren.

Abstract: There are many difficult conjectures about automorphic representations, many of which seem to be out of reach at the moment. It has therefore become increasingly popular to study instead families of automorphic representations and their statistical properties, which allows for additional analytic techniques to be used.

In my talk I want to discuss the distribution of Hecke eigenvalues or, in other words, Satake parameters in the family of spherical unramified automorphic representations of split classical groups. We obtain an effective distribution of the Satake parameters, when we order the family according to the size of analytic conductor. This has applications to various questions in number theory, for example, low-lying zeros in families of automorphic L-functions, but also yields an effective Weyl law for the underlying locally symmetric space. This is joint work with T. Finis.

Abstract: I will report on ongoing work, where I apply the Lefschetz fixed point theorem to local systems on the moduli space of abelian varieties of dimension at most 3, and use simple equalities in modular arithmetic, to study traces of Hecke operators on spaces of Siegel modular forms (of degree at most 3) modulo prime powers.

Abstract: Mirror symmetry, in a crude formulation, is usually presented as a correspondence between curve counting on a Calabi--Yau variety X, and some invariants extracted from a mirror family of Calabi--Yau varieties. After the physicists Bershadsky--Cecotti--Ooguri--Vafa (henceforth BCOV), this is organised according to the genus of the curves in X we wish to enumerate, and gives rise to an infinite recurrence of differential equations. In this talk, I will give a general introduction to these problems, and present a rigorous mathematical formulation of the BCOV conjecture at genus one, in terms of a lifting of the Grothendieck--Riemann--Roch. I will explain a proof of the conjecture for Calabi--Yau hypersurfaces in projective space, based on the Riemann--Roch theorem in Arakelov geometry. Our results generalise from dimension 3 to arbitrary dimensions previous work of Fang--Lu--Yoshikawa.