BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Ashay Burungale DTSTART:20231017T190000Z DTEND:20231017T200000Z DTSTAMP:20260314T081905Z UID:BC-MIT/3 DESCRIPTION:Title: Z eta elements for elliptic curves and some applications\nby Ashay Burun gale as part of BC-MIT number theory seminar\n\nLecture held in Maloney 56 0 at Boston College.\n\nAbstract\nThe talk plans to outline the existence of two-variable zeta element over an imaginary quadratic field for an elli ptic curve defined over Q. Its arithmetic consequences include proof of Ko bayashi's main conjecture for semistable curves and special cases of the B irch--Swinnerton-Dyer conjecture. (Joint with C. Skinner\, Y. Tian and X. Wan.)\n LOCATION:https://researchseminars.org/talk/BC-MIT/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Charlotte Chan (University of Michigan) DTSTART:20231017T203000Z DTEND:20231017T213000Z DTSTAMP:20260314T081905Z UID:BC-MIT/4 DESCRIPTION:Title: S upercuspidal representations and very regular elements\nby Charlotte C han (University of Michigan) as part of BC-MIT number theory seminar\n\nLe cture held in Maloney 560 at Boston College.\n\nAbstract\nIn the 1990s\, H enniart proved that certain supercuspidal\nrepresentations of p-adic GLn a re characterized by their character\nvalues on very regular elements\, a s pecial class of regular semisimple\nelements on which character formulae a re remarkably simple. Henniart's\nresult has seen many interesting applica tions---for example\, in\ndetermining algebraic descriptions of geometrica lly arising\nrepresentations. In this talk\, we'll discuss a generalizatio n of\nHenniart's theorem to general G. As a byproduct of our methods\, we\ nobtain an easy\, non-cohomological condition distinguishing unipotent\nsu percuspidal representations\, yielding a p-adic analogue of Lusztig's\ncri terion for finite fields. This is joint work with M. Oi.\n LOCATION:https://researchseminars.org/talk/BC-MIT/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Sarah Peluse (University of Michigan) DTSTART:20231114T200000Z DTEND:20231114T210000Z DTSTAMP:20260314T081905Z UID:BC-MIT/5 DESCRIPTION:Title: D ivisibility of character values of the symmetric group\nby Sarah Pelus e (University of Michigan) as part of BC-MIT number theory seminar\n\nLect ure held in MIT room 4-163.\n\nAbstract\nIn 2017\, Miller computed the cha racter tables of $S_n$ for all $n$ up to $38$ and looked at various statis tical properties of the entries. Characters of symmetric groups take only integer values\, and\, based on his computations\, Miller conjectured that almost all entries of the character table of $S_n$ are divisible by any f ixed prime power as $n$ tends to infinity. In this talk\, I will discuss j oint work with K. Soundararajan that resolves this conjecture\, and mentio n some related open problems.\n LOCATION:https://researchseminars.org/talk/BC-MIT/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Samit Dasgupta (Duke University) DTSTART:20231114T213000Z DTEND:20231114T223000Z DTSTAMP:20260314T081905Z UID:BC-MIT/6 DESCRIPTION:Title: O n the Brumer-Stark conjecture and refinements\nby Samit Dasgupta (Duke University) as part of BC-MIT number theory seminar\n\nLecture held in MI T room 2-449.\n\nAbstract\nIn this talk I will describe my recent work wit h Mahesh Kakde on the Brumer-Stark Conjecture and certain refinements. I w ill give a broad overview that motivates the conjecture and gives connecti ons to explicit class field theory. I will conclude with a description of recent work (joint w/ Kakde\, Jesse Silliman\, and Jiuya Wang) in which we complete the proof of the conjecture. Moreover\, we deduce a certain spec ial case of the Equivariant Tamagawa Number Conjecture\, which has importa nt corollaries. The key aspect of the most recent results\, which allows u s to handle the prime $p=2$\, is the proof of a version of Ribet's Lemma i n the case of characters that are congruent modulo $p$.\n LOCATION:https://researchseminars.org/talk/BC-MIT/6/ END:VEVENT BEGIN:VEVENT SUMMARY:Alexander D. Smith (UCLA) DTSTART:20240409T190000Z DTEND:20240409T200000Z DTSTAMP:20260314T081905Z UID:BC-MIT/7 DESCRIPTION:Title: S imple abelian varieties over finite fields with extreme point counts\n by Alexander D. Smith (UCLA) as part of BC-MIT number theory seminar\n\nLe cture held in Maloney 560 at Boston College.\n\nAbstract\nGiven a compactl y supported probability measure on the reals\, we will give a necessary an d sufficient condition for there to be a sequence of totally real algebrai c integers whose distribution of conjugates approaches the measure. We use this result to prove that there are infinitely many totally positive alge braic integers X satisfying tr(X)/deg(X) < 1.899\; previously\, there were only known to be infinitely many such integers satisfying tr(X)/deg(X) < 2. We also will explain how our method can be used in the search for simpl e abelian varieties with extreme point counts.\n LOCATION:https://researchseminars.org/talk/BC-MIT/7/ END:VEVENT BEGIN:VEVENT SUMMARY:Michael Bennett (University of British Columbia) DTSTART:20240409T203000Z DTEND:20240409T213000Z DTSTAMP:20260314T081905Z UID:BC-MIT/8 DESCRIPTION:Title: A rithmetic progressions in sumsets of geometric progressions\nby Michae l Bennett (University of British Columbia) as part of BC-MIT number theory seminar\n\nLecture held in Maloney 560 at Boston College.\n\nAbstract\nIf A and B are two geometric progressions\, we characterize all 3-term arith metic progressions in the sumset A+B. Somewhat surprisingly\, while mostly elementary\, this appears to require quite deep machinery from Diophantin e Approximation.\n LOCATION:https://researchseminars.org/talk/BC-MIT/8/ END:VEVENT BEGIN:VEVENT SUMMARY:Yiannis Sakellaridis (Johns Hopkins University) DTSTART:20240319T190000Z DTEND:20240319T200000Z DTSTAMP:20260314T081905Z UID:BC-MIT/9 DESCRIPTION:Title: R elative Langlands duality\, past and future\nby Yiannis Sakellaridis ( Johns Hopkins University) as part of BC-MIT number theory seminar\n\nLectu re held in MIT room 2-449.\n\nAbstract\nSince Riemann's 1859 report on the zeta function\, it is known that certain automorphic $L$-functions can be represented as ("period") integrals\, which often proves analytic propert ies such as the functional equation. The method was advanced by Jacquet\, Piatetski-Shapiro\, Rallis\, and many others since the 1970s\, giving rise to the "relative" Langlands program. It turns out that the relationship b etween periods and $L$-functions reflects a duality between certain Hamilt onian varieties for a reductive group and its Langlands dual group. I will set up this duality in a limited setting (joint work with David Ben-Zvi a nd Akshay Venkatesh)\, and speculate on how it might be expanded in the fu ture.\n LOCATION:https://researchseminars.org/talk/BC-MIT/9/ END:VEVENT BEGIN:VEVENT SUMMARY:David Ben-Zvi (University of Texas at Austin) DTSTART:20240319T203000Z DTEND:20240319T213000Z DTSTAMP:20260314T081905Z UID:BC-MIT/10 DESCRIPTION:Title: Geometric Arthur parameters\nby David Ben-Zvi (University of Texas at Austin) as part of BC-MIT number theory seminar\n\nLecture held in MIT roo m 2-449.\n\nAbstract\nArthur proposed a description of automorphic forms i n terms of tempered automorphic forms for centralizers of SL2 homomorphism s. I will explain a point of view on the Arthur parameterization in the se tting of function fields coming from relative Langlands duality\, emphasiz ing the role of $shearing$ (the symmetry of the derived category of graded vector spaces which simultaneously shifts weights and cohomological degre es). Shearing helps account for a deficit of Tannakian formalism in the de rived context - there are more eigenvalues for tensor actions than one mig ht expect. The talk reports on joint work with Yiannis Sakellaridis and Ak shay Venkatesh.\n LOCATION:https://researchseminars.org/talk/BC-MIT/10/ END:VEVENT BEGIN:VEVENT SUMMARY:Yunqing Tang (Berkeley) DTSTART:20240514T190000Z DTEND:20240514T200000Z DTSTAMP:20260314T081905Z UID:BC-MIT/11 DESCRIPTION:Title: The arithmetic of power series and applications to irrationality\nby Y unqing Tang (Berkeley) as part of BC-MIT number theory seminar\n\nLecture held in Maloney 560 at Boston College.\n\nAbstract\nIn this talk\, we will discuss various irrationality and linear independence problems including certain products of two (classical or p-adic) log values. The proofs use a n arithmetic holonomicity theorem\, the special case of which was used in the proof of the unbounded denominators conjecture\; our arithmetic holono micity theorem is inspired from Andre’s work on Grothendieck-Katz p-curv ature conjecture on arithmetic differential equations. A geometric version of our arithmetic holonomicity theorems have also been studied in recent work of Bost and Charles.\n\nThis is joint work in progress with Frank Cal egari and Vesselin Dimitrov.\n LOCATION:https://researchseminars.org/talk/BC-MIT/11/ END:VEVENT BEGIN:VEVENT SUMMARY:Ari Shnidman (Hebrew University of Jerusalem) DTSTART:20240514T203000Z DTEND:20240514T213000Z DTSTAMP:20260314T081905Z UID:BC-MIT/12 DESCRIPTION:Title: Vanishing criteria for Ceresa cycles\nby Ari Shnidman (Hebrew Universi ty of Jerusalem) as part of BC-MIT number theory seminar\n\nLecture held i n Maloney 560 at Boston College.\n\nAbstract\nThe Ceresa cycle of a curve is perhaps the simplest example of a\nhomologically trivial algebraic cycl e which need not be algebraically\ntrivial. Its vanishing in the Chow (res p. Griffiths) group has various\nimplications\, but the locus of vanishing Ceresa cycles in $M_g$ is quite\nmysterious\, beyond the fact that it con tains the hyperelliptic locus. I'll\npresent new vanishing criteria for th e Ceresa cycle of curves with\nautomorphisms\, one of them conditional on the Hodge conjecture. In certain\nlow genus cases the relevant Hodge conje cture is known\, and using this we\ndescribe the locus of Picard curves wi th vanishing Ceresa cycle. This is\njoint work with Jef Laga.\n LOCATION:https://researchseminars.org/talk/BC-MIT/12/ END:VEVENT BEGIN:VEVENT SUMMARY:Robert Lemke Oliver (Tufts University) DTSTART:20241029T190000Z DTEND:20241029T200000Z DTSTAMP:20260314T081905Z UID:BC-MIT/13 DESCRIPTION:Title: Enumerating Galois extensions of number fields\nby Robert Lemke Oliver (Tufts University) as part of BC-MIT number theory seminar\n\nLecture hel d in Room 2-449 at MIT.\n\nAbstract\nLet $k$ be a number field. We provide an asymptotic formula for the number of Galois extensions of $k$ with abs olute discriminant bounded by some $X \\geq 1$ as $X \\to \\infty$. We als o provide an asymptotic formula for the closely related count of extension s $K/k$ whose normal closure has discriminant bounded by $X$. The key behi nd these results is a new upper bound on the number of Galois extensions o f $k$ with a given Galois group $G$ and discriminant bounded by $X$\; we s how the number of such extensions is $O_{[k:Q]\,G}(X^{4/\\sqrt{|G|}})$. Th is improves over the previous best bound $O_{k\,G\,\\epsilon}(X^{3/8+\\eps ilon})$ due to Ellenberg and Venkatesh. In particular\, ours is the first bound for general $G$ with an exponent that decays as $|G| \\to \\infty$.\ n LOCATION:https://researchseminars.org/talk/BC-MIT/13/ END:VEVENT BEGIN:VEVENT SUMMARY:Hang Xue (University of Arizona) DTSTART:20241029T203000Z DTEND:20241029T213000Z DTSTAMP:20260314T081905Z UID:BC-MIT/14 DESCRIPTION:Title: Fourier--Jacobi periods on unitary groups\nby Hang Xue (University of Arizona) as part of BC-MIT number theory seminar\n\nLecture held in Room 2 -449 at MIT.\n\nAbstract\nWe explain a proof of the Gan--Gross--Prasad con jecture for Fourier--Jacobi periods on unitary groups via relative trace f ormulae. This is joint work with Paul Boisseau and Weixiao Lu.\n LOCATION:https://researchseminars.org/talk/BC-MIT/14/ END:VEVENT BEGIN:VEVENT SUMMARY:Stefan Patrikis (Ohio State University) DTSTART:20241203T200000Z DTEND:20241203T210000Z DTSTAMP:20260314T081905Z UID:BC-MIT/15 DESCRIPTION:Title: Compatibility of canonical l-adic local systems on Shimura varieties of no n-abelian type\nby Stefan Patrikis (Ohio State University) as part of BC-MIT number theory seminar\n\nLecture held in Maloney 560 at Boston Coll ege.\n\nAbstract\nLet $(G\, X)$ be a Shimura datum\, and let $K$ be a comp act open subgroup of $G(\\mathbb{A}_f)$. One hopes that under mild assumpt ions on $G$ and $K$\, the points of the Shimura variety $Sh_K(G\, X)$ para metrize a family of motives\; unlike in abelian type (moduli of abelian va rieties\, etc.)\, in non-abelian type this problem remains completely myst erious. I will discuss joint work with Christian Klevdal showing that for "superrigid\," including all non-abelian type\, Shimura varieties the poin ts (over number fields\, say) at least yield compatible systems of l-adic representations\, which should be the l-adic realizations of the conjectur al motives. Time permitting\, I will discuss some work in progress (with J ake Huryn\, Kiran Kedlaya\, and Klevdal) on a crystalline analogue.\n LOCATION:https://researchseminars.org/talk/BC-MIT/15/ END:VEVENT BEGIN:VEVENT SUMMARY:Myrto Mavraki (University of Toronto) DTSTART:20241203T213000Z DTEND:20241203T223000Z DTSTAMP:20260314T081905Z UID:BC-MIT/16 DESCRIPTION:Title: Uniformity in unlikely intersections and the dynamical André--Oort conjec ture\nby Myrto Mavraki (University of Toronto) as part of BC-MIT numbe r theory seminar\n\nLecture held in Maloney 560 at Boston College.\n\nAbst ract\nA rational map is postcritically finite (PCF) if its critical orbits are finite. Postcritically finite maps play an important role in dynamics . Further\, it was suggested by Silverman that they play a role analogous to CM elliptic elliptic curves. Inspired in part by the Pink-Zilber conjec tures in unlikely intersections\, Baker and DeMarco formulated a conjectur e aiming to describe the subvarieties of $M_d$ that contain a Zariski dens e set of PCF points. Their conjecture\, now known as dynamical André--Oor t conjecture (or DAO)\, was recently resolved in the case of curves by Ji- -Xie\, but remains open in higher dimensions. In this talk we will describ e recent work with DeMarco and Ye\, providing uniform bounds on the config urations of PCF points in families of subvarieties in $M_d$. We also provi de a gap principle in the spirit of Dimitrov--Gao--Habegger's\, Kühne's\, and Gao--Ge--Kühne's work on the uniform Mordell--Lang conjecture.\n LOCATION:https://researchseminars.org/talk/BC-MIT/16/ END:VEVENT BEGIN:VEVENT SUMMARY:Chen Wan (Rutgers University) DTSTART:20250225T200000Z DTEND:20250225T210000Z DTSTAMP:20260314T081905Z UID:BC-MIT/17 DESCRIPTION:Title: Some spherical varieties of Whittaker type\nby Chen Wan (Rutgers Unive rsity) as part of BC-MIT number theory seminar\n\nLecture held in Room 2-4 49 at MIT.\n\nAbstract\nIn this talk I will discuss a special category of spherical varieties whose L^2 space is exactly supported on all the temper ed generic representations. Then I will give two families of examples\, on e from the theory of double flag varieties\, and the other one from the th eory of Rankin-Selberg integrals. I will also explain how to use this theo ry to prove the local funcational equations for some Rankin-Selberg integr als. This is an ongoing joint work with Lei Zhang.\n LOCATION:https://researchseminars.org/talk/BC-MIT/17/ END:VEVENT BEGIN:VEVENT SUMMARY:Tonghai Yang (University of Wisconsin\, Madison) DTSTART:20250225T213000Z DTEND:20250225T223000Z DTSTAMP:20260314T081905Z UID:BC-MIT/18 DESCRIPTION:Title: (A)FL at infinity and arithmetic generating series of CM cycles\nby To nghai Yang (University of Wisconsin\, Madison) as part of BC-MIT number th eory seminar\n\nLecture held in Room 2-449 at MIT.\n\nAbstract\nIn this ta lk\, we propose a FL and AFL at the real place with a proof of FL (if time permits). We also define a generating series of arithmetic CM cycles inde xed by integer and conjecture it to be modular. Finally\, we explain the c onnection between the two. This is a preliminary report of my joint work w ith Andreas Mihatsch and Siddarth Sankaran.\n LOCATION:https://researchseminars.org/talk/BC-MIT/18/ END:VEVENT BEGIN:VEVENT SUMMARY:Trevor Wooley (Purdue University) DTSTART:20250401T190000Z DTEND:20250401T200000Z DTSTAMP:20260314T081905Z UID:BC-MIT/19 DESCRIPTION:Title: Optimal mean value estimates and function field arithmetic\nby Trevor Wooley (Purdue University) as part of BC-MIT number theory seminar\n\nLect ure held in Maloney 560 at Boston College.\n\nAbstract\nEssentially optima l estimates have been obtained for mean values of Vinogradov’s exponenti al sum as a consequence of the decoupling method (by Bourgain\, Demeter an d Guth)\, and the efficient congruencing method (by the speaker). Such wor k makes essential use of the fact that the system of Diophantine equations associated with these mean values is translation-dilation invariant. We r eport on progress for systems which are not translation-dilation invariant obtained by exploiting the arithmetic of function fields over the field o f p-adic numbers.\n LOCATION:https://researchseminars.org/talk/BC-MIT/19/ END:VEVENT BEGIN:VEVENT SUMMARY:Jacob Tsimerman (University of Toronto) DTSTART:20250401T203000Z DTEND:20250401T213000Z DTSTAMP:20260314T081905Z UID:BC-MIT/20 DESCRIPTION:Title: Good reductions of CM points for Exceptional Shimura Varieties\nby Jac ob Tsimerman (University of Toronto) as part of BC-MIT number theory semin ar\n\nLecture held in Maloney 560 at Boston College.\n\nAbstract\nGiven an Elliptic curve E with complex multiplication\, it is known that E has (po tentially) good reduction everywhere. Concretely\, this means that the j-i nvariant of E is an algebraic integer. The generalization of this result t o Abelian-Varieties follows from the Neron-Ogg-Shafarevich criterion for g ood reduction.\n\nWe generalize this result to Exceptional Shimura varieti es S. Concretely\, we show that there exists some integral model S_0 of S such that all special points of S extend to integral points of S_0. To pro ve this we establish a Neron-Ogg-Shafarevich criterion in this setting. Ou r methods are general and apply\, in particular\, to arbitrary variations of hodge structures with an immersive Kodaira-Spencer map.\n\nWe will expl ain the proof (which is largely in the realm of birational p-adic geometry ) and the open questions that remain. This is joint work with Ben Bakker.\ n LOCATION:https://researchseminars.org/talk/BC-MIT/20/ END:VEVENT BEGIN:VEVENT SUMMARY:Alexander Dunn (Georgia Institute of Technology) DTSTART:20250506T193000Z DTEND:20250506T203000Z DTSTAMP:20260314T081905Z UID:BC-MIT/21 DESCRIPTION:Title: Recent progress on Gauss sums and primes\nby Alexander Dunn (Georgia I nstitute of Technology) as part of BC-MIT number theory seminar\n\nLecture held in Room 2-449 at MIT.\n\nAbstract\nLarge sieve inequalities are a fu ndamental tool used to investigate prime numbers and exponential sums. In this lecture I will explain my work that resolves a 1978 conjecture of S. Patterson (conditional on the Generalized Riemann hypothesis) concerning t he bias of cubic Gauss sums over the prime numbers. This explains a well-k nown numerical bias first observed by Kummer in 1846. This bias was later the subject of testing on some of the first super computers in the 20th ce ntury. Time permitting\, results on higher order Gauss sums will be discus sed. This is joint work with Maksym Radziwill.\n LOCATION:https://researchseminars.org/talk/BC-MIT/21/ END:VEVENT BEGIN:VEVENT SUMMARY:Lue Pan (Princeton University) DTSTART:20250506T180000Z DTEND:20250506T190000Z DTSTAMP:20260314T081905Z UID:BC-MIT/22 DESCRIPTION:Title: Plectic Lie algebra action on the cohomology of Hilbert modular varieties< /a>\nby Lue Pan (Princeton University) as part of BC-MIT number theory sem inar\n\nLecture held in Room 2-449 at MIT.\n\nAbstract\nA result of Jan Ne kovář says that the Galois action on p-adic intersection cohomology of H ilbert modular varieties with coefficients in automorphic local systems is semisimple. We will explain a new proof of this result for the non-CM par t of the cohomology and extend it to the locally analytic completed cohomo logy. Interestingly\, Nekovář’s approach is based on the construction of partial Frobenii at places away from p\, while our method uses partial Sen operators at p to construct a plectic Lie algebra action (whose meanin g will be explained in the talk). This is joint work in progress with Yuan yang Jiang.\n LOCATION:https://researchseminars.org/talk/BC-MIT/22/ END:VEVENT BEGIN:VEVENT SUMMARY:Katherine Stange (University of Colorado Boulder) DTSTART:20251021T190000Z DTEND:20251021T200000Z DTSTAMP:20260314T081905Z UID:BC-MIT/23 DESCRIPTION:Title: The arithmetic of thin orbits\nby Katherine Stange (University of Colo rado Boulder) as part of BC-MIT number theory seminar\n\nLecture held in M aloney 560 at Boston College.\n\nAbstract\nWe consider the local-to-global question for orbits of thin groups/semigroups. We will discuss Apollonia n circle packings\, continued fractions\, and some related problems. In t he Apollonian case\, we ask about the integers which occur as curvatures i n a packing. We observe that they satisfy certain congruence restrictions \, and ask whether all sufficiently large integers otherwise occur. In th e case of continued fractions\, we consider variants of Zaremba's conjectu re on the rationals with bounded continued fractions. Joint work includes work with Haag\, Kertzer\, and Rickards.\n LOCATION:https://researchseminars.org/talk/BC-MIT/23/ END:VEVENT BEGIN:VEVENT SUMMARY:Daniil Rudenko DTSTART:20251021T203000Z DTEND:20251021T213000Z DTSTAMP:20260314T081905Z UID:BC-MIT/24 DESCRIPTION:Title: Multiple polylogarithms and homology of general linear groups\nby Dani il Rudenko as part of BC-MIT number theory seminar\n\nLecture held in Malo ney 560 at Boston College.\n\nAbstract\nGoncharov’s program is a chain o f constructions and conjectures connecting algebraic $K$-theory\, multiple polylogarithms\, and mixed Tate motives. I will describe a connection bet ween multiple polylogarithms and the homology of general linear groups\, a nd discuss its consequences for Goncharov’s program. The talk is based o n joint work with Alexander Kupers and Ismael Sierra.\n LOCATION:https://researchseminars.org/talk/BC-MIT/24/ END:VEVENT BEGIN:VEVENT SUMMARY:Ananth Shankar DTSTART:20251118T203000Z DTEND:20251118T213000Z DTSTAMP:20260314T081905Z UID:BC-MIT/25 DESCRIPTION:Title: $p$-adic hyperbolicity for Shimura varieties and period images\nby Ana nth Shankar as part of BC-MIT number theory seminar\n\nLecture held in 2-4 49 at MIT.\n\nAbstract\nBorel proved that every holomorphic map from a pro duct of punctured unit discs to a complex Shimura variety extends to a map from a product of discs to its Bailey-Borel compactification. In joint wo rk with Oswal\, Zhu\, and Patel\, we proved a p-adic version of this theor em over discretely valued fields for Shimura varieties of abelian type. I will speak about work with Bakker\, Oswal\, and Yao\, where we prove the a nalogous $p$-adic extension theorem for compact non-abelian Shimura variet ies and geometric period images for large primes $p$.\n LOCATION:https://researchseminars.org/talk/BC-MIT/25/ END:VEVENT BEGIN:VEVENT SUMMARY:Tony Feng (UC Berkeley) DTSTART:20251118T220000Z DTEND:20251118T230000Z DTSTAMP:20260314T081905Z UID:BC-MIT/26 DESCRIPTION:Title: Mirror symmetry and the Breuil—Mezard Conjecture: an update\nby Tony Feng (UC Berkeley) as part of BC-MIT number theory seminar\n\nLecture hel d in 2-449 at MIT.\n\nAbstract\nThe Breuil—Mezard Conjecture predicts a precise indexing of cycles in moduli spaces of local Galois representation s by modular representations of finite groups of Lie type. A couple years ago\, Bao Le Hung and I introduced a new approach to the Breuil—Mezard C onjecture based on a connection to an instance of mirror symmetry\, which in that instance predicts a precise indexing of Lagrangians in a symplecti c variety by representations of a quantum group. Recently\, we used this t o prove the Breuil—Mezard Conjecture in the generic range for arbitrary unramified groups\, including exceptional groups. My intent is to review t his and also work-in-progress with Le Hung and Zhongyipan Lin\, which aims to extend the result to ramified groups. The key new aspect of the ramifi ed case is a nascent theory of "Spectral Langlands functoriality"\, an ana logue of Langlands functoriality for the spectral (i.e.\, "Galois") side o f the Langlands correspondence.\n LOCATION:https://researchseminars.org/talk/BC-MIT/26/ END:VEVENT BEGIN:VEVENT SUMMARY:Jonathan Pila DTSTART:20251202T213000Z DTEND:20251202T223000Z DTSTAMP:20260314T081905Z UID:BC-MIT/27 DESCRIPTION:Title: Unlikely intersections with Bessel functions and Laguerre polynomials II\nby Jonathan Pila as part of BC-MIT number theory seminar\n\nLecture he ld in Maloney 560 at Boston College.\n\nAbstract\nSeveral problems in arit hmetic geometry identify a class of "special points" and seek to describe the set of special points satisfying a given system of algebraic equations . Two classical examples are the Manin-Mumford conjecture concerning torsi on points in abelian varieties\, and the Andre-Oort conjecture concerning CM points in Shimura varieties. We propose a new variation on this theme\, where the role of special points is played by the zeros of special functi ons such as the Bessel function\, or of classical orthogonal polynomials. The Bessel function problem is related to a conjecture of Fuglede in harmo nic analysis from 1974\, and the orthogonal polynomial problem to a conjec ture of Stieltjes from 1890.\n\nIn the first talk we will recall the Manin -Mumford conjecture and introduce the Bessel function and orthogonal polyn omial analogs. We will sketch the proof of classical Manin-Mumford using o -minimality and how it can be adapted to obtain various results in this ne w context. This naturally leads to the more "exotic" o-minimal structure o f multisummable functions\, as opposed to the structure R_{an\,exp} used i n classical applications.\nIn the second talk we will discuss new function al transcendence results that are needed to complete the argument: a varia nt of the Ax-Schanuel theorem for the special functions appearing in this context. This naturally leads to the study of differential Galois groups f or irregular-singular systems\, as opposed to the regular-singular systems used in classical applications.\n\nAll the results are based on joint wor k with Avner Kiro and some also with Gady Kozma.\n LOCATION:https://researchseminars.org/talk/BC-MIT/27/ END:VEVENT BEGIN:VEVENT SUMMARY:Gal Binyamini DTSTART:20251202T200000Z DTEND:20251202T210000Z DTSTAMP:20260314T081905Z UID:BC-MIT/28 DESCRIPTION:Title: Unlikely intersections with Bessel functions and Laguerre polynomials I\nby Gal Binyamini as part of BC-MIT number theory seminar\n\nLecture hel d in Maloney 560 at Boston College.\n\nAbstract\nSeveral problems in arith metic geometry identify a class of "special points" and seek to describe t he set of special points satisfying a given system of algebraic equations. Two classical examples are the Manin-Mumford conjecture concerning torsio n points in abelian varieties\, and the Andre-Oort conjecture concerning C M points in Shimura varieties. We propose a new variation on this theme\, where the role of special points is played by the zeros of special functio ns such as the Bessel function\, or of classical orthogonal polynomials. T he Bessel function problem is related to a conjecture of Fuglede in harmon ic analysis from 1974\, and the orthogonal polynomial problem to a conject ure of Stieltjes from 1890.\n\nIn the first talk we will recall the Manin- Mumford conjecture and introduce the Bessel function and orthogonal polyno mial analogs. We will sketch the proof of classical Manin-Mumford using o- minimality and how it can be adapted to obtain various results in this new context. This naturally leads to the more "exotic" o-minimal structure of multisummable functions\, as opposed to the structure R_{an\,exp} used in classical applications.\nIn the second talk we will discuss new functiona l transcendence results that are needed to complete the argument: a varian t of the Ax-Schanuel theorem for the special functions appearing in this c ontext. This naturally leads to the study of differential Galois groups fo r irregular-singular systems\, as opposed to the regular-singular systems used in classical applications.\n\nAll the results are based on joint work with Avner Kiro and some also with Gady Kozma.\n LOCATION:https://researchseminars.org/talk/BC-MIT/28/ END:VEVENT BEGIN:VEVENT SUMMARY:Christopher Skinner (Princeton University) DTSTART:20260210T203000Z DTEND:20260210T213000Z DTSTAMP:20260314T081905Z UID:BC-MIT/29 DESCRIPTION:Title: Euler systems and relative cohomology\nby Christopher Skinner (Princet on University) as part of BC-MIT number theory seminar\n\nLecture held in 2-449 at MIT.\n\nAbstract\nEuler systems -- organized collections of Galoi s cohomology classes for arithmetically interesting p-adic Galois represen tations -- have been a useful tool for establishing the conjectured relati on between special values of L-functions and the ranks and orders of Selme r groups when they exist. In this talk I will describe recent work provid ing new examples of Euler systems with cyclotomic variation\, including an Euler system for the symmetric square of a modular form. As a replacemen t for the motivic origin of prior examples\, we find the Galois extensions in the relative cohomology of Shimura varieties. The control needed to es tablish the norm relations and make connections with L-values is provided by recent results in integral p-adic Hodge theory\, allowing explicit conn ection with holomorphic automorphic forms.\n LOCATION:https://researchseminars.org/talk/BC-MIT/29/ END:VEVENT BEGIN:VEVENT SUMMARY:Romyar Sharifi (University of California\, Los Angeles) DTSTART:20260210T220000Z DTEND:20260210T230000Z DTSTAMP:20260314T081905Z UID:BC-MIT/30 DESCRIPTION:Title: Eisenstein cocycles for imaginary quadratic fields\nby Romyar Sharifi (University of California\, Los Angeles) as part of BC-MIT number theory s eminar\n\nLecture held in 2-449 at MIT.\n\nAbstract\nI will discuss the co nstruction of maps from the homology of Bianchi spaces for an imaginary qu adratic field F to second K-groups of ray class fields of F. These maps a re “Eisenstein” in the sense that they factor through the quotient by the action of an Eisenstein ideal way from the level. They are direct anal ogues of known explicit maps in the setting of modular curves and cyclotom ic fields. This is largely joint work with E. Lecouturier\, S. Shih\, and J. Wang\, though I intend to motivate this through the lens of work-in-pro gress on "artificial complexes" that aims to provide explicit formulas in terms of Steinberg symbols of elliptic units\, as in the cyclotomic settin g.\n LOCATION:https://researchseminars.org/talk/BC-MIT/30/ END:VEVENT BEGIN:VEVENT SUMMARY:Adrian Diaconu DTSTART:20260317T190000Z DTEND:20260317T200000Z DTSTAMP:20260314T081905Z UID:BC-MIT/31 DESCRIPTION:Title: Moments of quadratic L-functions over function fields\nby Adrian Diaco nu as part of BC-MIT number theory seminar\n\nLecture held in Maloney 560 at Boston College.\n\nAbstract\nIn 2001\, Conrey\, Farmer\, Keating\, Rubi nstein\, and Snaith developed a "recipe" utilizing heuristic arguments to predict the asymptotics of moments of various families of $L$-functions. T his heuristic was later extended by Andrade and Keating to include moments and ratios of the family of $L$-functions associated to hyperelliptic cur ves of genus $g$ over a fixed finite field. In joint work with Bergström\ , Petersen\, and Westerland\, we related the moment conjecture of Andrade and Keating to the problem of understanding the homology of the braid grou p with symplectic coefficients. We computed the stable homology groups of the braid groups with these coefficients\, together with their structure a s Galois representations\, and showed that the answer matches the number-t heoretic predictions. Our results\, combined with a recent homological sta bility theorem of Miller\, Patzt\, Petersen\, and Randal-Williams\, imply the conjectured asymptotics for all moments in the function field case\, f or all large enough odd prime powers $q$.\n LOCATION:https://researchseminars.org/talk/BC-MIT/31/ END:VEVENT BEGIN:VEVENT SUMMARY:David Zywina DTSTART:20260317T203000Z DTEND:20260317T213000Z DTSTAMP:20260314T081905Z UID:BC-MIT/32 DESCRIPTION:Title: Elliptic curves of low rank\nby David Zywina as part of BC-MIT number theory seminar\n\nLecture held in Maloney 560 at Boston College.\n\nAbstra ct\nFor an elliptic curve $E$ over a number field $K$\, the set $E(K)$ of $K$-points is a finitely generated abelian group whose rank is an importan t/mysterious invariant. It is an open and difficult problem to determine which ranks occur for elliptic curves over a fixed number field $K$. We wi ll discuss recent work which shows that there are infinitely many elliptic curves over $K$ of rank $r$ for each integer $0 \\leq r \\leq 4$. We wi ll construct our curves by specializing well-chosen nonisotrivial families . We will use a result of Kai\, which generalizes work of Green\, Tao and Ziegler to number fields\, to carefully choose our curves in the families .\n LOCATION:https://researchseminars.org/talk/BC-MIT/32/ END:VEVENT BEGIN:VEVENT SUMMARY:Giada Grossi DTSTART:20260414T193000Z DTEND:20260414T203000Z DTSTAMP:20260314T081905Z UID:BC-MIT/33 DESCRIPTION:by Giada Grossi as part of BC-MIT number theory seminar\n\nLec ture held in 2-449 at MIT.\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/BC-MIT/33/ END:VEVENT BEGIN:VEVENT SUMMARY:Gregorio Baldi DTSTART:20260414T210000Z DTEND:20260414T220000Z DTSTAMP:20260314T081905Z UID:BC-MIT/34 DESCRIPTION:by Gregorio Baldi as part of BC-MIT number theory seminar\n\nL ecture held in 2-449 at MIT.\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/BC-MIT/34/ END:VEVENT END:VCALENDAR