BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Jakub Konieczny (Hebrew University of Jerusalem) DTSTART:20200525T111500Z DTEND:20200525T121500Z DTSTAMP:20260422T225839Z UID:WarsawNT/1 DESCRIPTION:Title: Automatic multiplicative sequences\nby Jakub Konieczny (Hebrew Univer sity of Jerusalem) as part of Warsaw Number Theory Seminar\n\nLecture held in room 403 at IMPAN.\n\nAbstract\nAutomatic sequences - that is\, sequen ces computable by finite automata - give rise to one of the most basic mod els of computation. As such\, for any class of sequences it is natural to ask which sequences in it are automatic. In particular\, the question of c lassifying automatic multiplicative sequences has attracted considerable a ttention in the recent years. In the completely multiplicative case\, such classification was obtained independently by S. Li and O. Klurman and P. Kurlberg. The main topic of my talk will be the resolution of the general case\, obtained in a recent preprint with M. Lemańczyk and C. Müllner.\n LOCATION:https://researchseminars.org/talk/WarsawNT/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Alex Youcis (IMPAN Warsaw) DTSTART:20200601T111500Z DTEND:20200601T121500Z DTSTAMP:20260422T225839Z UID:WarsawNT/2 DESCRIPTION:Title: An approach to the characterization of the $p$-adic local Langlands corre spondence\nby Alex Youcis (IMPAN Warsaw) as part of Warsaw Number Theo ry Seminar\n\nLecture held in room 403 at IMPAN.\n\nAbstract\nIn 2013 P. S cholze provided an alternative proof of the local Langlands correspondence (LLC) for GLn and\, in doing so\, Scholze gave a new characterization of the LLC via a certain trace identity. In this talk the speaker will discus s joint work with A. Bertoloni Meli showing that a generalization of this trace identity characterizes the LLC for much more general groups if one a ssumes standard expected properties of such a correspondence.\n LOCATION:https://researchseminars.org/talk/WarsawNT/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Anna Szumowicz (Sorbonne Université\, IMJ-PRG) DTSTART:20200608T111500Z DTEND:20200608T121500Z DTSTAMP:20260422T225839Z UID:WarsawNT/3 DESCRIPTION:Title: Equidistribution in number fields\nby Anna Szumowicz (Sorbonne Univer sité\, IMJ-PRG) as part of Warsaw Number Theory Seminar\n\nLecture held i n room 403 at IMPAN.\n\nAbstract\nThe notion of $\\mathfrak{p}$-ordering c omes from the work of Bhargava on integer-valued polynomials. Let $k$ be a number field and let $\\mathcal{O}_{k}$ be its ring of integers. A sequen ce of elements in $\\mathcal{O}_{k}$ is a simultaneous $\\mathfrak{p}$-ord ering if it is equidistributed modulo every prime ideal in $\\mathcal{O} _{k}$ as well as possible. We prove that $\\mathbb{Q}$ the only number fie ld $k$ such that $\\mathcal{O}_{k}$ admits a simultaneous $\\mathfrak{p}$- ordering. It is a joint work with Mikołaj Frączyk.\n LOCATION:https://researchseminars.org/talk/WarsawNT/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Jolanta Marzec (TU Darmstadt) DTSTART:20200622T111500Z DTEND:20200622T121500Z DTSTAMP:20260422T225839Z UID:WarsawNT/4 DESCRIPTION:Title: Maass relations for Saito-Kurokawa lifts of higher levels\nby Jolanta Marzec (TU Darmstadt) as part of Warsaw Number Theory Seminar\n\nLecture held in room 403 at IMPAN.\n\nAbstract\nIt is known that a Siegel modular form is a (classical) Saito-Kurokawa\nlift of an elliptic modular form if and only if its Fourier coefficients\nsatisfy the so-called Maass relation s. The first construction of such a\nlift was given by Maass using corresp ondences between various modular\nforms. However\, in order to generalize this lift to higher levels it\nis easier to use a construction coming from representation theory. During\nthe talk we present history of this probl em and briefly discuss the\naforementioned constructions. We also indicate how one can read off the\nMaass relations from the latter. This work gene ralizes an approach\nof Pitale\, Saha and Schmidt from the classical to a higher level case.\n LOCATION:https://researchseminars.org/talk/WarsawNT/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Adam Keilthy (University of Oxford) DTSTART:20200629T111500Z DTEND:20200629T121500Z DTSTAMP:20260422T225839Z UID:WarsawNT/5 DESCRIPTION:Title: The block filtration and motivic multiple zeta values\nby Adam Keilth y (University of Oxford) as part of Warsaw Number Theory Seminar\n\nLectur e held in room 403 at IMPAN.\n\nAbstract\nMultiple zeta values are a class of transcendental numbers\, going back to Euler in the 1700s\nand with ti es to the Riemann zeta function. They are found arising naturally in many areas of mathematics and physics\, from algebraic geometry to Feynman ampl itudes. Unlike in the case of single zeta values\, we know many algebraic relations satisfied by multiple zeta values: the double shuffle relations\ , the associator relations\, the confluence relations. However it is unkno wn if any of these sets of relations are complete. Assuming Grothendieck's period conjecture\, a complete set of algebraic relations are given by th e motivic relations\, arising from a connection to P^1 minus three points. However these relations are inexplicit.\n\nIn this talk\, we introduce a new filtration\, called the block filtration\, on the space of multiple ze ta values. By considering the associated graded\, we describe several new families of motivic relations\, that provide a complete description of rel ations in low block degree. A generalisation of these results would thus p rovide a complete description of relations among multiple zeta values and aid in settling several open problems about the motivic Galois group of mi xed Tate motives.\n LOCATION:https://researchseminars.org/talk/WarsawNT/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Dorota Blinkiewicz (Adam Mickiewicz University\, Poznań) DTSTART:20201005T111500Z DTEND:20201005T121500Z DTSTAMP:20260422T225839Z UID:WarsawNT/6 DESCRIPTION:Title: Classes of extensions of commutative algebraic groups\nby Dorota Blin kiewicz (Adam Mickiewicz University\, Poznań) as part of Warsaw Number Th eory Seminar\n\n\nAbstract\nDuring the lecture I give explicit characteriz ation of $n$-torsion elements in the group of extensions of commutative\, smooth algebraic groups.\n LOCATION:https://researchseminars.org/talk/WarsawNT/6/ END:VEVENT BEGIN:VEVENT SUMMARY:Vesselin Dimitrov (University of Toronto) DTSTART:20201012T111500Z DTEND:20201012T121500Z DTSTAMP:20260422T225839Z UID:WarsawNT/7 DESCRIPTION:Title: Solution of the conjecture of Schinzel and Zassenhaus and some applicatio ns\nby Vesselin Dimitrov (University of Toronto) as part of Warsaw Num ber Theory Seminar\n\n\nAbstract\nWe will detail the full proof of an expl icit form of the Schinzel-Zassenhaus conjecture: an algebraic integer of d egree $n > 1$ is either a root of unity or else has at least one conjugate of modulus exceeding $2^{1/(4n)}$. We furthermore obtain an extension of the original conjecture over to the setting of holonomic functions\, with an application to the smallest critical value for (certain) rational funct ions.\n\nIn another application\, we would like to take the occasion to ra ise the apparently unsolved problem of the essential irreducibility (up-to the cyclotomic factor $X^2-X+1$ in degrees a multiple of 12) of $X^{2g} - X^g(1+X+1/X) + 1$\, the characteristic polynomial of the integer reciproc al Perron-Frobenius matrix of the smallest spectral radius in each given d imension. Our explicit Schinzel-Zassenhaus bound allows for at most $10$ f actors of each of these polynomials.\n LOCATION:https://researchseminars.org/talk/WarsawNT/7/ END:VEVENT BEGIN:VEVENT SUMMARY:Jakub Byszewski (Jagellonian University) DTSTART:20201019T111500Z DTEND:20201019T121500Z DTSTAMP:20260422T225839Z UID:WarsawNT/8 DESCRIPTION:Title: Finite order automorphisms of the ring of power series over a finite fiel d\nby Jakub Byszewski (Jagellonian University) as part of Warsaw Numbe r Theory Seminar\n\nLecture held in room 403 at IMPAN.\n\nAbstract\nThe No ttingham group at a prime $p$ is the group of (formal) power series $t+a_2 t^2+ a_3 t^3+ \\cdots$ in the variable $t$ with coefficients $a_i$ from t he field with $p$ elements with the group operation given by composition o f power series. This group is known to contain elements of order being an arbitrary power of $p$. Elements of order $p$ have been classified by Klop sch and have a nice description. For higher orders\, however\, only a hand ful of examples have been known explicitly.\n\nIn the talk we will show ho w to describe such series in closed computational form through finite auto mata. This allows us to construct many explicit examples and formulate a n umber of questions. The talk is based on joint work with Gunther Corneliss en and Djurre Tijsma.\n LOCATION:https://researchseminars.org/talk/WarsawNT/8/ END:VEVENT BEGIN:VEVENT SUMMARY:Danylo Radchenko (ETH Zürich) DTSTART:20201026T121500Z DTEND:20201026T131500Z DTSTAMP:20260422T225839Z UID:WarsawNT/9 DESCRIPTION:Title: Universal optimality of the E8 and Leech lattices\nby Danylo Radchenk o (ETH Zürich) as part of Warsaw Number Theory Seminar\n\nLecture held in room 403 at IMPAN.\n\nAbstract\nWe look at the problem of arranging point s in Euclidean space in order to minimize the potential energy of pairwise interactions. We show that the E8 lattice and the Leech lattice are unive rsally optimal in the sense that they have the lowest energy for all poten tials that are given by completely monotone potentials of squared distance . \nThe proof uses a new kind of interpolation formula for Fourier eigenfu nctions\, which is intimately related to the theory of modular forms.\nThe talk is based on a joint work with Henry Cohn\, Abhinav Kumar\, Stephen D . Miller\, and Maryna Viazovska.\n LOCATION:https://researchseminars.org/talk/WarsawNT/9/ END:VEVENT BEGIN:VEVENT SUMMARY:Julie Desjardins (University of Toronto) DTSTART:20201102T121500Z DTEND:20201102T131500Z DTSTAMP:20260422T225839Z UID:WarsawNT/10 DESCRIPTION:Title: Density of rational points on a family of del Pezzo surface of degree 1< /a>\nby Julie Desjardins (University of Toronto) as part of Warsaw Number Theory Seminar\n\nLecture held in room 403 at IMPAN.\n\nAbstract\nLet k be a number field and X an algebraic variety over k. We want to study the se t of k-rational points X(k). For example\, is X(k) empty? If not\, is it d ense with respect to the Zariski topology? Del Pezzo surfaces are classifi ed by their degrees d (an integer between 1 and 9). Manin and various auth ors proved that for all del Pezzo surfaces of degree >1 it is dense provid ed that the surface has a k-rational point (that lies outside a specific s ubset of the surface for d=2). For d=1\, the del Pezzo surface always has a rational point. However\, we don't know if the set of rational points is Zariski-dense. In this talk\, I present a result that is joint with Rosa Winter in which we prove the density of rational points for a specific fam ily of del Pezzo surfaces of degree 1 over k.\n LOCATION:https://researchseminars.org/talk/WarsawNT/10/ END:VEVENT BEGIN:VEVENT SUMMARY:Masha Vlasenko (IMPAN) DTSTART:20201109T121500Z DTEND:20201109T131500Z DTSTAMP:20260422T225839Z UID:WarsawNT/11 DESCRIPTION:Title: Dwork crystals\nby Masha Vlasenko (IMPAN) as part of Warsaw Number T heory Seminar\n\nLecture held in room 403 at IMPAN.\n\nAbstract\nIn his wo rk on rationality of zeta functions of algebraic varieties Bernard Dwork d iscovered a number of remarkable p-adic congruences. In this talk I will d emonstrate some of these congruences and overview our recent work with Fri ts Beukers which explains their underlying mechanism.\n LOCATION:https://researchseminars.org/talk/WarsawNT/11/ END:VEVENT BEGIN:VEVENT SUMMARY:Matija Kazalicki (University of Zagreb) DTSTART:20201116T121500Z DTEND:20201116T131500Z DTSTAMP:20260422T225839Z UID:WarsawNT/12 DESCRIPTION:Title: Congruences for sporadic sequences\, three fold covers of the elliptic m odular surfaces and modular forms for non-congruence subgroups\nby Mat ija Kazalicki (University of Zagreb) as part of Warsaw Number Theory Semin ar\n\nLecture held in room 403 at IMPAN.\n\nAbstract\nIn 1979\, in the cou rse of the proof of the irrationality of $\\zeta(2)$\n Ap\\'ery introduced numbers $b_n = \\sum_{k=0}^n {n \\choose k}^2{n+k\n\\choose k}$ that are\ , surprisingly\, integral solutions of recursive\nrelations $$(n+1)^2 u_{ n+1} - (11n^2+11n+3)u_n-n^2u_{n-1} = 0.$$\nZagier performed a computer sea rch on first 100 million triples\n$(A\,B\,C)\\in \\mathbb{Z}^3$ and found that the recursive relation\ngeneralizing $b_n$\n$$(n+1)u_{n+1} - (An^2+An +B)u_n + C n ^2 u_{n-1}=0\,$$\nwith the initial conditions $u_{-1}=0$ and $u_0=1$ has (non-degenerate\ni.e. $C(A^2-4C)\\ne 0$) integral solution for only six more triples\n(whose solutions are so called sporadic sequences) .\n\nStienstra and Beukers showed that the for prime $p\\ge 5$\n\\begin{e quation*}\nb_{(p-1)/2} \\equiv \\begin{cases} 4a^2-2p \\pmod{p} \\textrm{ if } p =\na^2+b^2\,\\textrm{ a odd}\\\\ 0 \\pmod{p} \\textrm{ if } p\\equi v 3\n\\pmod{4}.\\end{cases}\n\\end{equation*}\n\nRecently\, Osburn and Str aub proved similar congruences for all but one\nof the six Zagier's sporad ic sequences (three cases were already known\nto be true by the work of St ienstra and Beukers) and we proved the\ncongruence for the sixth sequence. \n\nIn this talk we describe congruences for the Ap\\'ery numbers\n$b_{(p- 1)/3}$ (and also for the other sporadic sequences).\nFor that we study Atk in and Swinnerton-Dyer type of congruences\nbetween Fourier coefficients o f cusp forms for non-congruence\nsubgroups\, $L$-functions of three covers of elliptic modular surfaces\nand Galois representations attached to thes e covers.\n LOCATION:https://researchseminars.org/talk/WarsawNT/12/ END:VEVENT BEGIN:VEVENT SUMMARY:Wadim Zudilin (Radboud University\, Nijmegen) DTSTART:20201123T121500Z DTEND:20201123T131500Z DTSTAMP:20260422T225839Z UID:WarsawNT/13 DESCRIPTION:Title: Dwork-type ($q$-)(super)congruences\nby Wadim Zudilin (Radboud Unive rsity\, Nijmegen) as part of Warsaw Number Theory Seminar\n\nLecture held in room 403 at IMPAN.\n\nAbstract\nThe "microscope" principle is this: If a rational function A(q) of variable q vanishes at every p-th root of unit y (for p prime)\, then A(q) == 0 modulo Φ_p(q)\, the p-th cyclotomic poly nomial\; assuming that A(1) is a well-defined rational number with a p-fre e denominator and specialising the congruence at q=1 we conclude with A(1) == 0 modulo p.\nIn other words\, behaviour of rational functions at p-th roots of unity may be instructive for gaining information about their valu es at 1 modulo p. With some "creative" extras\, we can further consider di visibility by higher powers of primes (and we can even deal with not neces sarily primes).\nIn my talk\, partly based on recent joint work with Victo r Guo\, I plan to highlight some novel outcomes of this "creative microsco pe" methodology -- examples of Dwork-type supercongruences for truncated h ypergeometric sums.\n LOCATION:https://researchseminars.org/talk/WarsawNT/13/ END:VEVENT BEGIN:VEVENT SUMMARY:Robert Osburn (University College Dublin) DTSTART:20201130T121500Z DTEND:20201130T131500Z DTSTAMP:20260422T225839Z UID:WarsawNT/14 DESCRIPTION:Title: Generalized Fishburn numbers\, torus knots and quantum modularity\nb y Robert Osburn (University College Dublin) as part of Warsaw Number Theor y Seminar\n\nLecture held in room 403 at IMPAN.\n\nAbstract\nThe Fishburn numbers are a sequence of positive integers with numerous combinatorial in terpretations and interesting asymptotic properties. In 2016\, Andrews and Sellers initiated the study of arithmetic properties of these numbers. In this talk\, we discuss a generalization of this sequence using knot theor y and the quantum modularity of the associated Kontsevich-Zagier series.\n \nThe first part is joint work with Colin Bijaoui (McMaster)\, Hans Boden (McMaster)\, Beckham Myers (Harvard)\, Will Rushworth (McMaster)\, Aaron T ronsgard (Toronto) and Shaoyang Zhou (Vanderbilt) while the second part is joint work with Ankush Goswami (RISC).\n LOCATION:https://researchseminars.org/talk/WarsawNT/14/ END:VEVENT BEGIN:VEVENT SUMMARY:Gunther Cornelissen (Utrecht University) DTSTART:20201207T121500Z DTEND:20201207T131500Z DTSTAMP:20260422T225839Z UID:WarsawNT/15 DESCRIPTION:Title: An analogy between number theory and spectral geometry\nby Gunther C ornelissen (Utrecht University) as part of Warsaw Number Theory Seminar\n\ n\nAbstract\nSunada’s construction of non-isometric\, isospectral manifo lds proceeds in the same way as Gassmann’s construction of non-isomorphi c number fields with the same zeta function\, using a group G with two non -conjugate subgroups H and K such that the permutation representations giv en by G acting on their cosets are isomorphic. In Gassmann’s example\, G was the permutation group on 6 letters and H and K the groups generated b y (12)(34) and (13)(24)\, and (12)(34) and (12)(56)\, respectively. These can be realized as covering groups of a compact Riemann surface of genus 2 . Recently\, the speaker and collaborators showed that isomophism of numbe r fields can be detected by equality of suitable L-series. This talk is ab out the finding the analogous result for manifolds. The result says that i f two manifolds are finite Riemannian covers of a developable orbifold\, a nd such that a certain homological condition is satisfied\, then the manif olds are isometric if and only if the spectra of finitely many Laplacians twisted by suitable unitary representations of the fundamental group are e qual. The result is explicit: in the above example\, one needs 56 spectral equalities corresponding to 180-dimensional representations. (Joint work with Norbert Peyerimhoff.)\n LOCATION:https://researchseminars.org/talk/WarsawNT/15/ END:VEVENT BEGIN:VEVENT SUMMARY:Tim Dokchitser (University of Bristol) DTSTART:20201214T121500Z DTEND:20201214T131500Z DTSTAMP:20260422T225839Z UID:WarsawNT/16 DESCRIPTION:Title: Curves with tame torsion\nby Tim Dokchitser (University of Bristol) as part of Warsaw Number Theory Seminar\n\nLecture held in currently onlin e.\n\nAbstract\nIn this talk I will sketch the proof of the fact that in e very\ngenus and for every prime p there are curves over Q with tame p-tors ion.\nIn genus 1\, this is something this is quite easy to deduce from the \ntheory of the Tate curve\, and I will explain how an explicit version of \nthe theory of hyperelliptic Mumford curves gives this in arbitrary\ngenu s. This is joint work with Matthew Bisatt.\n LOCATION:https://researchseminars.org/talk/WarsawNT/16/ END:VEVENT BEGIN:VEVENT SUMMARY:Bidisha Roy (IMPAN\, Warsaw) DTSTART:20201221T121500Z DTEND:20201221T131500Z DTSTAMP:20260422T225839Z UID:WarsawNT/17 DESCRIPTION:Title: Torsion groups of elliptic curves over number fields\nby Bidisha Roy (IMPAN\, Warsaw) as part of Warsaw Number Theory Seminar\n\nLecture held in currently online.\n\nAbstract\nComputing torsion groups of elliptic cur ves defined over number fields is a classical topic and it has a vast lite rature in algebraic number theory. Any elliptic curve of the form y^2 = x^ 3 + c is called a Mordell curve. Mordell curves are well studied elliptic curves with complex multiplication.\n\nIn this talk\, for Mordell curves defined over the field of rational numbers we will discuss the classificat ion of torsion groups over cubic and sextic fields. Also\, we present the classification of torsion groups of Mordell curves defined over cubic fiel ds. For Mordell curves over sextic fields\, we provide all possible torsio n groups. In the second part\, we briefly discuss torsion groups of Mordel l curves over higher degree number fields.\n LOCATION:https://researchseminars.org/talk/WarsawNT/17/ END:VEVENT BEGIN:VEVENT SUMMARY:Wojtek Wawrów (London School of Geometry and Number Theory) DTSTART:20210111T121500Z DTEND:20210111T131500Z DTSTAMP:20260422T225839Z UID:WarsawNT/18 DESCRIPTION:Title: Ranks of Jacobians over rational numbers\nby Wojtek Wawrów (London School of Geometry and Number Theory) as part of Warsaw Number Theory Semi nar\n\nLecture held in currently online.\n\nAbstract\nThe problem of findi ng out the possible Mordell-Weil ranks of abelian varieties of given dimen sion over a fixed number field has a long history. In this talk we shall s urvey various results which have appeared over the years bounding from bel ow the maximal rank that those varieties can have\, with particular emphas is on Jacobians of special families of curves over the field of rational n umbers.\n LOCATION:https://researchseminars.org/talk/WarsawNT/18/ END:VEVENT BEGIN:VEVENT SUMMARY:Netan Dogra (King's College London) DTSTART:20210118T121500Z DTEND:20210118T131500Z DTSTAMP:20260422T225839Z UID:WarsawNT/19 DESCRIPTION:Title: Some new results in the nonabelian Chabauty method\nby Netan Dogra ( King's College London) as part of Warsaw Number Theory Seminar\n\nLecture held in currently online.\n\nAbstract\nIn this talk I will discuss the non abelian Chabauty method\, which seeks to use p-adic analytic functions to determine the finite sets of rational points on higher genus curves\, and some new cases where it can be used to determine the solutions to Diophant ine equations.\n LOCATION:https://researchseminars.org/talk/WarsawNT/19/ END:VEVENT BEGIN:VEVENT SUMMARY:Oleksiy Klurman (University of Bristol) DTSTART:20210125T121500Z DTEND:20210125T131500Z DTSTAMP:20260422T225839Z UID:WarsawNT/20 DESCRIPTION:Title: Monotone chains of Hecke cusp forms\nby Oleksiy Klurman (University of Bristol) as part of Warsaw Number Theory Seminar\n\nLecture held in cur rently online.\n\nAbstract\nWe discuss a general joint equidistribution re sult for the Fourier coefficients of Hecke cusp forms. One simple conseque nce of such a result is that there exist infinitely many integers n (in fa ct an upper density of this set is positive) such that \n$\\tau (n)<\\tau (n+1)<\\tau(n+2)$ where $\\tau$ is a Ramanujan $\\tau$-function. This is b ased on a joint work with A. Mangerel (CRM).\n LOCATION:https://researchseminars.org/talk/WarsawNT/20/ END:VEVENT BEGIN:VEVENT SUMMARY:Mikołaj Frączyk (University of Chicago) DTSTART:20210301T160000Z DTEND:20210301T170000Z DTSTAMP:20260422T225839Z UID:WarsawNT/22 DESCRIPTION:Title: Sarnak’s density hypothesis in horizontal families\nby Mikołaj Fr ączyk (University of Chicago) as part of Warsaw Number Theory Seminar\n\n Lecture held in currently online.\n\nAbstract\nLet $G$ be a real semi simp le Lie group with an irreducible unitary representation $\\pi$. The non-te mperedness of $\\pi$ is measured by the real parameter $p(\\pi)$ which is defined as the infimum of $p$ such that $\\pi$ has non-zero matrix coeffic ients in $L^p(G)$. Sarnak and Xue conjectured that for any arithmetic latt ice $\\Gamma\\subset G$ and principal congruence subgroup $\\Gamma(q)\\sub set \\Gamma$\, the multiplicity of $\\pi$ in $L^2(G/\\Gamma(q))$ is at mos t $O(V(q)^{2/p(\\pi) +\\varepsilon})$\, where $V(q)$ is the covolume of $\ \Gamma(q)$. Sarnak and Xue proved this conjecture for $G=SL(2\,\\mathbb R) \,SL(2\,\\mathbb C)$. I will talk about the joint work with Gergely Harcos \, Peter Maga and Djordje Milicevic where we prove bounds of the same qual ity that hold uniformly for families of pairwise non-commensurable lattice s in $G=SL(2\,\\mathbb R)^a\\times SL(2\,\\mathbb C)^b$. These families of lattices\, which we call horizontal\, are given as unit groups of maximal orders of quaternion algebras over number fields.\n LOCATION:https://researchseminars.org/talk/WarsawNT/22/ END:VEVENT BEGIN:VEVENT SUMMARY:Sudhir Pujahari (University of Warsaw) DTSTART:20210308T121500Z DTEND:20210308T131500Z DTSTAMP:20260422T225839Z UID:WarsawNT/23 DESCRIPTION:Title: Arithmetic and statistics of sums of eigenvalues of Hecke operators\ nby Sudhir Pujahari (University of Warsaw) as part of Warsaw Number Theory Seminar\n\nLecture held in currently online.\n\nAbstract\nIn the first pa rt of the talk\, we will study about the distribution of gaps between eige nvalues of Hecke operators in both horizontal and vertical settings. As an application of this we will obtain a strong multiplicity one theorem and evidence towards Maeda conjecture. The horizontal setting is a joint work with M. Ram Murty. In the second part of the talk\, using recent developme nts in the theory of l-adic Galois representations we will study the norma l number of prime factors\nof sums of Fourier coefficients of eigenforms. Moreover\, we will see the distribution of distinct prime factors of sums of Fourier coefficients of eigenforms. The final part is a joint work with M. Ram Murty and V. Kumar Murty.\n LOCATION:https://researchseminars.org/talk/WarsawNT/23/ END:VEVENT BEGIN:VEVENT SUMMARY:Carlo Verschoor (University of Utrecht) DTSTART:20210315T121500Z DTEND:20210315T131500Z DTSTAMP:20260422T225839Z UID:WarsawNT/24 DESCRIPTION:Title: Bailey Type Factorizations of Horn Functions\nby Carlo Verschoor (Un iversity of Utrecht) as part of Warsaw Number Theory Seminar\n\nLecture he ld in currently online.\n\nAbstract\nA well-known identity by Bailey state s that Appell’s F4 function can be written as the product of two Gauss h ypergeometric functions under a suitable specialization of its parameters. Other identities of this type are known for Appell’s F2 and F3\, which are closely related to Bailey’s identity. The aim of this talk is to sho w that the same can be done for Horn’s H1\, H4 and H5 functions.\n LOCATION:https://researchseminars.org/talk/WarsawNT/24/ END:VEVENT BEGIN:VEVENT SUMMARY:Enis Kaya (University of Groningen) DTSTART:20210322T121500Z DTEND:20210322T131500Z DTSTAMP:20260422T225839Z UID:WarsawNT/25 DESCRIPTION:Title: Explicit Vologodsky Integration for Hyperelliptic Curves\nby Enis Ka ya (University of Groningen) as part of Warsaw Number Theory Seminar\n\nLe cture held in currently online.\n\nAbstract\nLet $X$ be a curve over a $p$ -adic field with semi-stable reduction and let $\\omega$ be a meromorphic $1$-form on $X$. There are two notions of $p$-adic integration one may ass ociate to this data: the Berkovich–Coleman integral which can be perform ed locally\; and the Vologodsky integral with desirable number-theoretic p roperties. In this talk\, we present a theorem comparing the two\, and des cribe algorithms for computing these integrals in the case that $X$ is a h yperelliptic curve. We also illustrate our algorithm with a numerical exam ple computed in Sage. This talk is partly based on joint work with Eric Ka tz.\n LOCATION:https://researchseminars.org/talk/WarsawNT/25/ END:VEVENT BEGIN:VEVENT SUMMARY:Boris Adamczewski (Institut Camille Jordan & CNRS) DTSTART:20210329T111500Z DTEND:20210329T121500Z DTSTAMP:20260422T225839Z UID:WarsawNT/26 DESCRIPTION:Title: Furstenberg's conjecture\, Mahler's method\, and finite automata\nby Boris Adamczewski (Institut Camille Jordan & CNRS) as part of Warsaw Numb er Theory Seminar\n\nLecture held in currently online.\n\nAbstract\nIt is commonly expected that expansions of numbers in multiplicatively independe nt bases\, such as 2 and 10\, should have no common structure. However\, i t seems extraordinarily difficult to confirm this naive heuristic principl e in some way or another. In the late 1960s\, Furstenberg suggested a seri es of conjectures\, which became famous and aim to capture this heuristic. The work I will discuss in this talk is motivated by one of these conject ures. Despite recent remarkable progress by Shmerkin and Wu\, it remains t otally out of reach of the current methods. While Furstenberg’s conjectu res take place in a dynamical setting\, I will use instead the language of automata theory to formulate some related problems that formalize and exp ress in a different way the same general heuristic. I will explain how the latter can be solved thanks to some recent advances in Mahler’s method\ ; a method in transcendental number theory initiated by Mahler at the end of the 1920s. This a joint work with Colin Faverjon.\n LOCATION:https://researchseminars.org/talk/WarsawNT/26/ END:VEVENT BEGIN:VEVENT SUMMARY:Anna Szumowicz (Caltech) DTSTART:20210412T140000Z DTEND:20210412T150000Z DTSTAMP:20260422T225839Z UID:WarsawNT/27 DESCRIPTION:Title: Cuspidal types on GL_p(O)\nby Anna Szumowicz (Caltech) as part of Wa rsaw Number Theory Seminar\n\nLecture held in currently online.\n\nAbstrac t\nLet $F$ be a non-Archimedean local field and let $O$ be its ring of\nin tegers. We describe the cupidal types on $\\mathrm{GL}_p(O)$ (where $p$ is a prime\nnumber) using Clifford theory. This gives some information and\n invariants attached to cuspidal types called orbits. We give an\nexample w hich shows that the orbit of a representation does not give\nenough inform ation to determine whether a representation is a cuspidal\ntype or not.\n LOCATION:https://researchseminars.org/talk/WarsawNT/27/ END:VEVENT BEGIN:VEVENT SUMMARY:Rusen Li (Shandong University) DTSTART:20210419T111500Z DTEND:20210419T121500Z DTSTAMP:20260422T225839Z UID:WarsawNT/28 DESCRIPTION:Title: Summation formulas of q-hyperharmonic numbers\nby Rusen Li (Shandong University) as part of Warsaw Number Theory Seminar\n\nLecture held in cu rrently online.\n\nAbstract\nIn 1990\, Spieß gave some identities includi ng the types of $\\sum_{\\ell=1}^n\\ell^k H_\\ell$\, $\\sum_{\\ell=1}^n\\e ll^k H_{n-\\ell}$ and $\\sum_{\\ell=1}^n\\ell^k H_\\ell H_{n-\\ell}$. In t his talk\, based upon a certain type of $q$-harmonic numbers $H_n^{(r)}(q) $\, several formulas of $q$-hyperharmonic numbers are derived as $q$-gener alizations. The main tools used in the talk are Abel’s identity and a q- version of the relation by Spieß.\nThis is based on a joint work with Tak ao Komatsu.\n LOCATION:https://researchseminars.org/talk/WarsawNT/28/ END:VEVENT BEGIN:VEVENT SUMMARY:Valentijn Karemaker (Universiteit Utrecht) DTSTART:20210426T111500Z DTEND:20210426T121500Z DTSTAMP:20260422T225839Z UID:WarsawNT/29 DESCRIPTION:Title: Polarisations of abelian varieties over finite fields via canonical lift ings\nby Valentijn Karemaker (Universiteit Utrecht) as part of Warsaw Number Theory Seminar\n\nLecture held in currently online.\n\nAbstract\nIn this talk we will give a widely applicable and computable description of polarisations of abelian varieties over finite field. More precisely\, we will describe all polarisations of all abelian varieties over a finite fie ld in a fixed isogeny class corresponding to a squarefree Weil polynomial\ , when one variety in the isogeny class admits a canonical lifting to char acteristic zero. The computability of the description relies on applying c ategorical equivalences between abelian varieties over finite fields and f ractional ideals in étale algebras. \nThis is joint work with Jonas Bergs tröm and Stefano Marseglia.\n LOCATION:https://researchseminars.org/talk/WarsawNT/29/ END:VEVENT BEGIN:VEVENT SUMMARY:Robert Slob (Ulm University) DTSTART:20210510T111500Z DTEND:20210510T121500Z DTSTAMP:20260422T225839Z UID:WarsawNT/30 DESCRIPTION:Title: Primitive divisors of sequences associated to elliptic curves over funct ion fields\nby Robert Slob (Ulm University) as part of Warsaw Number T heory Seminar\n\nLecture held in currently online.\n\nAbstract\nIn the fir st part of the talk\, we give a gentle introduction into the subject of di visibility sequences over the rational numbers and discuss the notion of a primitive divisor/Zsigmondy bound. We then explain how these notions can be extended to number fields and function fields\, and how to obtain a div isibility sequence from a non-torsion point on an elliptic curve over any of these fields. There will also be plenty of nice examples.\n\nIn the sec ond part of the talk\, we discuss the typical methods that are used to pro ve the existence of a Zsigmondy bound for a divisibility sequence obtained from a non-torsion point on an elliptic curve $E$ over a number or functi on field $K$. Let $P$ be this non-torsion point in $E(K)$\, and suppose Q is a torsion point in $E(K)$. We can also associate a sequence of divisors $\\{D_{nP+Q}\\}$ on $K$ to the sequence of points $\\{nP+Q\\}$. In my pre print\, we proved the existence of a Zsigmondy bound for this sequence $\\ {D_{nP+Q}\\}$ for $K$ a function field (under some minor conditions)\, ext ending the analogous result of Verzobio over number fields. I will provide the crucial ideas to apply the existing methods of the case $\\{nP\\}$ to my case $\\{nP+Q\\}$. Additionally\, I will highlight the differences wit h the number field case.\n LOCATION:https://researchseminars.org/talk/WarsawNT/30/ END:VEVENT BEGIN:VEVENT SUMMARY:Marco Streng (Universiteit Leiden) DTSTART:20210517T111500Z DTEND:20210517T121500Z DTSTAMP:20260422T225839Z UID:WarsawNT/31 DESCRIPTION:Title: Obtaining modular units via a recurrence relation\nby Marco Streng ( Universiteit Leiden) as part of Warsaw Number Theory Seminar\n\nLecture he ld in currently online.\n\nAbstract\nThe modular curve $Y^1(N)$ parametris es pairs $(E\,P)$\, where $E$ is an elliptic curve and $P$ is a point of o rder $N$ on $E$. One tool for studying this curve is the group of modular units on it\, that is\, the group of algebraic functions with no poles or zeroes.\n\nWe first review how a recurrence relation (related to elliptic divisibility sequences) gives rise to defining equations for the curves $Y ^1(N)$. We then show that the same recurrence relation also gives explicit algebraic formulae for a basis of the group of units on $Y^1(N)$.\n\nThis proves a conjecture of Maarten Derickx and Mark van Hoeij.\n LOCATION:https://researchseminars.org/talk/WarsawNT/31/ END:VEVENT BEGIN:VEVENT SUMMARY:Eric Delaygue (University Lyon 1) DTSTART:20210524T111500Z DTEND:20210524T121500Z DTSTAMP:20260422T225839Z UID:WarsawNT/32 DESCRIPTION:Title: On primary pseudo-polynomials and around Ruzsa's Conjecture\nby Eric Delaygue (University Lyon 1) as part of Warsaw Number Theory Seminar\n\nL ecture held in currently online.\n\nAbstract\nEvery polynomial $P(X)$ with integer coefficients satisfies the congruences $P(n+m)=P(n) \\mod m$ for all integers $n$ and $m$. An integer valued sequence is called a pseudo-po lynomial when it satisfies these congruences. Hall characterized pseudo-po lynomials and proved that they are not necessarily polynomials. A long sta nding conjecture of Ruzsa says that a pseudo-polynomial $a(n)$ is a polyno mial as soon as $\\limsup |a_n|^{1/n} < e$. A primary pseudo-polynomial is an integer valued sequence $a(n)$ such that $a(n+p)=a(n) \\mod p$ for all integers $n ≥ 0$ and all prime numbers $p$. The same conjecture has bee n formulated for them\, which implies Ruzsa’s\, and this talk will revol ve around this conjecture.\n LOCATION:https://researchseminars.org/talk/WarsawNT/32/ END:VEVENT BEGIN:VEVENT SUMMARY:Daniel Vargas-Montoya (University Lyon 1) DTSTART:20210531T111500Z DTEND:20210531T121500Z DTSTAMP:20260422T225839Z UID:WarsawNT/33 DESCRIPTION:Title: Algebraicity modulo $p$ of G functions\, hypergeometric series and stron g Frobenius structure\nby Daniel Vargas-Montoya (University Lyon 1) as part of Warsaw Number Theory Seminar\n\nLecture held in currently online. \n\nAbstract\nB. Dwork in his work about zeta function of a hypersurface o ver finite fields introduced the notion of strong Frobenius structure. In this talk we are going to take up this notion for the study of algebraicit y modulo $p$ of Siegel G functions\, where $p$ is a prime number. \nFirst ly\, we are going to see that if $f(t)$ is a power series (or Siegel G fun ction) with coefficients in the ring of integers $\\mathbb{Z}$ and if $f(t )$ is solution of a differential operator $L$ having strong Frobenius stru cture for $p$ of period $h$\, then the reduction of $f$ modulo $p$ is alge braic over $\\mathbb{F}_p(t)$ and its algebraicity degree is bounded by $p ^{n^2h}$\, where $n$ is the order of L and $\\mathbb{F}_p$ is the field of $p$ elements. Secondly\, we are going to show that\, under reasonable hyp otheses\, rigid differential operators have a strong Frobenius structure for almost every prime number $p$.\nFinally\, we are going to illustrate our results with several examples coming of hypergeometric series of type ${}_nF_n-1$.\n LOCATION:https://researchseminars.org/talk/WarsawNT/33/ END:VEVENT BEGIN:VEVENT SUMMARY:Michael J. Schlosser (Unviersity of Vienna) DTSTART:20210607T140000Z DTEND:20210607T150000Z DTSTAMP:20260422T225839Z UID:WarsawNT/34 DESCRIPTION:Title: On the infinite Borwein product raised to a real power\nby Michael J . Schlosser (Unviersity of Vienna) as part of Warsaw Number Theory Seminar \n\nLecture held in currently online.\n\nAbstract\nWe study the $q$-series coefficients appearing in the expansion of $\\prod_{n\\ge 1}[(1-q^n)/(1-q ^{pn})]^\\delta$\, the infinite Borwein product for an arbitrary prime $p$ \, raised to an arbitrary positive real power $\\delta$. Application of th e Hardy-Ramanujan-Rademacher circle method gives an asymptotic formula for the coefficients. For $p=3$ we give an estimate of their growth which ena bles us to partially confirm an earlier conjecture we made concerning an o bserved sign pattern of the coefficients when the exponent $\\delta$ is wi thin a specified range of positive real numbers. We then take a closer loo k at the cube of the infinite Borwein product\, for arbitrary $p$ (now a p ositive integer)\, and establish some vanishing and divisibility propertie s of the respective coefficients.\nThis is joint work with Nian Hong Zhou. \n LOCATION:https://researchseminars.org/talk/WarsawNT/34/ END:VEVENT BEGIN:VEVENT SUMMARY:Jason Bell (University of Waterloo) DTSTART:20210614T111500Z DTEND:20210614T121500Z DTSTAMP:20260422T225839Z UID:WarsawNT/35 DESCRIPTION:Title: Effective isotrivial Mordell-Lang in positive characteristic\nby Jas on Bell (University of Waterloo) as part of Warsaw Number Theory Seminar\n \nLecture held in currently online.\n\nAbstract\nThe Mordell-Lang conjectu re (now a theorem\, proved by Faltings\, Vojta\, McQuillan\, …) asserts that if $G$ is a semiabelian variety $G$ defined over an algebraically clo sed field of characteristic zero\, $X$ is a subvariety of $G$\, and $\\Gam ma$ is a finite rank subgroup of $G$\, then $X\\cap \\Gamma$ is a finite u nion of cosets of $\\Gamma$. In positive characteristic\, the naive trans lation of this theorem does not hold\, however Hrushovski\, using model th eoretic techniques\, showed that in some sense all counterexamples arise f rom semiabelian varieties defined over finite fields (the isotrivial case) . This was later refined by Moosa and Scanlon\, who showed in the isotriv ial case that the intersection of a subvariety of a semiabelian variety $G $ with a finitely generated subgroup $\\Gamma$ of $G$ that is invariant un der the Frobenius endomorphism $F:G\\to G$ is a finite union of sets of th e form $S+A$\, where $A$ is a subgroup of $\\Gamma$ and $S$ is a sum of or bits under the map $F$. We show how how one can use finite-state automat a to give a concrete description of these intersections $\\Gamma\\cap X$ i n the isotrivial setting\, by constructing a finite machine that identifie s all points in the intersection. In particular\, this allows us to give d ecision procedures for answering questions such as: is $X\\cap \\Gamma$ em pty? finite? does it contain a coset of an infinite subgroup? In addition\ , we are able to read off coarse asymptotic estimates for the number of po ints of height $\\le H$ in the intersection from the machine. This is joi nt work with Dragos Ghioca and Rahim Moosa.\n LOCATION:https://researchseminars.org/talk/WarsawNT/35/ END:VEVENT BEGIN:VEVENT SUMMARY:Lars Kühne (University of Copenhagen) DTSTART:20210621T111500Z DTEND:20210621T121500Z DTSTAMP:20260422T225839Z UID:WarsawNT/36 DESCRIPTION:Title: Equidistribution and Uniformity in Families of Curves\nby Lars Kühn e (University of Copenhagen) as part of Warsaw Number Theory Seminar\n\nLe cture held in currently online.\n\nAbstract\nIn the talk\, I will present an equidistribution result for fa milies of (non-degenerate) subvarieties in a (general) family of abelian varieties. This extends a result of DeMar co and Mavraki for curves in fibered products of elliptic surfaces. Using this result\, one can deduce a uniform version of the classical Bogomolov conjecture for curves embed ded in their Jacobians\, namely that the numbe r of torsion points lying on them is uniformly bounded in the genus of the curve. This has been previously only known in a few select cases by work of David–Philippon and DeMarco–Krieger–Ye. Finally\, one can obtain a rather uniform version of the Mordell-Lang conjecture as well by complem enting a re sult of Dimitrov–Gao–Habegger: The number of rational poin ts on a smooth algebraic curve defined over a number field can be bounded solely in terms of its genus and the Mordell-Weil rank of its Jacobian. Ag ain\, this was previously known only under additional assumptions (Stoll\, Katz–Rabinoff–Zureick-Brown).\n LOCATION:https://researchseminars.org/talk/WarsawNT/36/ END:VEVENT BEGIN:VEVENT SUMMARY:Vasily Golyshev (Steklov Institute) DTSTART:20210628T111500Z DTEND:20210628T121500Z DTSTAMP:20260422T225839Z UID:WarsawNT/37 DESCRIPTION:Title: Modularity proofs via fibered motives.\nby Vasily Golyshev (Steklov Institute) as part of Warsaw Number Theory Seminar\n\nLecture held in curr ently online.\n\nAbstract\nI will explain how techniques of fibered hyperg eometric motives can be used to provide `opportunistic' modularity proofs for conifold fibers in Calabi-Yau families. This is a report on joint work with Don Zagier\, and work in progress with Kilian Bönisch and Albrecht Klemm.\n LOCATION:https://researchseminars.org/talk/WarsawNT/37/ END:VEVENT BEGIN:VEVENT SUMMARY:Maciej Ulas (Jagiellonian University\, Kraków) DTSTART:20220307T120000Z DTEND:20220307T130000Z DTSTAMP:20260422T225839Z UID:WarsawNT/38 DESCRIPTION:Title: On solutions of certain meta-Fibonacci recurrences\nby Maciej Ulas ( Jagiellonian University\, Kraków) as part of Warsaw Number Theory Seminar \n\nLecture held in currently online.\n\nAbstract\nDuring the talk I will speak about recent findings concerning the solutions of the recurrence seq uence $h(n)=h(n-h(n-1))+h(n-2)$. This is a member of a class of so called meta-Fibonacci (or exotic) sequences. We show that for a broad class of in itial conditions the behavior of the solutions is easy\, i.e.\, governed b y sequences satisfying linear recurrence with constant coefficients or is closely related to certain functions counting binary partitions of special type. The talk is based on a joint work with Bartosz Sobolewski.\n LOCATION:https://researchseminars.org/talk/WarsawNT/38/ END:VEVENT BEGIN:VEVENT SUMMARY:Neelam Saikia (University of Virginia) DTSTART:20220307T131500Z DTEND:20220307T141500Z DTSTAMP:20260422T225839Z UID:WarsawNT/39 DESCRIPTION:Title: Traces of pth Hecke operators and p-adic hypergeometric functions\nb y Neelam Saikia (University of Virginia) as part of Warsaw Number Theory S eminar\n\nLecture held in currently online.\n\nAbstract\nMcCarthy defined hypergeometric functions in the p-adic setting using p-adic gamma function s. This function can be described as p-adic analogue of classical hypergeo metric function. In this talk we discuss the traces of pth Hecke operators acting on spaces of cusp forms of weight k and level 1 and their relation s with p-adic hypergeometric functions. As a consequence of this result we establish relations of Ramanujan’s tau-function and p-adic hypergeometr ic functions. This is a joint work with Sudhir Pujahari.\n LOCATION:https://researchseminars.org/talk/WarsawNT/39/ END:VEVENT BEGIN:VEVENT SUMMARY:Pierre Colmez (Institut de Mathématiques de Jussieu-Paris Rive Ga uche) DTSTART:20220404T110000Z DTEND:20220404T120000Z DTSTAMP:20260422T225839Z UID:WarsawNT/40 DESCRIPTION:Title: The upper half-planes\nby Pierre Colmez (Institut de Mathématiques de Jussieu-Paris Rive Gauche) as part of Warsaw Number Theory Seminar\n\nL ecture held in IMPAN\, Warsaw\, conference room 6\, ground floor.\n\nAbstr act\nWe will give a short introduction to the geometric part of the p-adic Langlands program.\n LOCATION:https://researchseminars.org/talk/WarsawNT/40/ END:VEVENT BEGIN:VEVENT SUMMARY:Piotr Miska (Jagiellonian University) DTSTART:20220404T121500Z DTEND:20220404T131500Z DTSTAMP:20260422T225839Z UID:WarsawNT/41 DESCRIPTION:Title: (R)-dense and (N)-dense subsets of positive integers and generalized quo tient sets\nby Piotr Miska (Jagiellonian University) as part of Warsaw Number Theory Seminar\n\nLecture held in currently online (via Zoom).\n\n Abstract\nA subset $A$ of the set of positive integers is (R)-dense if it s quotient set $ R(A)=\\{a/b \\colon a\, b \\in A\\}$ is dense in the posi tive real half-line (with respect to natural topology on real numbers). It is a classical result that the set of prime numbers is (R)-dense. The pro of of this fact is based on the property of counting function of prime num bers. Actually\, this proof shows something more. Namely\, for each infini te subset $B$ of the set of positive integers\, the set $R(P\,B)=\\{p/b \\ colon p \\in P\, b \\in B\\}$ is dense in the set of positive real numbers . This motivates to introduce the notion of (N)-denseness. We say that a s et $A$ of positive integers is (N)-dense if the set $R(A\,B)$ is dense in the set of positive real numbers for every set $B$ of positive integers. D uring the talk we will consider characterizations of (N)-dense sets and co nnections between (N)-denseness of a given set.\nIn 2019 Leonetti and Sann a introduced the notion of direction sets $D^k(A)=\\{(a_1/\\|a\\|^2\, \\ld ots\, a_k/\\|a\\|^2)\\colon a=(a_1\,\\ldots\, a_k) \\in A^k\\}$ that allow s us to generalize the property of (R)-denseness. Indeed\, $A$ is (R)-dens e if and only if $D^2(A)$ is dense in the set of points of unit circle wit h all the coordinates positive. We will see that denseness of $D^k(A)$ in the set of points of unit sphere with all the coordinates positive is equi valent to denseness of the generalized quotient set $R^k(A)=\\{(a_1/a_k\,\ \ldots\, a_{k-1}/a_k)\\colon a_1\,\\ldots\, a_k \\in A\\}$ in the set of p oints of $R^{k-1}$ with all the coordinates positive.\nWe will also show s ome connections between (N)-denseness of a given set $A$ and denseness of sets $R^k(A)$ with the counting function of $A$ and its dispersion.\nThe t alk is based on a joint work with János T. Tóth.\n LOCATION:https://researchseminars.org/talk/WarsawNT/41/ END:VEVENT BEGIN:VEVENT SUMMARY:Gwladys Fernandes (Université de Versailles Saint-Quentin-en-Yvel ines) DTSTART:20220509T110000Z DTEND:20220509T120000Z DTSTAMP:20260422T225839Z UID:WarsawNT/42 DESCRIPTION:Title: Hypertranscendence of solutions of linear difference equations\nby G wladys Fernandes (Université de Versailles Saint-Quentin-en-Yvelines) as part of Warsaw Number Theory Seminar\n\nLecture held in currently online.\ n\nAbstract\nThe general question of this talk is the classification of di fferentially algebraic solutions of linear difference equations of the fol lowing type: \n\n$$ (\\ast) \\qquad a_0(z)f(z) + a_1(z)f(R(z)) + ... + a_ n(z)f(R_n(z))=0$$\n\nwhere\, for every $i$\, $a_i(z) \\in \\C(z)$\, $R(z) \\in \\C(z)$ and $R_n(z)$ is the $n$-th composition of $R(z)$ with itself. We say that such a function is differentially algebraic over $\\C(z)$ if there exist an non-zero integer $n$ and a non-zero polynomial $P \\in \\C( z)[X_0\,...\, X_n]$ such that $P(f(z)\,...\, f^{(n)}(z))=0$\, where $f^{(i )}$ is the $i$-th derivative of $f$ with respect to $z$. Otherwise\, it is hypertranscendental over $\\C(z)$.\n\nThe classification of differentiall y algebraic solutions is known for three types of non-linear difference eq uations : the Schröder's\, Böttcher's and Abel's equations : $f(qz)=R(f( z))$\, $f(z^d)=R(f(z))$\, $f(R(z))=f(z)+1$\, respectively\, where $q \\in \\C^{\\ast}$\, $d \\in \n$\, $d \\geq 2$. A classification of the differen tial algebraicity of solutions of linear difference equations of the above type $(\\ast)$ is made in an article of B. Adamczewski\, T. Dreyfus\, C. Hardouin\, for these same operators : q-differences $z \\to qz$\, mahleria n $z \\to z^d$\, and shift $z \\to z+1$\, by the means of an adapted diff erence Galois theory.\n\nIn this talk\, we discuss the generalisation of t hese results to any function $R$ (rational or algebraic over $\\C(z)$)\, i n the case where $(\\ast)$ is of order $1$. This is a work in progress wit h L. Di Vizio. Natural applications appear in examples of generating serie s of random walks\, which satisfy this kind of equation of order $1$.\n LOCATION:https://researchseminars.org/talk/WarsawNT/42/ END:VEVENT BEGIN:VEVENT SUMMARY:Matthias Storzer (MPIM Bonn) DTSTART:20220509T121500Z DTEND:20220509T131500Z DTSTAMP:20260422T225839Z UID:WarsawNT/43 DESCRIPTION:Title: Modularity of Nahm sums\nby Matthias Storzer (MPIM Bonn) as part of Warsaw Number Theory Seminar\n\nLecture held in currently online.\n\nAbstr act\nThe modularity properties of so-called Nahm sums are known to be rela ted to certain elements in the Bloch group. Nevertheless\, a first conject ure about the characterisation of modular Nahm sums in terms of these elem ents turned out to be false. In this talk\, we will review the motivation behind the conjecture and discuss why it fails\, which could lead to a ref ined version.\n LOCATION:https://researchseminars.org/talk/WarsawNT/43/ END:VEVENT BEGIN:VEVENT SUMMARY:Eugenia Rosu (TU Darmstadt and University​ of Leiden) DTSTART:20220606T110000Z DTEND:20220606T120000Z DTSTAMP:20260422T225839Z UID:WarsawNT/44 DESCRIPTION:Title: Twists of elliptic curves with CM\nby Eugenia Rosu (TU Darmstadt and University​ of Leiden) as part of Warsaw Number Theory Seminar\n\nLectu re held in IMPAN Warsaw\, ground floor\, room 6.\n\nAbstract\nWe consider certain families of sextic twists of the elliptic curve $y^2=x^3+1$ that a re not defined over $\\mathbb{Q}$\, but over $\\mathbb{Q}[\\sqrt{-3}]$. We compute a formula that relates the central value of their L-functions $L( E\, 1)$ to the square of a trace of a modular function evaluated at a CM p oint. Assuming the Birch and Swinnerton-Dyer conjecture\, when the value a bove is non-zero\, we should recover the order of the Tate-Shafarevich gro up\, and we show that the value is indeed an integer square.\n LOCATION:https://researchseminars.org/talk/WarsawNT/44/ END:VEVENT BEGIN:VEVENT SUMMARY:Daniel Vargas Montoya (IMPAN) DTSTART:20220606T121500Z DTEND:20220606T131500Z DTSTAMP:20260422T225839Z UID:WarsawNT/45 DESCRIPTION:Title: q-strong Frobenius structure\nby Daniel Vargas Montoya (IMPAN) as pa rt of Warsaw Number Theory Seminar\n\nLecture held in IMPAN\, Warsaw\, con ference room 6\, ground floor\,.\n\nAbstract\nThe notion of strong Frobeni us structure is classically studied in the theory of $p$-adic differential equations. This notion was introduced by B.Dwork in his study of zeta fun ctions. Recently\, we propose a new definition of this notion for q-diffe rence operators. The relevance of this definition is supported for two res ults. The first one deals with confluence and the second one deals with co ngruence modulo the cyclotomic polynomial. So the first part of the talk i s devoted to presenting our definition of q-strong Frobenius structure and the second part we are going to present the two previous results.\n LOCATION:https://researchseminars.org/talk/WarsawNT/45/ END:VEVENT END:VCALENDAR