Publisher: Cambridge University Press   (Total: 388 journals)

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 Forum of Mathematics, PiNumber of Followers: 1     Open Access journal ISSN (Online) 2050-5086 Published by Cambridge University Press  [388 journals]
• ENDOSCOPY FOR HECKE CATEGORIES, CHARACTER SHEAVES AND REPRESENTATIONS

• Authors: GEORGE LUSZTIG; ZHIWEI YUN
Abstract: For a reductive group $G$ over a finite field, we show that the neutral block of its mixed Hecke category with a fixed monodromy under the torus action is monoidally equivalent to the mixed Hecke category of the corresponding endoscopic group $H$ with trivial monodromy. We also extend this equivalence to all blocks. We give two applications. One is a relationship between character sheaves on $G$ with a fixed semisimple parameter and unipotent character sheaves on the endoscopic group $H$ , after passing to asymptotic versions. The other is a similar relationship between representations of $G(\mathbb{F}_{q})$ with a fixed semisimple parameter and unipotent representations of $H(\mathbb{F}_{q})$ .
PubDate: 2020-05-28T00:00:00.000Z
DOI: 10.1017/fmp.2020.9
Issue No: Vol. 8 (2020)

• ++++++++ ++++++++ ++++++++++++ ++++++++++++++++$p$ ++++++++++++ ++++++++ +++++GROUPS&rft.title=Forum+of+Mathematics,+Pi&rft.issn=2050-5086&rft.date=2020&rft.volume=8&rft.aulast=BARNEA&rft.aufirst=YIFTACH&rft.au=YIFTACH+BARNEA&rft.au=JAN-CHRISTOPH+SCHLAGE-PUCHTA&rft_id=info:doi/10.1017/fmp.2020.8">BRANCH GROUPS, ORBIT GROWTH, AND SUBGROUP GROWTH TYPES FOR PRO- $p$ GROUPS

• Authors: YIFTACH BARNEA; JAN-CHRISTOPH SCHLAGE-PUCHTA
Abstract: In their book Subgroup Growth, Lubotzky and Segal asked: What are the possible types of subgroup growth of the pro- $p$ group? In this paper, we construct certain extensions of the Grigorchuk group and the Gupta–Sidki groups, which have all possible types of subgroup growth between $n^{(\log n)^{2}}$ and $e^{n}$ . Thus, we give an almost complete answer to Lubotzky and Segal’s question. In addition, we show that a class of pro- $p$ branch groups, including the Grigorchuk group and the Gupta–Sidki groups, all have subgroup growth type $n^{\log n}$ .
PubDate: 2020-05-26T00:00:00.000Z
DOI: 10.1017/fmp.2020.8
Issue No: Vol. 8 (2020)

• HALF-SPACE MACDONALD PROCESSES

• Authors: GUILLAUME BARRAQUAND; ALEXEI BORODIN, IVAN CORWIN
Abstract: Macdonald processes are measures on sequences of integer partitions built using the Cauchy summation identity for Macdonald symmetric functions. These measures are a useful tool to uncover the integrability of many probabilistic systems, including the Kardar–Parisi–Zhang (KPZ) equation and a number of other models in its universality class. In this paper, we develop the structural theory behind half-space variants of these models and the corresponding half-space Macdonald processes. These processes are built using a Littlewood summation identity instead of the Cauchy identity, and their analysis is considerably harder than their full-space counterparts.We compute moments and Laplace transforms of observables for general half-space Macdonald measures. Introducing new dynamics preserving this class of measures, we relate them to various stochastic processes, in particular the log-gamma polymer in a half-quadrant (they are also related to the stochastic six-vertex model in a half-quadrant and the half-space ASEP). For the polymer model, we provide explicit integral formulas for the Laplace transform of the partition function. Nonrigorous saddle-point asymptotics yield convergence of the directed polymer free energy to either the Tracy–Widom (associated to the Gaussian orthogonal or symplectic ensemble) or the Gaussian distribution depending on the average size of weights on the boundary.
PubDate: 2020-05-26T00:00:00.000Z
DOI: 10.1017/fmp.2020.3
Issue No: Vol. 8 (2020)

• ++++++++ ++++++++ ++++++++++++ ++++++++++++++++$p$ ++++++++++++ ++++++++ ++++-ADIC+ ++++++++ ++++++++ ++++++++++++ ++++++++++++++++$L$ ++++++++++++ ++++++++ ++++-FUNCTIONS+FOR+UNITARY+GROUPS&rft.title=Forum+of+Mathematics,+Pi&rft.issn=2050-5086&rft.date=2020&rft.volume=8&rft.aulast=EISCHEN&rft.aufirst=ELLEN&rft.au=ELLEN+EISCHEN&rft.au=MICHAEL+HARRIS,+JIANSHU+LI,+CHRISTOPHER+SKINNER&rft_id=info:doi/10.1017/fmp.2020.4">$p$ -ADIC $L$ -FUNCTIONS FOR UNITARY GROUPS

• Authors: ELLEN EISCHEN; MICHAEL HARRIS, JIANSHU LI, CHRISTOPHER SKINNER
Abstract: This paper completes the construction of $p$ -adic $L$ -functions for unitary groups. More precisely, in Harris, Li and Skinner [‘ $p$ -adic $L$ -functions for unitary Shimura varieties. I. Construction of the Eisenstein measure’, Doc. Math.Extra Vol. (2006), 393–464 (electronic)], three of the authors proposed an approach to constructing such $p$ -adic $L$ -functions (Part I). Building on more recent results, including the first named author’s construction of Eisenstein measures and $p$ -adic differential operators [Eischen, ‘A $p$ -adic Eisenstein measure for unitary groups’, J. Reine Angew. Math.699 (2015), 111–142; ‘ $p$ -adic differential operators on automorphic forms on unitary groups’, Ann. Inst. Fourier (Grenoble)62(1) (2012), 177–243], Part II of the present paper provides the calculations of local $\unicode[STIX]{x1D701}$ -integrals occurring in the Euler product (including at
PubDate: 2020-05-06T00:00:00.000Z
DOI: 10.1017/fmp.2020.4
Issue No: Vol. 8 (2020)

• THE EXACT MINIMUM NUMBER OF TRIANGLES IN GRAPHS WITH GIVEN ORDER AND SIZE

• Authors: HONG LIU; OLEG PIKHURKO, KATHERINE STADEN
Abstract: What is the minimum number of triangles in a graph of given order and size? Motivated by earlier results of Mantel and Turán, Rademacher solved the first nontrivial case of this problem in 1941. The problem was revived by Erdős in 1955; it is now known as the Erdős–Rademacher problem. After attracting much attention, it was solved asymptotically in a major breakthrough by Razborov in 2008. In this paper, we provide an exact solution for all large graphs whose edge density is bounded away from  $1$ , which in this range confirms a conjecture of Lovász and Simonovits from 1975. Furthermore, we give a description of the extremal graphs.
PubDate: 2020-04-20T00:00:00.000Z
DOI: 10.1017/fmp.2020.7
Issue No: Vol. 8 (2020)

• ON THE COHOMOLOGY OF TORELLI GROUPS

• Authors: ALEXANDER KUPERS; OSCAR RANDAL-WILLIAMS
Abstract: We completely describe the algebraic part of the rational cohomology of the Torelli groups of the manifolds $\#^{g}S^{n}\times S^{n}$ relative to a disc in a stable range, for $2n\geqslant 6$ . Our calculation is also valid for $2n=2$ assuming that the rational cohomology groups of these Torelli groups are finite-dimensional in a stable range.
PubDate: 2020-04-13T00:00:00.000Z
DOI: 10.1017/fmp.2020.5
Issue No: Vol. 8 (2020)

• THE DE BRUIJN–NEWMAN CONSTANT IS NON-NEGATIVE

• Authors: BRAD RODGERS; TERENCE TAO
Abstract: For each $t\in \mathbb{R}$ , we define the entire function $$\begin{eqnarray}H_{t}(z):=\int _{0}^{\infty }e^{tu^{2}}\unicode[STIX]{x1D6F7}(u)\cos (zu)\,du,\end{eqnarray}$$ where $\unicode[STIX]{x1D6F7}$ is the super-exponentially decaying function $$\begin{eqnarray}\unicode[STIX]{x1D6F7}(u):=\mathop{\sum }_{n=1}^{\infty }(2\unicode[STIX]{x1D70B}^{2}n^{4}e^{9u}-3\unicode[STIX]{x1D70B}n^{2}e^{5u})\exp (-\unicode[STIX]{x1D70B}n^{2}e^{4u}).\end{eqnarray}$$ Newman showed that there exists a finite constant $\unicode[STIX]{x1D6EC}$ (the de Bruijn–Newman constant) such that the zeros of $H_{t}$ are all real precisely when $t\geqslant \unicode[STIX]{x1D6EC}$ . The Riemann hypothesis is equivalent to the assertion $\unicode[STIX]{x1D6EC}\leqslant 0$ , and Newman conjectured the complementary bound $\unicode[STIX]{x1D6EC}\geqslant 0$ . In this paper, we establish Newman’s conjecture. The argument proceeds by assuming for contradiction that $\unicode[STIX]{x1D6EC} PubDate: 2020-04-06T00:00:00.000Z DOI: 10.1017/fmp.2020.6 Issue No: Vol. 8 (2020) • SERRE WEIGHTS AND BREUIL’S LATTICE CONJECTURE IN DIMENSION THREE • Authors: DANIEL LE; BAO V. LE HUNG, BRANDON LEVIN, STEFANO MORRA Abstract: We prove in generic situations that the lattice in a tame type induced by the completed cohomology of a$U(3)$-arithmetic manifold is purely local, that is, only depends on the Galois representation at places above$p$. This is a generalization to$\text{GL}_{3}$of the lattice conjecture of Breuil. In the process, we also prove the geometric Breuil–Mézard conjecture for (tamely) potentially crystalline deformation rings with Hodge–Tate weights$(2,1,0)$as well as the Serre weight conjectures of Herzig [‘The weight in a Serre-type conjecture for tame$n$-dimensional Galois representations’, Duke Math. J. 149(1) (2009), 37–116] over an unramified field extending the results of Le et al. [‘Potentially crystalline deformation 3985 rings and Serre weight conjectures: shapes and shadows’, Invent. Math. 212(1) (2018), 1–107]. We also prove results in modular representation theory about lattices in Deligne–Lusztig representations for the group$\text{GL}_{3}(\mathbb{F}_{q})$. PubDate: 2020-03-25T00:00:00.000Z DOI: 10.1017/fmp.2020.1 Issue No: Vol. 8 (2020) • ENUMERATION OF MEANDERS AND MASUR–VEECH VOLUMES • Authors: VINCENT DELECROIX; ÉLISE GOUJARD, PETER ZOGRAF, ANTON ZORICH Abstract: A meander is a topological configuration of a line and a simple closed curve in the plane (or a pair of simple closed curves on the 2-sphere) intersecting transversally. Meanders can be traced back to H. Poincaré and naturally appear in various areas of mathematics, theoretical physics and computational biology (in particular, they provide a model of polymer folding). Enumeration of meanders is an important open problem. The number of meanders with$2N$crossings grows exponentially when$N$grows, but the long-standing problem on the precise asymptotics is still out of reach.We show that the situation becomes more tractable if one additionally fixes the topological type (or the total number of minimal arcs) of a meander. Then we are able to derive simple asymptotic formulas for the numbers of meanders as$N$tends to infinity. We also compute the asymptotic probability of getting a simple closed curve on a sphere by identifying the endpoints of two arc systems (one on each of the two hemispheres) along the common equator.The new tools we bring to bear are based on interpretation of meanders as square-tiled surfaces with one horizontal and one vertical cylinder. The proofs combine recent results on Masur–Veech volumes of moduli spaces of meromorphic quadratic differentials in genus zero with our new observation that horizontal and vertical separatrix diagrams of integer quadratic differentials are asymptotically uncorrelated. The additional combinatorial constraints we impose in this article yield explicit polynomial asymptotics. PubDate: 2020-03-23T00:00:00.000Z DOI: 10.1017/fmp.2020.2 Issue No: Vol. 8 (2020) • PRIMES REPRESENTED BY INCOMPLETE NORM FORMS • Authors: JAMES MAYNARD Abstract: Let$K=\mathbb{Q}(\unicode[STIX]{x1D714})$with$\unicode[STIX]{x1D714}$the root of a degree$n$monic irreducible polynomial$f\in \mathbb{Z}[X]$. We show that the degree$n$polynomial$N(\sum _{i=1}^{n-k}x_{i}\unicode[STIX]{x1D714}^{i-1})$in$n-k$variables takes the expected asymptotic number of prime values if$n\geqslant 4k$. In the special case$K=\mathbb{Q}(\sqrt[n]{\unicode[STIX]{x1D703}})$, we show that$N(\sum _{i=1}^{n-k}x_{i}\sqrt[n]{\unicode[STIX]{x1D703}^{i-1}})$takes infinitely many prime values, provided$n\geqslant 22k/7$.Our proof relies on using suitable ‘Type I’ and ‘Type II’ esti... PubDate: 2020-02-06T00:00:00.000Z DOI: 10.1017/fmp.2019.8 Issue No: Vol. 8 (2020) • CHARACTER LEVELS AND CHARACTER BOUNDS • Authors: ROBERT M. GURALNICK; MICHAEL LARSEN, PHAM HUU TIEP Abstract: We develop the concept of character level for the complex irreducible characters of finite, general or special, linear and unitary groups. We give characterizations of the level of a character in terms of its Lusztig label and in terms of its degree. Then we prove explicit upper bounds for character values at elements with not-too-large centralizers and derive upper bounds on the covering number and mixing time of random walks corresponding to these conjugacy classes. We also characterize the level of the character in terms of certain dual pairs and prove explicit exponential character bounds for the character values, provided that the level is not too large. Several further applications are also provided. Related results for other finite classical groups are obtained in the sequel [Guralnick et al. ‘Character levels and character bounds for finite classical groups’, Preprint, 2019, arXiv:1904.08070] by different methods. PubDate: 2020-01-24T00:00:00.000Z DOI: 10.1017/fmp.2019.9 Issue No: Vol. 8 (2020) • MOMENTS OF RANDOM MULTIPLICATIVE FUNCTIONS, I: LOW MOMENTS, BETTER THAN SQUAREROOT CANCELLATION, AND CRITICAL MULTIPLICATIVE CHAOS • Authors: ADAM J. HARPER Abstract: We determine the order of magnitude of$\mathbb{E}|\sum _{n\leqslant x}f(n)|^{2q}$, where$f(n)$is a Steinhaus or Rademacher random multiplicative function, and$0\leqslant q\leqslant 1$. In the Steinhaus case, this is equivalent to determining the order of$\lim _{T\rightarrow \infty }\frac{1}{T}\int _{0}^{T}|\sum _{n\leqslant x}n^{-it}|^{2q}\,dt$.In particular, we find that$\mathbb{E}|\sum _{n\leqslant x}f(n)|\asymp \sqrt{x}/(\log \log x)^{1/4}$. This proves a conjecture of Helson that one should have better than squareroot cancellation in the first moment and disproves counter-conjectures of various other authors. We deduce some consequences for the distribution and large deviations of$\sum _{n\leqslant x}f(n)$.The proofs develop a connection between$\mathbb{E}|\sum _{n\leqslant x}f(n)|^{2q}$and the$q\$ th moment of a critical, approximately Gaussian, multiplicative chaos and then establish the required estimates for that. We include some general introductory discussion about critical multiplicative chaos to help readers unfamiliar with that area.
PubDate: 2020-01-20T00:00:00.000Z
DOI: 10.1017/fmp.2019.7
Issue No: Vol. 8 (2020)

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