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  • $p$ -ADIC $L$ -FUNCTIONS FOR UNITARY GROUPS
    Forum Math. Pi (IF 3.857) Pub Date : 2020-05-06
    ELLEN EISCHEN; MICHAEL HARRIS; JIANSHU LI; CHRISTOPHER SKINNER

    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

    更新日期:2020-07-07
  • ENDOSCOPY FOR HECKE CATEGORIES, CHARACTER SHEAVES AND REPRESENTATIONS
    Forum Math. Pi (IF 3.857) Pub Date : 2020-05-28
    GEORGE LUSZTIG; ZHIWEI YUN

    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

    更新日期:2020-05-28
  • BRANCH GROUPS, ORBIT GROWTH, AND SUBGROUP GROWTH TYPES FOR PRO- $p$ GROUPS
    Forum Math. Pi (IF 3.857) Pub Date : 2020-05-26
    YIFTACH BARNEA; JAN-CHRISTOPH SCHLAGE-PUCHTA

    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

    更新日期:2020-05-26
  • HALF-SPACE MACDONALD PROCESSES
    Forum Math. Pi (IF 3.857) Pub Date : 2020-05-26
    GUILLAUME BARRAQUAND; ALEXEI BORODIN; IVAN CORWIN

    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

    更新日期:2020-05-26
  • THE EXACT MINIMUM NUMBER OF TRIANGLES IN GRAPHS WITH GIVEN ORDER AND SIZE
    Forum Math. Pi (IF 3.857) Pub Date : 2020-04-20
    HONG LIU; OLEG PIKHURKO; KATHERINE STADEN

    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

    更新日期:2020-04-20
  • ON THE COHOMOLOGY OF TORELLI GROUPS
    Forum Math. Pi (IF 3.857) Pub Date : 2020-04-13
    ALEXANDER KUPERS; OSCAR RANDAL-WILLIAMS

    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.

    更新日期:2020-04-13
  • THE DE BRUIJN–NEWMAN CONSTANT IS NON-NEGATIVE
    Forum Math. Pi (IF 3.857) Pub Date : 2020-04-06
    BRAD RODGERS; TERENCE TAO

    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

    更新日期:2020-04-06
  • SERRE WEIGHTS AND BREUIL’S LATTICE CONJECTURE IN DIMENSION THREE
    Forum Math. Pi (IF 3.857) Pub Date : 2020-03-25
    DANIEL LE; BAO V. LE HUNG; BRANDON LEVIN; STEFANO MORRA

    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

    更新日期:2020-03-25
  • ENUMERATION OF MEANDERS AND MASUR–VEECH VOLUMES
    Forum Math. Pi (IF 3.857) Pub Date : 2020-03-23
    VINCENT DELECROIX; ÉLISE GOUJARD; PETER ZOGRAF; ANTON ZORICH

    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

    更新日期:2020-03-23
  • PRIMES REPRESENTED BY INCOMPLETE NORM FORMS
    Forum Math. Pi (IF 3.857) Pub Date : 2020-02-06
    JAMES MAYNARD

    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 $K=\mathbb{Q}(\sqrt[n]{\unicode[STIX]{x1D703}})$

    更新日期:2020-02-06
  • CHARACTER LEVELS AND CHARACTER BOUNDS
    Forum Math. Pi (IF 3.857) Pub Date : 2020-01-24
    ROBERT M. GURALNICK; MICHAEL LARSEN; PHAM HUU TIEP

    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

    更新日期:2020-01-24
  • MOMENTS OF RANDOM MULTIPLICATIVE FUNCTIONS, I: LOW MOMENTS, BETTER THAN SQUAREROOT CANCELLATION, AND CRITICAL MULTIPLICATIVE CHAOS
    Forum Math. Pi (IF 3.857) Pub Date : 2020-01-20
    ADAM J. HARPER

    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

    更新日期:2020-01-20
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