Consider an elliptic curve defined over the rational numbers and embedded in projective space. The rational points on the curve are viewed as integer vectors with coprime coordinates. What can be said about the rational points for which the number of prime factors dividing a fixed coordinate does not exceed a fixed bound? If the bound is zero, then Siegel's theorem guarantees that there are only finitely many such points. We consider, theoretically and computationally, two conjectures: one is a generalization of Siegel's theorem, and the other is a refinement that resonates with Hall's conjecture.

This paper provides a proof of a $p$-adic character formula by means of motivic integration. We use motivic integration to produce virtual Chow motives that control the values of the characters of all depth-zero supercuspidal representations on all topologically unipotent elements of $p$}-adic $\SL(2)$; likewise, we find motives for the values of the Fourier transform of all regular elliptic orbital integrals having minimal nonnegative depth in their own Cartan subalgebra, on all topologically nilpotent elements of $p$-adic $\mathfrak{sl}(2)$. We then find identities in the ring of virtual Chow motives over $\mathbb{Q}$ that relate these two classes of motives. These identities provide explicit expressions for the values of characters of all depth-zero supercuspidal representations of $p$}-adic $\SL(2)$ as linear combinations of Fourier transforms of semisimple orbital integrals, thus providing a proof of a $p$-adic character formula.

A discrete Morse function on a simplicial complex describes how to construct a homotopy-equivalent CW-complex with possibly fewer cells. We associate a Boolean function with a given simplicial complex and construct a discrete Morse function using its Fourier transform.

Methods from theoretical computer science by O’Donnell, Saks, Schramm, and Servedio, together with experimental data on complexes from Hachimori’s library and on chessboard complexes, provide some evidence that the constructed discrete Morse functions are efficient.

In this paper, we consider the sequence of Frobenius conjugacy classes for a Galois extension $K/\QQ$, ordered by the increasing sequence of rational primes. For a given $K$, we look at the frequencies of nonoverlapping consecutive $k$-tuples in this sequence. We compare these frequencies to what would be expected by the Cebotarev density theorem if there were statistical independence between successive Frobenius classes. We find striking variations of behavior as $K$ varies.

On the one hand, it is well known that Jacobians of (hyper)elliptic curves defined over $\Q$ having a rational point of order l can be used in many applications, for instance in the construction of class groups of quadratic fields with a nontrivial l-rank. On the other hand, it is also well known that 11 is the least prime number that is not the order of a rational point of an elliptic curve defined over $\Q$. It is therefore interesting to look for curves of higher genus whose Jacobians have a rational point of order 11. This problem has already been addressed, and Flynn found such a family $\Fl_t$ of genus-2 curves. Now it turns out that the Jacobian $J_0(23)$ of the modular genus-2 curve $X_0(23)$ has the required property, but does not belong to $\Fl_t$. The study of $X_0(23)$ leads to a method giving a partial solution of the considered problem. Our approach allows us to recover $X_0(23)$ and to construct another 18 distinct explicit curves of genus 2 defined over $\Q$ whose Jacobians have a rational point of order 11. Of these 19 curves, 10 do not have any rational Weierstrass point, and 9 have a rational Weierstrass point. None of these curves are $\Qb$-isomorphic to each other, nor $\Qb$-isomorphic to an element of Flynn's family $\Fl_t$. Finally, the Jacobians of these new curves are absolutely simple.

In this paper we present two algorithms: the first tests the projectivity of a smooth complete toric variety and the second determines the extremal classes of the Mori cone of a smooth projective toric variety. The crucial fact is that we are able to give a complete description of $\aunox$, determining a basis $B$ of $\aunox$ and the coordinates with respect to $B$ of any element of $\aunox$. The computational condition testing the projectivity is obtained by Kleiman's criterion of ampleness, while the condition determining the extremality of a class comes directly from the definition of a nonextremal class. The algorithms are used to study the Mori cone of Fano toric $n$-folds with dimension $n\leq 4$ and Picard number $\rho \geq 3$, computing all extremal rays of the Mori cone. Moreover, we describe a toric almost Fano variety of dimension $3$ and Picard number $35$ together with its Mori cone.

We introduce a way of describing cohomology of the symmetric groups $\Sig n$ with coefficients in Specht modules. We study $\HlR i$ for $i \in \{0,1,2\}$ and $R = \Z$, $\Fp$. The focus lies on the isomorphism type of $\Hlz 2$. Unfortunately, only in few cases can we determine this exactly. In many cases we obtain only some information about the prime divisors of $|\Hlz 2|$. The most important tools we use are the Zassenhaus algorithm, the branching rules, Bockstein-type homomorphisms, and the results from Burichenko et al., 1996.

We present some conjectures and open problems on partition hook lengths motivated by known results on the subject. The conjectures were suggested by extensive experimental calculations using a computer algebra system. The first conjecture unifies two classical results on the number of standard Young tableaux and the number of pairs of standard Young tableaux of the same shape. The second unifies the classical hook formula and the marked hook formula. The third includes the longstanding Lehmer conjecture, which says that the Ramanujan tau function never assumes the value zero. The fourth is a more precise version of the third in the case of $3$-cores. We also list some open problems on partition hook lengths.

Herein we develop connections between zeta functions and some recent ``mysterious'' constants of nonlinear physics. In an important analysis of coupled Winfree oscillators, Quinn, Rand, and Strogatz developed a certain $N$-oscillator scenario whose bifurcation phase offset $\phi$ is implicitly defined, with a conjectured asymptotic behavior $\sin \phi \sim 1 - c_1/N$, with experimental estimate $c_1 = 0.605443657\dotsc$. We are able to derive the exact theoretical value of this ``QRS constant'' $c_1$ as a real zero of a particular Hurwitz zeta function. This discovery enables, for example, the rapid resolution of $c_1$ to extreme precision. Results and conjectures are provided in regard to higher-order terms of the $\sin \phi$ asymptotic, and to yet more physics constants emerging from the original QRS work.

Donaldson gives three operators on a space of Hermitian metrics on a complex projective manifold: $T, T_{\nu}, T_K$. Iterations of these operators converge to balanced metrics, and these themselves approximate constant scalar curvature metrics. In this paper we investigate the convergence properties of these iterations by examining the case of the Riemann sphere as well as higher-dimensional $\mathbb{CP}^n$.