As early as the 1930s, P\'al Erd\H{o}s conjectured that: {\em for any multiplicative function $f:\mathbb{N}\to\{-1,1\}$, the partial sums $\sum_{n\leq x}f(n)$ are unbounded.} Considering this conjecture, in this paper we consider multiplicative functions $f$ satisfying $$\sum_{p\leq x}f(p)=c\cdot\frac{x}{\log x}(1+o(1)).$$ We prove that if $c>0$ then the partial sums of $f$ are unbounded, and if $c<0$ then the partial sums of $\mu f$ are unbounded. Extensions of this result are also discussed.

Let $f$ be a cuspidal newform of weight $k$ on $\Gamma_0(Np)$ with $(p,N)=1$. On one hand, we may associate to $f$ an automorphic representation $\pi(f)=\bigotimes_\ell \pi_\ell(f)$ of $\mathrm{GL}_2(\mathbb{A})$. On the other, Deligne taught us how to attach a continuous, $2$-dimensional a $p$-adic Galois representation $V_f$ to $f$. The local Langlands correspondence describes how, for $(\ell, pN)=1$, the local factor $\pi_\ell(f)$ and the restriction $V_{f,\ell}$ of $V_f$ to $\mathrm{Gal}(\bar{\mathbb{Q}}_\ell/\mathbb{Q}_\ell)$ determine one another. The situation is different when $\ell=p$: $V_{f,p}$ is a much richer object than $\pi_p(f)$. The $p$-adic Langlands philosophy pioneered by Breuil predicts that there is a refined $p$-adic correspondence under which $V_{f,p}$ is completely determined by a suitable $p$-adic refinement of $\pi_p(f)$. Breuil proposed a candidate for this refined representation and showed that it can be found in the completed cohomology of a tower of modular curves. In this report on ongoing joint work with Samit Dasgupta, I will explain how Breuil's results can be extended to the situation of Shimura curves.

The speaker plans to sketch some ideas which lead to the following, weaker and conditional Theorem. If the level of distribution of primes exceeds ½ then there exists a bounded even number d (different from zero) with the property that there are arbitrarily long arithmetic progressions of primes p such that p+d is also a prime for each element of the progression. The bound for d depends on the level of distribution of primes. If we suppose the Elliott-Halberstam conjecture (that is a level equal to 1, or even at least 0.971) then the theorem is true with a d not exceeding 16.

Several other results will be mentioned, among with a condition (requiring a hypothetical distribution level at least ¾ for the primes and some other sequences involving the Liouville function and/or the primes) which imply an affirmative answer for the question in the title. Also some unconditional results of similar type will be mentioned for primes, products of two primes and some other numbers with special multiplicative constraints like d(n)=d(n+1).

It is believed an automorphic forms of large spectral parameter is, in some sense, approximately uniformly distributed over the locally symmetric space. I will discuss some results towards this conjecture.

An amicable pair for an elliptic curve E/Q is a pair of primes (p,q) of good reduction for E satisfying #E(F_{p}) = q and #E(F_{q}) = p.
Aliquot cycles are analogously defined longer cycles. Although rare for non-CM curves, amicable pairs are -- surprisingly -- relatively abundant in the CM
case. We present heuristics and conjectures for the frequency of amicable pairs and aliquot cycles, and some results for the CM case (including the
especially intricate j=0 case). This is joint work with Joseph H. Silverman.

Slides: pdf.

Loic Merel proved that for every positive integer d, there
is a bound B_{d} such that if E is an elliptic curve over a number field
K of degree d, then #E(K)_{tor} ≤ B_{d}. A natural question is then to
classify all possibilities for E(K)_{tor}, as E/K varies over all
elliptic curves over degree d number fields. When d=1, Barry Mazur
answered this question, in his famous Steele-prize winning 1977 paper
"Modular Curves and the Eisenstein Ideal". When d=2 and d=3 this
question has also been answered, due to work of Kenku-Momose,
Abromovich, Kamienny, Merel, Parent and others. This talk will be
mainly about the still
unresolved case d=4, toward which much recent progress has been made.
This is joint work with Sheldon Kamienny.

Slides: pdf or sage worksheet.

I will discuss to what extent it is possible with currently available methods to explicitly determine the (finite) set of rational points on a given curve of genus at least 2 that is defined over the rational numbers. I will present a number of examples where these methods have been successful, and I will explain what is still missing to turn these methods into an algorithm.

Slides: pdf.

The zeros of Eisenstein series satisfy many intriguing properties. A classical result of F. K. C. Rankin and Swinnerton-Dyer shows that the zeros all lie on a certain arc of the fundamental domain for the full modular group. More recently, Nozaki demonstrated an interlacing property for zeros of Eisentstein series whose weights differ by 12. We extend these results to certain Eisenstein series of level 2, and highlight a connection between these zeros and the Jacobi elliptic function cn(u). This is joint work with S. Garthwaite, L. Long, and H. Swisher.

Slides: pdf.