15 problems

Given a modular polynomial equation $f(x_1,\ldots,x_n)\equiv 0\pmod N$ and the associated lattice-based Coppersmith method, determine the optimal asymptotic region of exponents…

Let $\xi$ and $\xi'$ be the cycles associated with Algorithm, and suppose $n\leq4$ and $\|\xi\|>1$. For each modular symbol ${\mathbf v}$ in step A of the algorithm, choose…

Let ${\mathbf v}=[v_1,\dots,v_n]$ be a modular symbol with $\|{\mathbf v}\|>1$, and define … Let $E$ be the standard basis of ${\mathbb Z}^n$. The quadratic form $b({\mathbf v})$ i…

Let ${\mathbf v}=[v_1,\dots,v_n]$ be a modular symbol with $\|{\mathbf v}\|>1$. Define … For a top-dimensional cone $\sigma$ in the Voronoi decomposition containing…

Let $A>0$ be an integer. Suppose that $(M_1x+a_1)(M_2x+a_2)$ has no fixed prime divisors as $x$ ranges over the integers, with $M_1,M_2>0$ and $0\leq a_i<M_i$ for $i=1,2$. Strong H…

Non-negligible-prime-ideal conjecture. The proportion of such coefficient vectors is inverse-polynomial in $d$ and $\log(B)$:

Let $m$ be a positive integer, let $\mathbb{Q}(\zeta_m+\zeta_m^{-1})$ be the maximal real subfield of the $m$th cyclotomic field, and let $h^+(m)$ denote its class number. Polynomi…

Let $a,b,d$ be the minimizers of Problem, and let $\mathbf A,\mathbf B\in\mathbb R[[X,Y]]$ be the series associated to $a$ and $b$, respectively. Let $\mathcal C^{[1/3,2/3]}$ and…

The computational refinement of the 1-3-5 conjecture. Except for $m\in\{31,43,111,151,168,200,248,263,319,456,479,871,1752,1864,3544\}$, every such $m$ has a representation for whi…

Let $T\in A_5$, and let $C_T$ be the corresponding genus-two curve, with $r(T)$ denoting the rank quantity used in the source. Rational-points conjecture. For $r(T)>1$, the values…

For each even weight $k$ and level $N$, let $Q_k(N)$ be the set of quadratic-twist classes of classical newforms with rational coefficients and no complex multiplication, represent…

Let $B$, $\sigma$, $t$, $m$, and $\nu$ be as above, and let $\mathcal{K}(t,\sigma)$ be the set of integers with exactly $t$ distinct prime divisors, each having signature matching…

Denominator conjecture. The polynomial $L'_p$ is the denominator of every coefficient of the three modular polynomials in the functions $b'_1,b'_2,b'_3$.

Generalized Borchardt mean identity. With the notation above, the displayed identity holds for all such $\tau$ and $\gamma$. This extends the preceding Borchardt mean formula from…

Let $N$ be large, and let $[a,b]$ be an interval contained in $[N,2N]$ whose length is at most $N^{1/2+c}$ for some absolute constant $c>0$. Square-root barrier conjecture. There e…