Mathematics Problem Archive

How often is the group of a random elliptic curve over $\mathbb{F}_q$ cyclic?...

Characterize the typical arithmetic structure of $\#E(\mathbb{F}_q)$ for elliptic curves over finite fields....

Prove that there are sufficiently many prime-order elliptic curves over every finite field $\mathbb{F}_q$....

Prove that sufficiently many elliptic curves $E/\mathbb{F}_p$ have smooth group order $\#E(\mathbb{F}_p)$....

Quantify elliptic curves over finite fields whose group order has a large prime divisor....

Prove the conjecture that the EC-LCG, EC-PG, and EC-NRG sequences defined in the slides are very well distributed....

Given $n$, efficiently choose a prime power $q$ and construct an elliptic curve $E/\mathbb{F}_q$ with $\#E(\mathbb{F}_q)=n$....

Given the Weil polynomial of an abelian-threefold isogeny class over a finite field, determine whether the class contains a Jacobian....

Decide whether a given principally polarized abelian threefold over a field $k$ is the Jacobian of a curve over $k$....

Find more MNT curves, including usable larger embedding degrees, more curve families, and smaller cofactors....

Construct pairing-friendly hyperelliptic curves suitable for cryptography....

Determine the computational hardness of the Pairing Inversion Problem....

Prove an analogous small-polynomial-factor worst-case hardness result for SVP and SIVP on ideal lattices....

Is the shortest vector problem on ideal or cyclic lattices NP-hard, either exactly or under approximation?...

Reduce computational problems on arbitrary lattices to corresponding problems on cyclic or ideal lattices....

Does SVP reduce to CVP while remaining inside the class of cyclic or ideal lattices?...

Exhibit cyclic or ideal lattices on which LLL achieves its worst-case approximation factor....

Develop an algebraic analogue of the LLL lattice-reduction algorithm that exploits ideal-lattice structure....

Construct efficient pseudorandom functions from ideal-lattice problems....

Use algebraic tools to solve computational problems on ideal lattices efficiently....

Develop an efficient quantum algorithm for the shortest vector problem on ideal lattices....

Use a baby-step/giant-step infrastructure framework to speed infrastructure discrete logarithms or point counting by a polynomial factor....

Give efficient reductions in both directions between the degree-zero divisor-class-group discrete logarithm problem and the infrastructure discrete lo...

For a compact convex domain $\Omega$, the values of $F_\Omega$ at the vertices of its corner locus $C_\Omega$ give complete coordinates. How are these...

Give another proof of the paper's identities (Ж) and (ж) using the methods for identity (1). Can $f(a,b,c,d)$ be interpreted as a residue at $(a+b)+(c...

Büchi's problem on sufficiently large sequences of square numbers with constant second difference....

Exponent pair conjecture: for all $\varepsilon > 0$, is the pair $(\varepsilon, 1/2 + \varepsilon)$ an exponent pair?...

Let $K$ be a number field and let $\rho$ be an irreducible $p$-adic representation of $\operatorname{Gal}(\overline K/K)$ that is unramified outside f...

For every totally real number field $F$ and prime $p$, do the Iwasawa invariants $\lambda(F_\infty/F)$ and $\mu(F_\infty/F)$ of the cyclotomic $\mathb...

Hermite's problem: is it possible, for any natural number $n$, to assign a sequence of natural numbers to each real number such that the sequence for ...

Lang and Trotter's conjecture on supersingular primes that the number of supersingular primes less than a constant $X$ is within a constant multiple o...

For an Artin $L$-function attached to a Galois extension of number fields, is its leading Taylor coefficient at $s=0$ the product of the corresponding...

For a motive (or the cohomology of a smooth projective variety) and an appropriate integer argument, is the order of vanishing of its L-function the p...

Find the value of the De Bruijn–Newman constant....

For every $\varepsilon>0$, is there a $T_0(\varepsilon)$ such that, whenever $T\geq T_0$ and $H=T^{1/4+\varepsilon}$, the interval $(T,T+H]$ contains ...

For fixed admissible $k$, does $T^{-1}\int_0^T|\zeta(1/2+it)|^{2k}\,dt$ have the Keating–Snaith asymptotic $a(k)G(k+1)^2G(2k+1)^{-1}(\log T)^{k^2}$, w...

The density hypothesis for zeroes of the Riemann zeta function....

Piltz divisor problem on bounding $\Delta_k(x) = D_k(x) - xP_k(\log(x))$...

Let $K$ be a number field and let $\pi$ be a cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb A_K)$ with unitary central character. Is ev...

Selberg's 1/4 conjecture: the eigenvalues of the Laplace operator on Maass wave forms of congruence subgroups are at least $1/4$....

Bombieri–Lang conjecture: $K$-rational points on a variety of general type over a number field $K$ are not a dense set in Zariski topology....

Manin conjecture: if K-rational points on Fano variety are Zariski-dense subset, then the distribution of points of height: $H(x)\leq B$ in any Zarisk...

For an abelian variety or suitable motive over a number field, are its normalized Frobenius conjugacy classes equidistributed in the associated compac...

Vojta's conjecture: points on non-singular algebraic variety over algebraic number field that not satisfy certain height inequality are contained in s...

Fix $n\geq3$. If coprime nonzero integers $a_1,\ldots,a_n$ have sum zero and no proper subsum zero, is it true that for every $\varepsilon>0$ there is...

Szpiro's conjecture: for any $\varepsilon > 0$, there is some constant $C(\varepsilon)$ such that, for any elliptic curve $E$ defined over $\mathbb{Q}...

Congruent number problem (a corollary to Birch and Swinnerton-Dyer conjecture, per Tunnell's theorem): determine precisely what rational numbers are c...

Erdős–Moser problem: is $1^1 + 2^1 = 3^1$ the only solution to the Erdős–Moser equation?...

Which integers can be written as the sum of three perfect cubes?...

For coprime integers $1\leq a<d$, is the least prime $p(a,d)$ congruent to $a\pmod d$ always less than $d^2$?...