34 problems

Let $f$ be a weight-$2$ newform of level $\Gamma_0(p)$ with integer Fourier coefficients, let $E=E_f$ be the associated elliptic curve with multiplicative reduction at $p$, and let…

For odd $n$, let $\Delta=n^2+4$ and let the maximal order of discriminant $\Delta$ have class number one. Yokoi's conjecture. All such maximal orders must satisfy … This is a class…

Let $F$ be a real quadratic field, let Z_\\A(s) be the difference of two ray class partial zeta functions associated with a ray class, and let $\exp(-Z_\A'(0))$ be the correspond…

Let $p$ be a prime, let $\tau\in\mathcal{H}_p$ be a real quadratic point of discriminant $D$ prime to $p$, and let $H_\tau$ be the narrow ring class field of the order defined by…

Let $K$ be a real quadratic number field, and let $X(\mathfrak f)$ denote Shintani's invariant associated with an ideal or modulus $\mathfrak f$, defined in terms of the double sin…

Let $d>0$ range over fundamental discriminants, let $h(d)$ denote the narrow class number of the real quadratic field of discriminant $d$, and let $\varepsilon_d$ be its fundamenta…

Let $K$ be a real quadratic field, let $\mathfrak{C}$ be a narrow ideal class of $K$, and define … where $\mathfrak{C}_1$ and $\mathfrak{C}_2$ are the ideal classes used in this co…

Let $K$ be a real quadratic field, and let Stark numbers and Galois action refer to the algebraic quantities and Galois-theoretic data sought in Hilbert's twelfth problem. Manin's…

Let $\Delta=n^2-4$ be squarefree, with $n\ge 21$, and write $h_{n^2-4}$ for the class number of the corresponding quadratic order. Mollin's conjecture. … The supplied status eviden…

Consider the $34$ ternary lattices listed in the theorem immediately preceding the conjecture, over the relevant real quadratic fields. A universal ternary lattice is a ternary qua…

Let $D$ be a positive squarefree integer, let … and let $\mathcal{O}_F$ denote the ring of integers of $F$. A universal ternary $\mathcal{O}_F$-lattice is a universal quadratic lat…

Let $\beta\in\mathbb R$ satisfy $a\beta^2+b\beta+c=0$ with $a,b,c\in\mathbb Z$ and $b^2-4ac$ nonsquare, and let $\r\in\mathbb Q^2$. Let \shin^{\r}[\beta] and…

Stark's conjecture. Except in the trivial case $\e_\A=1$, one has

Let $p$ be an odd prime with $p\equiv 3\pmod{4}$, and let … be the fundamental unit of $\mathbb{Q}(\sqrt{p})$, where $(t,u)$ is the minimal positive solution of $x^2-py^2=\pm4$. Mo…

Let $K=\mathbb{Q}(\sqrt{D})$, where $D=pq$ with $p\equiv 5\pmod 8$ and $q\equiv 1\pmod 8$, and suppose … Assume also … Let $X_{\infty}$ be the Iwasawa module of the cyclotomic…

Class-number infinitude conjecture. There are infinitely many $r\in\mathbb{Z}^+$ such that $p$ does not divide the class number of $K_r$.

Deutsch's conjecture. The form $q_1$ is not universal over any real quadratic field other than $\mathbb{Q}(\sqrt{5})$.

Let $K$ be a real quadratic field, let $p$ be a finite prime that is ramified or inert in $K$, and let $\tau$ be a real multiplication point in the Drinfeld $p$-adic upper half pla…

Let $K$ range over real quadratic fields, with regulator $R_K$, discriminant $d_K$, and parameter $m_K$ as used in the source. Regulator-growth conjecture. There exists an infinite…

Let $D>1$ be a square-free positive integer, let $K=\mathbb{Q}(\sqrt{D})$ be the associated real quadratic field, and let $M_1(K)$ and $M_2(K)$ denote its first and second Euclidea…

Let $\Gamma$ be a congruence subgroup of $\operatorname{SL}_2(\mathbb{Z})$, let $E$ be a real quadratic field, and let $H(\Gamma,E)$ denote the image of the connecting homomorphism…

Let $N$ be a level, let $E$ be a real quadratic field, and let $\psi_{\Gamma,E}$ denote the map associated with a subgroup $\Gamma$ as in the conjecture. Write $A_E(N)$ and…

Let $D$ range over fundamental discriminants. For the values of the Klein modular invariant $j$ at real quadratic irrationalities, define … where the sum is over classes of indefin…

Let $U$ be a sufficiently small level and let $\kappa=[k_1,k_2]\psi$ be a locally algebraic weight near the boundary. Let $V\in\{U_p,U_{\mathfrak p}\}$. For each…

Let $\uptheta\in\mathbb R-\mathbb Q$ be a fundamental quadratic unit and let $K=\mathbb Q(\uptheta)$, with fundamental discriminant $D$. The multivalued quantum modular invariant…