Since 2008 we have the following remarkable correspondance:

Odd irreducible 2-dim Galois repn $\longleftrightarrow$ weight 1 newforms

note: all Galois representations in this question are ment to be continuous complex linear representations.

That this is a bijection is the consequence of three deep results.

• Serre-Deligne: for each weight 1 newform there is an irreducible 2-dim Galois representation.

• Weil-Langlands: the converse is true if the Artin conjecture holds for those representations.

• Khare-Wintenberger: the Artin conjecture holds for odd irreducible 2-dim Galois representation (from Serre's modularity conjecture).

Now, we also famously have (Wiles-Breuil-Conrad-Diamond-Taylor):

Elliptic curves over $\mathbb{Q}$ $\longrightarrow$ weight 2 newforms

My first question is, what is known/conjectured on the converse direction? This is, does every newform of weight 2 arises from an elliptic curve over $\mathbb{Q}$? (Note that this correspondence can only hold up to isogeny) Are there any other source for newforms of this weight? How can we tell apart those that came from elliptic curves?

For the other sister correspondance, we are one case short of something like this:

Even irreducible 2-dim Galois repn $\longrightarrow$ Maass cusp forms with $\lambda = 1/4$

Only the icosahedral case of the (2-dim, even) strong Artin conjecture is still open. The second question is equivalent to the first: is this expected to be a bijection, as in the odd case? You can freely assume the full 2-dim strong Artin conjecture to answer this.

Since 2008 we have the following remarkable correspondence:

Odd irreducible 2-dim Galois repn $\longleftrightarrow$ weight 1 newforms

note: all Galois representations in this question are ment to be continuous complex linear representations.

That this is a bijection is the consequence of three deep results.

• Serre-Deligne: for each weight 1 newform there is an irreducible 2-dim Galois representation.

• Weil-Langlands: the converse is true if the Artin conjecture holds for those representations.

• Khare-Wintenberger: the Artin conjecture holds for odd irreducible 2-dim Galois representation (from Serre's modularity conjecture).

Now, we also famously have (Wiles-Breuil-Conrad-Diamond-Taylor):

Elliptic curves over $\mathbb{Q}$ $\longrightarrow$ weight 2 newforms

My question is, what is known/conjectured on the converse direction? This is, does every newform of weight 2 arises from an elliptic curve over $\mathbb{Q}$? (Note that this correspondence can only hold up to isogeny) Are there any other source for newforms of this weight? How can we tell apart those that came from elliptic curves?

Since 2008 we have the following remarkable correspondance:

Odd irreducible 2-dim Galois repn $\longleftrightarrow$ weight 1 newforms

note: all Galois representations in this question are ment to be continuous complex linear representations.

That this is a bijection is the consequence of three deep results.

• Weil-Langlands: the converse is true if the Artin conjecture holds for those representations.

• Serre-Deligne: for each weight 1 newform there is an irreducible 2-dim Galois representation.

• Khare-Wintenberger: the Artin conjecture holds for odd irreducible 2-dim Galois representation (from Serre's modularity conjecture).

Now, we also famously have (Wiles-Breuil-Conrad-Diamond-Taylor):

Elliptic curves over $\mathbb{Q}$ $\longrightarrow$ weight 2 newforms

My first question is, what is known/conjectured on the converse direction? This is, does every newform of weight 2 arises from an elliptic curve over $\mathbb{Q}$? (Note that this correspondence can only hold up to isogeny) Are there any other source for newforms of this weight? How can we tell apart those that came from elliptic curves?

For the other sister correspondance, we are one case short of something like this:

Even irreducible 2-dim Galois repn $\longrightarrow$ Maass cusp forms with $\lambda = 1/4$

Only the icosahedral case of the (2-dim, even) strong Artin conjecture is still open. The second question is equivalent to the first: is this expected to be a bijection, as in the odd case? You can freely assume the full 2-dim strong Artin conjecture to answer this.

Since 2008 we have the following remarkable correspondance:

Odd irreducible 2-dim Galois repn $\longleftrightarrow$ weight 1 newforms

note: all Galois representations in this question are ment to be continuous complex linear representations.

That this is a bijection is the consequence of three deep results.

• Serre-Deligne: for each weight 1 newform there is an irreducible 2-dim Galois representation.

• Weil-Langlands: the converse is true if the Artin conjecture holds for those representations.

• Khare-Wintenberger: the Artin conjecture holds for odd irreducible 2-dim Galois representation (from Serre's modularity conjecture).

Now, we also famously have (Wiles-Breuil-Conrad-Diamond-Taylor):

Elliptic curves over $\mathbb{Q}$ $\longrightarrow$ weight 2 newforms

My first question is, what is known/conjectured on the converse direction? This is, does every newform of weight 2 arises from an elliptic curve over $\mathbb{Q}$? (Note that this correspondence can only hold up to isogeny) Are there any other source for newforms of this weight? How can we tell apart those that came from elliptic curves?

For the other sister correspondance, we are one case short of something like this:

Even irreducible 2-dim Galois repn $\longrightarrow$ Maass cusp forms with $\lambda = 1/4$

Only the icosahedral case of the (2-dim, even) strong Artin conjecture is still open. The second question is equivalent to the first: is this expected to be a bijection, as in the odd case? You can freely assume the full 2-dim strong Artin conjecture to answer this.

Since 2008 we have the following remarkable correspondance:

Odd irreducible 2-dim Galois repn $\longleftrightarrow$ weight 1 newforms

That this is a bijection is the consequence of three deep results.

• Weil-Langlands: for each weight 1 newform there is an irreducible 2-dim Galois representation.

• Serre-Deligne: the converse is true if the Artin conjecture holds for those representations.

• Khare-Wintenberger: the Artin conjecture holds for odd irreducible 2-dim Galois representation (from Serre's modularity conjecture).

Now, we also famously have (Wiles-Breuil-Conrad-Diamond-Taylor):

Elliptic curves over $\mathbb{Q}$ $\longrightarrow$ weight 2 newforms

My first question is, what is known/conjectured on the converse direction? This is, does every newform of weight 2 arises from an elliptic curve over $\mathbb{Q}$? (Note that this correspondence can only hold up to isogeny) Are there any other source for newforms of this weight? How can we tell apart those that came from elliptic curves?

For the other sister correspondance, we are one case short of something like this:

Even irreducible 2-dim Galois repn $\longrightarrow$ Maass cusp forms with $\lambda = 1/4$

Only the icosahedral case of the (2-dim, even) strong Artin conjecture is still open. The second question is equivalent to the first: is this expected to be a bijection, as in the odd case? You can freely assume the full 2-dim strong Artin conjecture to answer this.

Since 2008 we have the following remarkable correspondance:

Odd irreducible 2-dim Galois repn $\longleftrightarrow$ weight 1 newforms

note: all Galois representations in this question are ment to be continuous complex linear representations.

That this is a bijection is the consequence of three deep results.

• Weil-Langlands: the converse is true if the Artin conjecture holds for those representations.

• Serre-Deligne: for each weight 1 newform there is an irreducible 2-dim Galois representation.

• Khare-Wintenberger: the Artin conjecture holds for odd irreducible 2-dim Galois representation (from Serre's modularity conjecture).

Now, we also famously have (Wiles-Breuil-Conrad-Diamond-Taylor):

Elliptic curves over $\mathbb{Q}$ $\longrightarrow$ weight 2 newforms

My first question is, what is known/conjectured on the converse direction? This is, does every newform of weight 2 arises from an elliptic curve over $\mathbb{Q}$? (Note that this correspondence can only hold up to isogeny) Are there any other source for newforms of this weight? How can we tell apart those that came from elliptic curves?

For the other sister correspondance, we are one case short of something like this:

Even irreducible 2-dim Galois repn $\longrightarrow$ Maass cusp forms with $\lambda = 1/4$

Only the icosahedral case of the (2-dim, even) strong Artin conjecture is still open. The second question is equivalent to the first: is this expected to be a bijection, as in the odd case? You can freely assume the full 2-dim strong Artin conjecture to answer this.